# Symmetries of Differential equations and Applications in Relativistic
Physics.

Andronikos Paliathanasis

(Athens, 2014)

###

#### Preface

This thesis is part of the PhD program of the Department of Astronomy,
Astrophysics and Mechanics of the Faculty of Physics of the University of
Athens, Greece.

In memory of my grandmother Amalia

#### Abstract

In this thesis, we study the one parameter point transformations which leave
invariant the differential equations. In particular we study the Lie and the
Noether point symmetries of second order differential equations. We establish
a new geometric method which relates the point symmetries of the differential
equations with the collineations of the underlying manifold where the motion
occurs. This geometric method is applied in order the two and three
dimensional Newtonian dynamical systems to be classified in relation to the
point symmetries; to generalize the Newtonian Kepler-Ermakov system in
Riemannian spaces; to study the symmetries between classical and quantum
systems and to investigate the geometric origin of the Type II hidden
symmetries for the homogeneous heat equation and for the Laplace equation in
Riemannian spaces. At last but not least, we apply this geometric approach in
order to determine the dark energy models by use the Noether symmetries as a
geometric criterion in modified theories of gravity.

#### Acknowledgement

This doctoral thesis would not have been possible without the guidance and
support of numerous people and I wish to express my gratitude here.

Professor Peter G.L. Leach deserves a special mention. It was your research
that inspired me to wade into the symmetries of differential equations and
your encouragement when I first set about my own research proved invaluable.

I am grateful for my collaboration with my Supervisor, Professor Michael
Tsamparlis. Your faith in my abilities and your advice, on both my research
and my career, throughout the years, has allowed me to grow to the scientist I
am today. You introduced me to the subject of differential geometry and taught
me the geometric thought process.

I would especially like to thank both my advisors Dr. Spyros Basilakos, who
opened up to me the subject of cosmology, and Dr. Christos Efthimiopoulos; the
discussions we had on the topic have been priceless. Also, I feel fortunate
and grateful for my examination committee, Prof. S. Capozziello, Prof. P. J.
Ioannou, Prof T. Apostolatos and Prof A. H. Kara. Without your important
comments I could not have completed this thesis.

Above all, I feel grateful for my friends and family. Stelios, Sifis, Venia
and Thanos your friendship and encouragement all these years have been my most
valuable support mechanism.

Last but not least, I would not be who I am today without my family. My
parents Maria and Sotiris, and my grandfather Ioannis, you have always
believed in and you have supported me in all the important decisions of my
life. Marianthi, your patience, your kindness and your help on all levels, is
more than any brother deserves.

###### Contents

  1. I Introduction
    1. 1 Introduction
      1. 1.1 Summary
        1. 1.1.1 Summary of Part I: Introduction
        2. 1.1.2 Summary of Part II: Symmetries of ODEs
        3. 1.1.3 Summary of Part III: Symmetries of PDEs
        4. 1.1.4 Summary of Part IV: Noether symmetries and theories of gravity
    2. 2 Point transformations and Invariant functions
      1. 2.1 Introduction
      2. 2.2 Point Transformations
        1. 2.2.1 Infinitesimal Transformations
        2. 2.2.2 Invariant Functions
        3. 2.2.3 Lie Algebras
      3. 2.3 Lie symmetries of differential equations
        1. 2.3.1 Prolongation of point transformations
        2. 2.3.2 Lie symmetries of ODEs
      4. 2.4 Lie symmetries of PDEs
      5. 2.5 Lie Bäcklund symmetries
      6. 2.6 Noether point symmetries
        1. 2.6.1 Noether symmetries of ODEs
        2. 2.6.2 Noether point symmetries of PDEs
      7. 2.7 Collineations of Riemannian manifolds
        1. 2.7.1 Lie Derivative
        2. 2.7.2 Motions of Riemannian spaces
        3. 2.7.3 Symmetries of the connection
  2. II Symmetries of ODEs
    1. 3 Lie symmetries of geodesic equations
      1. 3.1 Introduction
      2. 3.2 The Lie symmetry conditions in an affine space
      3. 3.3 Lie symmetries of autoparallel equation
      4. 3.4 Lie and Noether symmetries of geodesic equations
        1. 3.4.1 Noether symmetries and conservation laws
      5. 3.5 Applications
        1. 3.5.1 The geodesic symmetries of Einstein spaces
        2. 3.5.2 The geodesic symmetries of Gödel spacetime
        3. 3.5.3 The geodesic symmetries of Taub spacetime
        4. 3.5.4 The geodesic symmetries of a 1+3 decomposable spacetime metric
        5. 3.5.5 The geodesic symmetries of the FRW metrics
      6. 3.6 Conclusion
      7. 3.A The determining equations
    2. 4 Motion on a curved space
      1. 4.1 Introduction
      2. 4.2 Lie symmetries of a dynamical system in a Riemannian space
      3. 4.3 Noether symmetries of a dynamical system in a Riemannian space
      4. 4.4 2D autonomous systems which admit Lie/Noether point symmetries
        1. 4.4.1 2D autonomous Newtonian systems which admit Noether symmetries
      5. 4.5 3D autonomous Newtonian systems admit Lie/Noether point symmetries
        1. 4.5.1 3D autonomous Newtonian systems admit Lie point symmetries
        2. 4.5.2 3D autonomous Newtonian systems which admit Noether point symmetries
      6. 4.6 Motion on the two dimensional sphere
        1. 4.6.1 Noether Symmetries
      7. 4.7 Applications
        1. 4.7.1 Lie point symmetries of the Kepler-Ermakov system.
        2. 4.7.2 Point symmetries of the Hènon - Heiles potential
      8. 4.8 Conclusion
      9. 4.A Proof of main Theorem
      10. 4.B Tables of Newtonian systems admit Lie and Noether symmetries
    3. 5 The autonomous Kepler Ermakov system in a Riemannian space
      1. 5.1 Introduction
      2. 5.2 The two dimensional autonomous Kepler Ermakov system
      3. 5.3 Generalizing the autonomous Kepler Ermakov system
      4. 5.4 The three dimensional autonomous Euclidian Kepler Ermakov system
        1. 5.4.1 The 3D autonomous Hamiltonian Kepler Ermakov system of type I $(\mu\neq 0)$
        2. 5.4.2 The 3D autonomous Hamiltonian Kepler Ermakov system of type II $(\mu=0)$
      5. 5.5 Integrability of 3D autonomous Euclidian Kepler Ermakov system
        1. 5.5.1  Noether symmetries generated from the translation group
        2. 5.5.2 Noether symmetries generated from $so\left(3\right)$
      6. 5.6 The autonomous Riemannian Kepler Ermakov system
        1. 5.6.1 The non Hamiltonian autonomous Riemannian Kepler Ermakov system
        2. 5.6.2 The autonomous conservative Riemannian Kepler Ermakov system
      7. 5.7 The autonomous Riemannian Kepler Ermakov system in General Relativity
        1. 5.7.1 The Riemannian Kepler Ermakov system on a 4D FRW spacetime
        2. 5.7.2 The Riemannian Kepler Ermakov system on a 3D FRW spacetime
      8. 5.8 The Riemannian Kepler Ermakov system in cosmology
        1. 5.8.1 The case of scalar field cosmology
        2. 5.8.2 The case of $f\left(R\right)$ Cosmology
      9. 5.9 Conclusion
      10. 5.A Appendix
  3. III Symmetries of PDEs
    1. 6 Lie symmetries of a general class of PDEs
      1. 6.1 Introduction
      2. 6.2 The case of the second order PDEs
      3. 6.3 The Lie symmetry conditions for a linear function $F(x,u,u_{i})$
      4. 6.4 Symmetries of the Poisson equation in a Riemannian space
        1. 6.4.1 Lie symmetries of Laplace equation
        2. 6.4.2 Symmetries of conformal Poisson equation in a Riemannian space
        3. 6.4.3 Lie symmetries of the conformal Laplace equation
      5. 6.5 The heat conduction equation with a flux in a Riemannian space
        1. 6.5.1 The homogeneous heat equation
        2. 6.5.2 Case $\ q\left(t,x,u\right)=q\left(u\right)$
      6. 6.6 Conclusion
      7. 6.A Appendix A
      8. 6.B Appendix B
    2. 7 Point symmetries of Schrödinger and the Klein Gordon equations
      1. 7.1 Introduction
      2. 7.2 Noether symmetries of Conformal Lagrangians
      3. 7.3 Lie point symmetries of Schrödinger and the Klein Gordon equation
        1. 7.3.1 Symmetries of the Schrödinger equation
        2. 7.3.2 Symmetries of the Klein Gordon equation
        3. 7.3.3 Symmetries of the Conformal Klein Gordon equation
      4. 7.4 $sl\left(2,R\right)$ and the Klein Gordon equation
        1. 7.4.1 The oscillator
        2. 7.4.2 The Kepler Ermakov potential with an oscillator term
      5. 7.5 Applications
        1. 7.5.1 Euclidean central force
        2. 7.5.2 Lie symmetry classification of Schrödinger and the Klein Gordon equations in Euclidian space
      6. 7.6 The Klein Gordon equation in a spherically symmetric space-time
      7. 7.7 WKB approximation
      8. 7.8 Conclusion
    3. 8 The geometric origin of Type II hidden symmetries
      1. 8.1 Introduction
      2. 8.2 Lie symmetries of Laplace equation in certain Riemannian spaces
      3. 8.3 Reduction of the Laplace equation in certain Riemannian spaces
        1. 8.3.1 Riemannian spaces admitting a gradient KV
        2. 8.3.2 Riemannian spaces admitting a gradient HV.
        3. 8.3.3 Riemannian spaces admitting a sp.CKV
      4. 8.4 Type II hidden symmetries of the 3D and the 4D wave equation
        1. 8.4.1 Laplacian in $M^{4}$
        2. 8.4.2 Laplacian in $M^{3}$
      5. 8.5 The Laplace equation in the Petrov type III and in the FRW-like spacetime
        1. 8.5.1 The Laplace equation in the Petrov Type III spacetime
        2. 8.5.2 The Laplace equation in the $n~{}$dimensional FRW-like spacetime.
      6. 8.6 Reduction of the homogeneous heat equation in certain Riemannian spaces
        1. 8.6.1 The heat equation in a space which admits a gradient KV
        2. 8.6.2 The heat equation in a space which admits a gradient homothetic vector
      7. 8.7 Applications of the reduction of the homogeneous heat equation
        1. 8.7.1 The heat equation in a $1+n$ decomposable space
        2. 8.7.2 FRW space-time with a gradient HV
      8. 8.8 The Heat equation in Petrov type III spacetime
      9. 8.9 Conclusion
      10. 8.A Proof of Corollary
      11. 8.B The homogeneous heat equation in the Petrov spacetimes
        1. 8.B.1 Petrov type N
        2. 8.B.2 Petrov type D
        3. 8.B.3 Petrov type II
  4. IV Noether symmetries and theories of gravity
    1. 9 Noether symmetries in Scalar field Cosmology
      1. 9.1 Introduction
      2. 9.2 Conformally equivalent Lagrangians and scalar field Cosmology
      3. 9.3 Generalization to dimension $n$
      4. 9.4 Conformal Lagrangians in scalar field cosmology
      5. 9.5 Noether point symmetries of a non minimally coupled Scalar field.
        1. 9.5.1 Case A. $R_{KM}=0$
        2. 9.5.2 Case A: Analytic solutions for $k=1$
        3. 9.5.3 Case A: Analytic solutions for $\left|k\right|\neq 1$
        4. 9.5.4 Case B: The 2d metric is conformally flat
      6. 9.6 Noether point symmetries of a minimally coupled Scalar field.
        1. 9.6.1 Hyperbolic - UDM Potential
        2. 9.6.2 Exponential Potential
      7. 9.7 The Lie and Noether symmetries of Bianchi class A homogeneous cosmologies with a scalar field.
        1. 9.7.1 Bianchi I
      8. 9.8 Conclusion
      9. 9.A Relating the ranges of the constants $F_{0}$ and $\left|k\right|$
      10. 9.B Computation of the gradient functions $S_{1}\left(r,\theta\right),$ $S_{2}\left(r,\theta\right)$
    2. 10 Using Noether point symmetries to specify $f(R)$ gravity
      1. 10.1 Introduction
      2. 10.2 Cosmology with a modified gravity
      3. 10.3 Modified gravity versus symmetries
      4. 10.4 Noether point symmetries of $f\left(R\right)$ gravity
      5. 10.5 Analytic Solutions
        1. 10.5.1 Power law model $R^{\mu}$ with $\mu=\frac{3}{2}$
        2. 10.5.2 Power law model $R^{\mu}$ with $\mu=\frac{7}{8}$
        3. 10.5.3 $\Lambda_{bc}$CDM model with $(b,c)=(1,\frac{3}{2})$
        4. 10.5.4 $\Lambda_{bc}$CDM model with $(b,c)=(1,\frac{7}{8})$
      6. 10.6 Noether point symmetries in spatially non-flat $f(R)$ models
      7. 10.7 Conclusion
      8. 10.A Special solutions for the power law model $R^{n}$
    3. 11 Noether symmetries in $f(T)$ gravity
      1. 11.1 Introduction
      2. 11.2 Unholonomic frames and Connection Coefficient
      3. 11.3 $f(T)$-gravity
      4. 11.4 Generalized Lagrangian formulation of $f(T)$ gravity
      5. 11.5 The Noether Symmetry Approach for $f(T)$ gravity
        1. 11.5.1 Searching for Noether point symmetries in general spacetimes
      6. 11.6 Spatially flat FRW
        1. 11.6.1 Noether symmetries
        2. 11.6.2 Exact cosmological solutions
        3. 11.6.3 Cosmological analogue to other models
      7. 11.7 Static spherically symmetric spacetimes
        1. 11.7.1 Exact Solutions
      8. 11.8 Conclusion
    4. 12 Discussion
      1. 12.1 Discussion

###### List of Tables

  1. 2.1 Collineations of a Riemannian space
  2. 3.1 Lie algebra of the G del geodesic symmetries
  3. 3.2 Lie algebra of the Taub geodesic symmetries
  4. 3.3 The conformal algebra of the 1+3 metric
  5. 3.4 The conformal algebra of a flat 3d metric
  6. 3.5 The special forms of the scale factor for —K—=1
  7. 3.6 The special forms of the scale factor for K=0
  8. 4.1 Special projective algebra of the 2d Euclidian space
  9. 4.2 Two dimensional Newtonian systems admit Lie symmetries (1/4)
  10. 4.3 Two dimensional Newtonian systems admit Lie symmetries (2/4)
  11. 4.4 Two dimensional conservative Newtonian systems admit Lie symmetries (1/3)
  12. 4.5 Two dimensional autonomous Newtonian systems admiting Noether symmetries (1/2)
  13. 4.6 Two dimensional autonomous Newtonian systems admiting Noether symmetries (2/2)
  14. 4.7 Three dimensional autonomous Newtonian systems admit Lie symmetries (1/2)
  15. 4.8 Three dimensional autonomous Newtonian systems admit Lie symmetries (1/2)
  16. 4.9 Three dimensional conservative Newtonian systems admiting Lie symmetries
  17. 4.10 Three dimensional conservative Newtonian systems admit Noether symmetries (1/5)
  18. 4.11 Three dimensional conservative Newtonian systems admit Noether symmetries (2/5)
  19. 4.12 Noether symmetries/Integrals and potentials for the Lagrangian of the 2D sphere
  20. 4.13 Noether symmetries of H non - Heiles potential
  21. 4.14 Two dimensional Newtonian systems admiting Lie symmetries (3/4)
  22. 4.15 Two dimensional Newtonian systems admiting Lie symmetries (4/4)
  23. 4.16 Two dimensional conservative Newtonian systems admiting Lie symmetries (2/3)
  24. 4.17 Two dimensional conservative Newtonian systems admiting Lie symmetries (3/3)
  25. 4.18 Three dimensional conservative Newtonian systems admitingt Noether symmetries (3/5)
  26. 4.19 Three dimensional conservative Newtonian systems admiting Noether symmetries (4/5)
  27. 4.20 Three dimensional conservative Newtonian systems admiting Noether symmetries (5/5)
  28. 5.1 Potentials for which the Kepler Ermakov in the 4D FRW space admit Noether symmetries
  29. 5.2 Potentials for which the Kepler Ermakov in the 3D FRW space admit Noether symmetries
  30. 6.1 Functions of $q(u)$ where the Heat equation admits Lie symmetries
  31. 7.1 The 2D Klein Gordon admitting Lie symmetries from the homothetic group
  32. 7.2 The 3D Klein Gordon admitting Lie symmetries from the homothetic group
  33. 9.1 Lie and Noether Symmetries of Bianchi I scalar field
  34. 9.2 Lie and Noether Symmetries of Bianchi II scalar field
  35. 9.3 Lie and Noether Symmetries of Bianchi VI/VII scalar field
  36. 9.4 Lie and Noether Symmetries of Bianchi VIII/IX scalar field
  37. 11.1 Noether Symmetries and Noether Integrals for arbitary f(T)
  38. 11.2 Extra Noether Symmetries and Noether Integrals for $f(T)=T^{n}$
  39. 11.3 Extra Noether Symmetries and Noether Integrals for $f(T)=T^{\frac{1}{2}}$

##  Part I Introduction

###  Chapter 1 Introduction

####  1.1 Summary

In this thesis, we study the geometric properties of the Lie and the Noether
point symmetries of second order differential equations. In particular, we
find a connection between the point symmetries of some class of second order
differential equations with the collineations of the underlying manifold where
the ”motion” occurs.

The novelty here is that we provide a geometrical method to determine the
symmetries of dynamical systems. The importance of Lie and Noether symmetries
is that they offer invariant functions which can be used to find analytic
solutions of the dynamical system.

The above mentioned geometric method is applied in order the two and three
dimensional Newtonian dynamical systems to be classified in relation to the
point symmetries; to generalize the Newtonian Kepler-Ermakov system in
Riemannian spaces; to study the symmetries between classical and quantum
systems and to investigate the geometric origin of the Type II hidden
symmetries for the homogeneous heat equation and for the Laplace equation in
Riemannian spaces. At last but not least, we apply this geometric approach in
order to determine the dark energy models by use the Noether symmetries as a
geometric criterion in modified theories of gravity.

The plan of the thesis is as follows.

#####  1.1.1 Summary of Part I: Introduction

In Part I we give the basic properties and definitions of one parameter point
transformations.

In Chapter 2, we study the geometry of the one parameter point transformation,
the properties of Lie algebras and the invariant functions. Moreover, the Lie
and the Noether symmetries of ordinary and partial differential equations are
analyzed and two schemes for using Lie symmetries to construct solutions are
presented. Furthermore, we study the action of point transformation on linear
differential geometry object.

#####  1.1.2 Summary of Part II: Symmetries of ODEs

In Part II we study the geometric origin of the Lie and the Noether point
symmetries of second order ordinary differential equations.

In Chapter 3, we consider the set of autoparallels - not necessarily affinely
parameterized - of a symmetric connection. We find that the major symmetry
condition relates the Lie symmetries with the special projective symmetries of
the connection. We derive the Lie symmetry conditions for a general system of
second order ODE polynomial in the first derivatives and we apply these
conditions in the special case of geodesic equations of Riemannian spaces.
Furthermore we give the generic Lie symmetry vector of the geodesic equations
in terms of the special projective collineations of the metric and their
degenerates and the generic Noether symmetry vector of the geodesic Lagrangian
in terms of the homothetic algebra of the Riemannian space. Finally we apply
the results to various cases and eventually we give the Lie symmetries, the
Noether symmetries and the associated conserved quantities of Einstein spaces,
the Gödel spacetime, the Taub spacetime and the Friedman Robertson Walker
spacetimes.

In Chapter 4, we generalize the results of the previous chapter in the case of
the equations of motion of a particle moving in a Riemannian space under the
action of a general force $F^{i}$. We apply these results in order to
determine all two dimensional and and all three dimensional Newtonian
dynamical systems which admit Lie and Noether point symmetries. We demonstrate
the use of the results in two cases. The non-conservative Kepler - Ermakov
system and the case of the Hènon Heiles type potentials.

In Chapter 5, we generalize the two-dimensional autonomous
Hamiltonian–Kepler–Ermakov dynamical system to three dimensions using the
$sl(2,R)$ invariance of Noether symmetries and determine all three-dimensional
autonomous Hamiltonian–Kepler–Ermakov dynamical systems which are Liouville
integrable via Noether symmetries. Subsequently, we generalize the autonomous
Kepler–Ermakov system in a Riemannian space which admits a gradient homothetic
vector by the requirements (a) that it admits a first integral (the Riemannian
Ermakov invariant) and (b) it has $sl(2,R)$ invariance. We consider both the
non-Hamiltonian and the Hamiltonian systems. In each case, we compute the
Riemannian–Ermakov invariant and the equations defining the dynamical system.
We apply the results in general relativity and determine the autonomous
Hamiltonian-Riemannian–Kepler–Ermakov system in the spatially flat Friedman
Robertson Walker spacetime. We consider a locally rotational symmetric
spacetime of class A and discuss two cosmological models. The first
cosmological model consists of a scalar field with an exponential potential
and a perfect fluid with a stiff equation of state. The second cosmological
model is the$f(R)$-modified gravity model of $\Lambda_{bc}$CDM. It is shown
that in both applications the gravitational field equations reduce to those of
the generalized autonomous Riemannian–Kepler–Ermakov dynamical system which is
Liouville integrable via Noether integrals.

#####  1.1.3 Summary of Part III: Symmetries of PDEs

In Part III we study the geometric origin of the Lie and the Noether point
symmetries of second order ordinary differential equations.

In Chapter 6, we attempt to extend this correspondence of point symmetries and
collineations of the space to the case of second order partial differential
equations. We examine the PDE of the form
$A^{ij}u_{ij}-F(x^{i},u,u_{i})=0$,where $u=u\left(x^{i}\right)$ and $u_{ij}$
stands for the second partial derivative. We find that if the coefficients
$A_{ij}$ are independent of $u$ then the Lie point symmetries of the PDE form
a subgroup of the conformal symmetries of the metric defined by the
coefficients $A_{ij}$. We specialize the study to linear forms of
$F(x^{i},u,u_{i})~{}$and write the Lie symmetry conditions for this case. We
apply this result to two cases. The Poisson/Yamabe equation for which we
derive the Lie symmetry vectors. Subsequently we consider the heat equation
with a flux in an n-dimensional Riemannian space and show that the Lie
symmetry algebra is a subalgebra of the homothetic algebra of the space. We
discuss this result in the case of de Sitter space time and in flat space.

In Chapter 7, we determine the Lie point symmetries of the Schrödinger and the
Klein Gordon equations in a general Riemannian space. It is shown that these
symmetries are related with the homothetic and the conformal algebra of the
metric of the space respectively. We consider the kinematic metric defined by
the classical Lagrangian and show how the Lie point symmetries of the
Schrödinger equation and the Klein Gordon equation are related with the
Noether point symmetries of this Lagrangian. The general results are applied
to two practical problems a. The classification of all two and three
dimensional potentials in a Euclidian space for which the Schrödinger equation
and the Klein Gordon equation admit Lie point symmetries and b. The
application of Lie point symmetries of the Klein Gordon equation in the
exterior Schwarzschild spacetime and the determination of the metric by means
of conformally related Lagrangians.

In Chapter 8, we study the geometric origin of Type II hidden symmetries for
the Laplace equation and for the homogeneous heat equation in certain
Riemannian spaces. As concerns the homogeneous heat equation, we study the
reduction of the heat equation in Riemannian spaces which admit a gradient
Killing vector, a gradient homothetic vector and in Petrov Type D, N, II and
Type III spacetimes. In each reduction we identify the source of the Type II
hidden symmetries. More specifically we find that (a) if we reduce the heat
equation by the symmetries generated by the gradient KV the reduced equation
is a linear heat equation in the nondecomposable space. (b) If we reduce the
heat equation via the symmetries generated by the gradient HV the reduced
equation is a Laplace equation for an appropriate metric. In this case the
Type II hidden symmetries are generated from the proper CKVs. (c) In the
Petrov space–times the reduction of the heat equation by the symmetry
generated from the nongradient HV gives PDEs which inherit the Lie symmetries
hence no Type II hidden symmetries appear. For the reduction of the Laplace
equation we consider Riemannian spaces which admit a gradient Killing vector,
a gradient Homothetic vector and a special Conformal Killing vector. In each
reduction we identify the source of Type II hidden symmetries. We find that in
general the Type II hidden symmetries of the Laplace equation are directly
related to the transition of the CKVs from the space where the original
equation is defined to the space where the reduced equation resides. In
particular we consider the reduction of the Laplace equation (i.e., the wave
equation) in the Minkowski space and obtain the results of all previous
studies in a straightforward manner. We consider the reduction of Laplace
equation in spaces which admit Lie point symmetries generated from a non-
gradient HV and a proper CKV and we show that the reduction with these vectors
does not produce Type II hidden symmetries. We apply the results to general
relativity and consider the reduction of Laplace equation in locally
rotational symmetric space times (LRS) and in algebraically special vacuum
solutions of Einstein’s equations which admit a homothetic algebra acting
simply transitively. In each case we determine the Type II hidden symmetries.
We apply the general results to cases in which the initial metric is
specified.

#####  1.1.4 Summary of Part IV: Noether symmetries and theories of gravity

In Part IV, we apply the Noether symmetry approach as a geometric criterion,
in order to probe the nature of dark energy in modified theories of gravity.

In Chapter 9, we discuss the conformal equivalence of Lagrangians for scalar
fields in a Riemannian space of dimension $4$ and $n$ respectively. In
particular we enunciate a theorem which proves that the field equations for a
non-minimally coupled scalar field are the same at the conformal level with
the field equations of the minimally coupled scalar field. The necessity to
preserve Einstein’s equations in the context of Friedmann Robertson Walker
spacetime leads us to apply, the current general analysis to the scalar field
(quintessence or phantom) in spatially flat FRW cosmologies. Furthermore, we
apply the Noether symmetry approach in non minimally coupled scalar field in a
spatially flat FRW spacetime and by using the Noether invariants we determine
analytical solutions for the field equations. Moreover we apply the same
procedure for a minimally coupled scalar field in a spatially flat FRW
spacetime and in Biachi Class A homogeneous spacetimes.

In Chapter 10, a detailed study of the modified gravity, $f(R)~{}$models is
performed, using that the Noether point symmetries of these models are
geometric symmetries of the mini superspace of the theory. It is shown that
the requirement that the field equations admit Noether point symmetries
selects definite models in a self-consistent way. As an application in
Cosmology we consider the Friedman -Robertson-Walker spacetime and show that
the only cosmological model which is integrable via Noether point symmetries
is the $\Lambda_{bc}$CDM model, which generalizes the Lambda Cosmology.
Furthermore using the corresponding Noether integrals we compute the analytic
form of the main cosmological functions.

In Chapter 11, we apply the Noether symmetry approach in the $f\left(T\right)$
modified theory of gravity in a spatially flat FRW spacetime and in static
spherically symmetric spacetime. First, we present a full set of Noether
symmetries for some minisuperspace models and we find that only the
$f\left(T\right)=T^{n}$ model admits extra Noether symmetries. The existence
of extra Noether integrals can be used in order to simplify the system of
differential equations as well as to determine the integrability of the model.
Then, we compute analytical solutions and find that spherically symmetric
solutions in $f(T)$ gravity can be recast in terms of Schwarzschild-like
solutions modified by a distortion function depending on a characteristic
radius.

Finally, in Chapter 12 we discuss our results.

###  Chapter 2 Point transformations and Invariant functions

####  2.1 Introduction

Lie symmetry of a differential equation is a one parameter point
transformation which leaves the differential equation invariant. Lie
symmetries111In the following sections, by Lie symmetry we mean point
symmetry. There are also generalized Lie symmetries which are not point
symmetries. is the main tool to study nonlinear differential equations. Indeed
Lie symmetries provide invariant functions which can be used to construct
analytic solutions for a differential equation. These solutions we call
invariant solutions. One such example is the solution
$u\left(x,y\right)=e^{k\left(t-x\right)}~{}$of the wave equation

$u_{xx}-u_{tt}=0$

which is found by applying the Lie symmetry $X=\partial_{x}+ku\partial_{u}.$

The structure of the chapter is as follows. In section 2.2, we study the
geometry of the one parameter point transformation, the properties of Lie
algebras and the invariant functions. Invariant functions are functions which
remain unchanged under the action of a point transformation. In section 2.3,
the Lie symmetries of ordinary and partial differential equations are analyzed
and two schemes for using Lie symmetries to construct solutions are presented.
In section 2.6, Noether symmetries, a special class of Lie symmetries, are
discussed. Noether symmetries are admited only by systems whose equation of
motion result from a variational principle. Noether symmetries are important
because they produce conservation laws and can be used to simplify the
differential equations.

In section 2.7, the action of point transformation on linear differential
geometry object is examined. Collineations are point transformations which do
not leave necessary invariant a geometric object. In particular we study the
collineations of the metric (Conformal motions) and of the Christoffel symbols
(Projective collineations) of a Riemannian space.

####  2.2 Point Transformations

Let $M~{}$be a manifold of class$~{}~{}C^{p}~{}$with$~{}p\succeq 2$ and let
$U$ be a neighborhood in $M.$ Consider two points $P,Q\in U~{}$with
coordinates $\left(x_{P},y_{P}\right)$ and
$\left(x_{Q}^{\prime},y_{Q}^{\prime}\right)$ respectively. A point
transformation on $U$ is a relation among the coordinates of the points
$P,Q\in U$ which is defined by the transformation equations

$x_{Q}^{\prime}=x^{\prime}\left(x_{P},y_{P}\right)~{}~{},~{}~{}y_{Q}^{\prime}=y^{\prime}\left(x_{P},y_{P}\right)$

where the functions
$x^{\prime}\left(x,y\right),~{}y^{\prime}\left(x,y\right)~{}$are $C^{p-1}$ and

$\det\left|\frac{\partial\left(x^{\prime},y^{\prime}\right)}{\partial\left(x,y\right)}\right|\neq
0.$ (2.1)

Condition (2.1 ) means that the functions
$x^{\prime}\left(x,y\right),~{}y^{\prime}\left(x,y\right)$ are independent. A
special class of point transformations are the one parameter point
transformations defined as follows [1].

###### Definition 2.2.1

The one parameter point transformations are point transformations that depend
on one arbitrary parameter as follows

$x^{\prime}=y^{\prime}\left(x,y,\varepsilon\right)~{},~{}y^{\prime}=y^{\prime}\left(x,y,\varepsilon\right)$
(2.2)

where $\varepsilon\in\mathbb{R}$ and the transformation satisfies the
following conditions.

a) They are well defined, that is, that if

$x^{\prime}\left(x_{1},y_{1},\varepsilon\right)=x^{\prime}\left(x_{2},y_{2},\varepsilon\right),~{}y^{\prime}\left(x_{1},y_{1},\varepsilon\right)=y^{\prime}\left(x_{2},y_{2},\varepsilon\right)$

then $x_{1}=y_{1}$ and $y_{1}=y_{2}.$

b) They can be composed, that is, that if

$x^{\prime}=y^{\prime}\left(x,y,\varepsilon\right)~{}~{},~{}~{}y^{\prime}=y^{\prime}\left(x,y,\varepsilon\right)$

and

$x^{\prime\prime}=x^{\prime\prime}\left(x^{\prime},y^{\prime},\varepsilon^{\prime}\right)~{},~{}~{}y^{\prime\prime}=y^{\prime\prime}\left(x^{\prime},y^{\prime},\varepsilon^{\prime}\right)$

are two successive one parameter point transformations, there is a one
parameter point transformation parametrized by the real parameter
$\varepsilon^{\prime\prime}=\left(\varepsilon,\varepsilon^{\prime}\right)$ so
that
$x^{\prime\prime}=x^{\prime\prime}\left(x,y,\varepsilon^{\prime\prime}\right),~{}~{}y^{\prime\prime}=y^{\prime\prime}\left(x,y,\varepsilon^{\prime\prime}\right).$

c) They are invertible, that is, for each one parameter point transformation

$x^{\prime}=y^{\prime}\left(x,y,\varepsilon\right)~{}~{},~{}~{}y^{\prime}=y^{\prime}\left(x,y,\varepsilon\right)$

there exists the inverse transformation

$x^{\prime\prime}\left(x^{\prime},y^{\prime},\varepsilon_{inv}\right)=x~{}~{},~{}~{}y^{\prime\prime}\left(x^{\prime},y^{\prime},\varepsilon_{inv}\right)=y$

d)There is the identity transformation defined by the value222Without
abandoning generality, we can take $\varepsilon_{0}=0$.
$\varepsilon=\varepsilon_{0},~{}$that is

$x^{\prime}\left(x,y,\varepsilon_{0}\right)=x~{}~{},~{}~{}y^{\prime}\left(x,y,\varepsilon_{0}\right)=y.$

From the above it follows that the one parameter point transformations form a
group. A group of one parameter point transformation defines a family of
curves in $M,$ which are parametrized by the parameter $\varepsilon$ and are
called the orbits of the group of transformations. These curves may be viewed
as the integral curves of a differentiable vector field $X\in M.$

#####  2.2.1 Infinitesimal Transformations

Let
$\bar{x}\left(x,y,\varepsilon\right),~{}\bar{y}\left(x,y,\varepsilon\right)$
be the parametric equations of a group orbit through the point
$P\left(x,y,0\right)$. The tangent vector at the point $P=P\left(x,y,0\right)$
is given by

$X_{P}=\frac{\partial\bar{x}}{\partial\varepsilon}|_{\varepsilon\rightarrow
0}\partial_{x}|_{P}+\frac{\partial\bar{y}}{\partial\varepsilon}|_{\varepsilon\rightarrow
0}\partial_{y}|_{P}.$

The vector $X_{P}$ defines near the point $P\left(x,y,0\right)$ a point
transformation

$\bar{x}=x+\varepsilon\xi_{P}~{}~{},~{}~{}\bar{y}=y+\varepsilon\eta_{P}$ (2.3)

where we have set

$\xi_{P}=\frac{\partial\bar{x}}{\partial\varepsilon}|_{\varepsilon\rightarrow
0}~{}~{},\text{~{}~{}}\eta=\frac{\partial\bar{y}}{\partial\varepsilon}|_{\varepsilon\rightarrow
0}.$ (2.4)

The point transformation (2.3) is a one parameter a point transformation which
is called an infinitesimal point transformation. The vector field $X_{P}$ is
called the generator of the infinitesimal transformation (2.3) along the orbit
through the point $P$. Evidently, the infinitesimal transformation moves a
point along the orbit of the group through that point.

###### Example 2.2.2

Compute the generator of the infinitesimal transformation for the one
parameter point transformation

$\displaystyle\bar{x}$ $\displaystyle=$ $\displaystyle
x\cos\varepsilon-y\sin\varepsilon$ $\displaystyle\bar{y}$ $\displaystyle=$
$\displaystyle x\sin\varepsilon+y\cos\varepsilon.$

Solution: We have

$\displaystyle\xi\left(x,y\right)$ $\displaystyle=$
$\displaystyle\frac{\partial\bar{x}}{\partial\varepsilon}|_{\varepsilon\rightarrow
0}=-\left(x\sin\varepsilon+y\cos\varepsilon\right)|_{\varepsilon\rightarrow
0}=-y$ $\displaystyle\eta\left(x,y\right)$ $\displaystyle=$
$\displaystyle\frac{\partial\bar{y}}{\partial\varepsilon}|_{\varepsilon\rightarrow
0}=\left(x\cos\varepsilon-y\sin\varepsilon\right)|_{\varepsilon\rightarrow
0}=x$

from which follows that the generator of the infinitesimal transformation is
$X=-y\partial_{x}+x\partial_{y}.$

It has been showed that a one parameter point transformation fixes an
infinitesimal generator up to a constant depending on the parametrization of
the group orbit. In the following section, it will be shown that the converse
holds true, that is, for an infinitesimal generator there always exists a
unique one parameter point transformation.

###### Integral curves

Consider a differentiable vector field $X\in M$ given
by$~{}X=X^{i}\partial_{i}$. At each point of $P\in M$, $X$ determines a smooth
curve
$\gamma_{X}\left(\varepsilon,P\right)=c_{X}^{i}\left(\varepsilon,P\right)\partial_{i}$,
where $\varepsilon\in J_{\varepsilon}~{}$and $J_{\varepsilon}~{}$is an open
intevral of $\mathbb{R},$ as follows

$\frac{dc_{X}^{i}\left(\varepsilon,P\right)}{d\varepsilon}=X^{i}\left(c_{X}^{i}\left(\varepsilon,P\right)\right)~{},~{}~{}c_{X}^{i}\left(0,P\right)=x^{i}\left(P\right).$
(2.5)

The curve $\gamma_{X}\left(J_{\varepsilon},P\right)$ is called the integral
curve of $X$ through $P$.

Equation (2.5) defines an autonomous system of ordinary differential equations
(ODEs) with solutions $\gamma_{X}\left(t,P\right)$ subject to the initial
conditions $c_{X}^{i}\left(0,P\right)=x^{i}\left(P\right)$. The existence and
uniqueness of integral curves is given by, the following theorem [2].

###### Theorem 2.2.3

Let $\gamma_{1}\left(J_{1},P\right)~{}$and $\gamma_{2}\left(J_{2},P\right)$ be
two integral curves of the vector field $X$ on $M,$ with the same initial
condition $x^{i}\left(P\right)$. Then $\gamma_{1}\left(J_{1},P\right)~{}$and
$\gamma_{2}\left(J_{2},P\right)$ are equal on $J_{1}\cap J_{2}$, where
$J_{1},J_{2}$ are two open intervals of $\mathbb{R}.$

In the case where $\gamma_{X}\left(J_{\varepsilon},P\right)$ defines a one
parameter point transformation, the system (2.5) defines the generator of the
infinitesimal transformation. Due to the uniqueness of the solution there
exists only one parameter point transformation for each vector field $X$.

###### Example 2.2.4

Consider the space $\mathbb{R}^{2}$ with coordinates $\left(x,y\right)$ and
the vector field $X=y\partial_{x}+x\partial_{y}$. Find the integral curve of
$X~{}$through the point $P=\left(x_{0},y_{0}\right).$

Solution: Let
$\gamma_{P}\left(\varepsilon\right)=\left(x\left(\varepsilon\right),y\left(\varepsilon\right)\right)~{}$be
the integral curve of $X$ passing through $P$. The system of autonomous first
order equations defining the integral curves are

$\frac{dx}{d\varepsilon}=y~{}~{},~{}~{}\frac{dy}{d\varepsilon}=x$

with the initial condition
$x\left(0\right)=x_{0}~{},~{}y\left(0\right)=y_{0}$. The solution of this
system is

$\displaystyle x\left(\varepsilon\right)$ $\displaystyle=$ $\displaystyle
x_{0}\cosh\varepsilon+y_{0}\sinh\varepsilon$ (2.6) $\displaystyle
y\left(\varepsilon\right)$ $\displaystyle=$ $\displaystyle
x_{0}\sinh\varepsilon+y_{0}\cosh\varepsilon.$ (2.7)

Equations (2.6),(2.7) define the rotation in the hyperbolic space.

#####  2.2.2 Invariant Functions

Let $F\left(x,y\right)$ be a function in $M$. Under the one parameter point
transformation

$\bar{x}=\bar{x}\left(x,y,\varepsilon\right),~{}\bar{y}=\bar{y}\left(x,y,\varepsilon\right)$

the function becomes $\bar{F}\left(\bar{x},\bar{y}\right).$

###### Definition 2.2.5

The function $F$ is invariant under the one parameter point transformation if
and only if $\bar{F}\left(\bar{x},\bar{y}\right)=0$
when$~{}F\left(x,y\right)=0$ at all points where the one parameter point
transformation acts. Equivalently, the generator $X$ of the point
transformation is a symmetry of the function $F$ if

$X\left(F\right)=0~{}~{},~{}~{}modF=0$ (2.8)

The symmetry condition (2.8) is equivalent to the first order partial
differential equation (PDE)

$\xi\frac{\partial F}{\partial x}+\eta\frac{\partial F}{\partial y}=0.$ (2.9)

In order to determine all functions which are invariant under the
infinitesimal generator $X$ one has to solve the associated Lagrange system

$\frac{dx}{\xi\left(x,y\right)}=\frac{dy}{\eta\left(x,y\right)}.$

The characteristic function or zero order invariant $W$ of $X$ is defined as
follows

$dW=\frac{dx}{\xi\left(x,y\right)}-\frac{dy}{\eta\left(x,y\right)}.$ (2.10)

The zero order invariant is indeed invariant under the $X,$ that is
$X\left(W\right)=0.$ Therefore, any function of the form $F=F\left(W\right),$
where $W$ is the zero order invariant satisfies (2.9) and it is invariant
under the one parameter point transformation with generator $X.$

#####  2.2.3 Lie Algebras

In section 2.2 we considered the one parameter point transformation which
depend on one parameter $\varepsilon$. However transformations can depend on
more that one parameter, as follows

$\bar{x}=\bar{x}\left(x,y,\mathbf{E}\right)~{}~{},~{}~{}\bar{y}=\bar{y}\left(x,y,\mathbf{E}\right)$
(2.11)

where $\mathbf{E}=\varepsilon^{\beta}\partial_{\beta},$ is a vector field in
the $\mathbb{R}^{\kappa},~{}\beta=1...\kappa$ with the same properties of
definition 2.2.1, is a multi parameter point transformation,

For every parameter $\varepsilon^{\beta}~{}$of the multi parameter point
transformation (2.11) an infinitesimal generator can be defined

$X_{\beta}=\xi_{\beta}\left(x,y\right)\partial_{x}+\eta_{\beta}\left(x,y\right)\partial_{y}$

where

$\xi_{\beta}=\frac{\partial\bar{x}}{\partial\varepsilon_{\beta}}|_{\varepsilon_{\beta}\rightarrow
0}~{}~{},~{}~{}\eta_{\beta}=\frac{\partial\bar{y}}{\partial\varepsilon_{\beta}}|_{\varepsilon_{\beta}\rightarrow
0}.$

Let $F\left(x,y\right)$ be a function in $M$ which is invariant under a
multiparameter point transformation. Since the multi parameter transformation
can be described as $m$ one parameter point transformations, $F$ is invariant
under $m$ infinitesimal generators.

###### Definition 2.2.6

A Lie algebra is a finite dimensional linear space $G,$ in which a binary
operator, denoted $\left[~{},~{}\right]$ has been defined which has the
following properties

i) $\left[X,X\right]=0$ for all $X\in G$

ii)
$\left[X,\left[Y,Z\right]\right]+\left[Y,\left[Z,X\right]\right]+\left[Z,\left[X,Y\right]\right]=0$
for all $X,Y,Z\in G$.

iii)If $X_{A},X_{B}\in G$ then$~{}\left[X_{A},X_{B}\right]=C_{AB}^{C}X_{C}$,
$X_{C}\in G$. The quantities $C_{AB}^{C}$ are constants and are called the
structure constants of the Lie algebra.

The operator $\left[~{},~{}\right]$ is called the commutator and it is defined
by the following expression

$\left[X_{A},X_{B}\right]=X_{A}X_{B}-X_{B}X_{A}=-\left[X_{B},X_{N}\right].$
(2.12)

From the definition of the commutator (2.12) and from the requirements of the
Lie algebra, it follows that the structure constants are antisymmetric in the
two lower indices, i.e.

$C_{AB}^{C}+C_{BA}^{C}=0$ (2.13)

and they have to satisfy the Jacobi identity

$C_{AB}^{E}C_{DE}^{C}+C_{BD}^{E}C_{AE}^{C}+C_{DA}^{E}C_{BE}^{C}=0.$ (2.14)

The structure constants characterize the Lie algebra because every set of
constants $C_{AB}^{C}$ which satisfy (2.13) and (2.14) defines _locally_ a
unique Lie group. An important property of the structure constants is that
they do not change under a coordinate transformation. The structure constants
do change under a transformation of the basis; this property is useful because
it can be used to simplify the structure constants of the given group.

###### Definition 2.2.7

Let $G,H$ be closed Lie algebras with elements
$\left\\{X_{A}\right\\},~{}\left\\{Y_{a}\right\\}$ respectively. If $\dim
H\leq\dim G$ and $Y_{a}\in G$ then $H$ is called a Lie subalgebra of $G.$

Suppose that the vector fields $X,Y$ leave invariant a function
$F=F\left(x,y\right)$. If $\left[X,Y\right]=Z~{}$with $Z\neq X,Y$, i.e. the
generators $X,Y$ do not form a closed Lie algebra, then $F$ is also invariant
under the action of $Z$. This process can be used to find extra symmetries.

###### Example 2.2.8

The vector fields

$X_{1}=\sin\theta\partial_{\phi}+\cos\theta\cot\phi\partial_{\theta}~{}\
,~{}X_{2}=\cos\theta\partial_{\phi}-\sin\theta\cot\phi\partial_{\theta}~{}~{},~{}~{}X_{3}=\partial_{\theta}$

span the $so\left(3\right)$ Lie algebra with structure constants
$C_{12}^{3}=C_{31}^{2}=C_{23}^{1}=1,$ i.e. the commutators are

$\left[X_{1},X_{2}\right]=X_{3}~{}~{},~{}~{}\left[X_{3},X_{1}\right]=X_{2}~{}~{},~{}~{}\left[X_{2},X_{3}\right]=X_{1}$

If the function $F=F\left(\theta,\phi\right)$ is invariant under the
infinitesimal generators $X_{1},X_{3}$, then it is also invariant under the
action of $X_{2}$. In that case, it is easy to see that $F=F_{0},$ where
$F_{0}$ is a constant.

####  2.3 Lie symmetries of differential equations

Previously, we studied the case when a function $F\in M~{}$is invariant under
the action of a one parameter point transformation. In the following we
consider the case of differential equations (DEs) which are invariant under a
group of one parameter point transformations.

#####  2.3.1 Prolongation of point transformations

In order to study the action of a point transformation to a differential
equation $H\left(x,y,y^{\prime},...,y^{\left(n\right)}\right)~{}$where
$y=y\left(x\right)$, we have to prolong the point transformation to the
derivatives $y^{\left(n\right)}$. The infinitesimal transformation (2.3) in
the jet space $B_{M}=\left\\{x,y,y^{\prime},...,y^{\left(n\right)}\right\\}$
is

$\displaystyle\bar{x}$ $\displaystyle=$ $\displaystyle x+\varepsilon\xi$
$\displaystyle\bar{y}$ $\displaystyle=$ $\displaystyle x+\varepsilon\eta$
$\displaystyle\bar{y}^{\left(1\right)}$ $\displaystyle=$ $\displaystyle
y^{\left(1\right)}+\varepsilon\eta^{\left[1\right]}$ $\displaystyle...$
$\displaystyle\bar{y}^{\left(n\right)}$ $\displaystyle=$ $\displaystyle
y^{\left(n\right)}+\varepsilon\eta^{\left[n\right]}$

where
$y^{\left(n\right)}=\frac{d^{n}y}{dx^{n}},~{}\bar{y}^{\left(n\right)}=\frac{d^{n}\bar{y}}{d\bar{x}^{n}}$
and

$\eta^{\left[1\right]}=\frac{\partial\bar{y}^{\left(1\right)}}{\partial\varepsilon}~{}~{},...,~{}~{}\eta^{\left[n\right]}=\frac{\partial\bar{y}^{\left(n\right)}}{\partial\varepsilon}.$

That means that the variation equals the difference of the derivatives before
and after the action of the one parameter transformation. For the first
prolongation function $\eta^{\left[1\right]}~{}$we have

$\eta^{\left[1\right]}\equiv\lim_{\varepsilon\rightarrow
0}\left[\frac{1}{\varepsilon}\left(\bar{y}^{\left(1\right)}-y^{\left(1\right)}\right)\right]=\frac{d\eta}{dx}-y^{\left(1\right)}\frac{d\xi}{dx}.$

Similarly for$~{}\eta^{\left[n\right]}$ we have the expression

$\eta^{\left[n\right]}=\frac{d\eta^{n-1}}{dx}-y^{\left(n\right)}\frac{d\xi}{dx}=\frac{d^{n}}{dx^{n}}\left(\eta-y^{\left(1\right)}\xi\right)+y^{\left(n+1\right)}\xi.$
(2.15)

Finally, the extension of the infinitesimal generator in the jet space
$B_{M}~{}$is

$X^{\left[n\right]}=X+\eta^{\left[1\right]}\partial_{y^{\left(1\right)}}+...+\eta^{\left[n\right]}\partial_{y^{\left[n\right]}}.$

The field $X^{\left[n\right]}\in B_{M}$ is called the nth prolongation of the
generator $X,~{}$where

$X=\xi\left(x,y\right)\partial_{x}+\eta\left(x,y\right)\partial_{y}$

is the infinitesimal point generator in the space $\left\\{x,y\right\\}$.

It is possible to write the prolongation coefficients in terms of the partial
derivatives of the components $\xi\left(x,y\right),~{}\eta\left(x,y\right)$.
The first and the second prolongation are expressed as follows

$X^{\left[1\right]}=X+\left[\eta_{,x}+y^{\left(1\right)}\left(\eta_{,y}-\xi_{,x}\right)-y^{\left(1\right)^{2}}\xi_{,y}\right]\partial_{y^{\left(1\right)}}$
$X^{\left[2\right]}=X^{\left[1\right]}+\left[\begin{array}[]{c}\eta_{,xx}+2\left(\eta_{,xy}-\xi_{,xx}\right)y^{\left(1\right)}+\left(\eta_{,yy}-2\xi_{,xy}\right)y^{\left(1\right)^{2}}+\\\
-y^{\left(1\right)^{3}}\xi_{,yy}+\left(\eta_{,y}-2\xi_{,x}-3\xi_{,y}y^{\left(1\right)}\right)y^{\left(2\right)}\end{array}\right]\partial_{y^{\left(2\right)}}$
(2.16)

where the comma $","$ denotes partial derivative.

Some important observations [3] for the prolongation coefficient
$\eta^{\left[n\right]}$ are:

(a) $\eta^{\left[n\right]}$ is linear in $y^{\left(n\right)}$

(b) $\eta^{\left[n\right]}$ is a polynomial in the derivatives
$y^{\left(1\right)},...,y^{\left(n\right)}$ whose coefficients are linear
homogeneous in the functions $\xi\left(x,y\right),$ $\eta\left(x,y\right)$ up
to nth order partial derivatives.

###### Multiparameter prolongation

In the case the differential equation $H$ depends on $n$ independent variables
$\left\\{x^{i}:i=1..n\right\\}$ and $m$ dependent variables
$\left\\{u^{A}:A=1...m\right\\}$, i.e.
$H=H\left(x^{i},u^{A},u_{,i}^{A},u_{,ij}^{A},..\right),$ we consider the one
parameter point transformation

$\bar{x}^{i}=\Xi^{i}\left(x^{i},u^{A},\varepsilon\right)~{}~{},~{}~{}\bar{u}^{A}=\Phi^{A}\left(x^{i},u^{A},\varepsilon\right).$

In this case the generating vector is

$X=\xi^{i}\left(x^{k},u^{A}\right)\partial_{i}+\eta^{A}\left(x^{k},u^{A}\right)\partial_{A}$
(2.17)

where

$\xi^{i}\left(x^{k},u^{A}\right)=\frac{\partial\Xi^{i}\left(x^{i},u^{A},\varepsilon\right)}{\partial\varepsilon}|_{\varepsilon\rightarrow
0}~{}~{},~{}~{}\eta^{A}\left(x^{k},u^{A}\right)=\frac{\partial\Phi\left(x^{i},u^{A},\varepsilon\right)}{\partial\varepsilon}|_{\varepsilon\rightarrow
0}.$

To extend the generator vector in the jet space
$\bar{B}_{\bar{M}}=\left\\{x^{i},u^{A},u_{,i}^{A},u_{,ij}^{A},..,u_{ij...i_{n}}^{A}\right\\}$
we apply the same procedure as in section 2.3.1. Therefore the vector field
$X^{\left[n\right]}\in\bar{B}_{\bar{M}}$

$X^{\left[n\right]}=X+\eta_{i}^{A}\partial_{u_{i}}+...+\eta_{ij..i_{n}}^{A}\partial_{u_{ij..i_{n}}}$

is defined as the nth prolongation of the generator (2.17), where333Where
$D_{i}=\frac{\partial}{\partial x^{i}}+u_{i}^{A}\frac{\partial}{\partial
u^{A}}+u_{ij}^{A}\frac{\partial}{\partial
u_{j}^{A}}+...+u_{ij..i_{n}}^{A}\frac{\partial}{\partial u_{jk..i_{n}}^{A}}.$

$\eta_{i}^{A}=D_{i}\eta^{A}-u_{,j}^{A}D_{i}\xi^{j}$ (2.18)
$\eta_{ij..i_{n}}^{A}=D_{i_{n}}\eta_{ij..i_{n-1}}^{A}-u_{ij..k}D_{i_{n}}\xi^{k}.$
(2.19)

In terms of the partial derivatives of the components
$\xi^{i}\left(x^{k},u^{A}\right),~{}\eta^{A}\left(x^{k},u^{A}\right)$ of the
generator vector (2.17), the first and the second prologations of (2.17) are
expressed as follows

$X^{\left[1\right]}=X+\left(\eta_{,i}^{A}+u_{,i}^{B}\eta_{,B}^{A}-\xi_{,i}^{j}u_{,j}^{A}-u_{,i}^{A}u_{,j}^{B}\xi_{,B}^{j}\right)\partial_{u_{i}^{A}}$
(2.20)
$X^{\left[2\right]}=X^{\left[1\right]}+\left[\begin{array}[]{c}\eta_{,ij}^{A}+2\eta_{,B(i}^{A}u_{,j)}^{B}-\xi_{,ij}^{k}u_{,k}^{A}+\eta_{,BC}^{A}u_{,i}^{B}u_{,j}^{C}-2\xi_{,(i\left|B\right|}^{k}u_{j)}^{B}u_{,k}^{A}+\\\
-\xi_{,BC}^{k}u_{,i}^{B}u_{,j}^{A}u_{,k}^{A}+\eta_{,B}^{A}u_{,ij}^{B}-2\xi_{,(j}^{k}u_{,i)k}^{A}+-\xi_{,B}^{k}\left(u_{,k}^{A}u_{,ij}^{B}+2u_{(,j}^{B}u_{,i)k}^{A}\right)\end{array}\right]\partial_{u_{ij}}.$
(2.21)

#####  2.3.2 Lie symmetries of ODEs

In the previous sections we analyzed the invariance of functions under the
action of a point transformation. In the following we define the invariance of
ordinary differential equations (ODEs) under a one parameter point
transformation.

Consider the $N-$dimensional system of ODEs444In the following equations, $t$
is the independent parameter and $x^{i}=x^{i}\left(t\right)$ the dependent
parameters.

$x^{\left(n\right)i}=\omega^{i}\left(t,x^{k},\dot{x}^{k},\ddot{x}^{k},...,x^{\left(n-1\right)i}\right)$
(2.22)

where $\dot{x}^{i}=\frac{dx^{i}}{dt}$ ,
$x^{\left(n\right)}=\frac{d^{n}x}{dt^{n}}$ and the infinitesimal point
transformation with infinitesimal generator $X$ is

$\displaystyle\bar{t}$ $\displaystyle=$ $\displaystyle
t+\varepsilon\xi\left(t,x^{k}\right)$ (2.23) $\displaystyle\bar{x}^{i}$
$\displaystyle=$ $\displaystyle x+\varepsilon\eta^{i}\left(t,x^{k}\right)$
(2.24)

###### Theorem 2.3.1

Let

$X=\xi\left(t,x^{k}\right)\partial_{t}+\eta^{i}\left(t,x^{k}\right)\partial_{i}$
(2.25)

be the infinitesimal generator of point transformation (2.23)-(2.24) and

$X^{\left[n\right]}=X+\eta_{\left[1\right]}^{i}\partial_{\dot{x}^{i}}+...+\eta_{\left[n\right]}^{i}\partial_{x^{\left(n\right)i}}$

be the nth prolongation of $X$, where $\eta_{\left[n\right]}^{i}$ is given by
(2.19). We shall say that the $N-$dimensional system of ODEs (2.22) is
invariant under the point transformation (2.23), (2.24) if and only if there
exists a function $\lambda$ such as the following condition holds

$\left[X^{\left[n\right]},A\right]=\lambda A$ (2.26)

where555For Hamiltonian systems, the operator $\mathbf{A}$ is called
Hamiltonian vector field.

$A=\frac{\partial}{\partial t}+\dot{x}^{i}\frac{\partial}{\partial
x^{i}}+...+\omega^{i}\left(t,x^{k},\dot{x}^{k},\ddot{x}^{k},...,x^{\left(n-1\right)i}\right)\frac{\partial}{\partial
x^{\left(n\right)i}}.$

In that case we say that $X$ is a Lie point symmetry of the $N-$dimensional
system of ODEs (2.22).

If $f^{i}$ is a solution of the system (2.22), i.e. $Af^{i}=0,$ then condition
(2.26) becomes $X\left(Af^{i}\right)=0,~{}$that is,

$\eta_{\left[n\right]}^{i}=X^{\left[n-1\right]}\omega^{i}\left(t,x^{k},\dot{x}^{k},\ddot{x}^{k},...,x^{\left(n-1\right)i}\right).$
(2.27)

Equations (2.27) are called the determining equations. The solution of the
determining equations (2.27) gives the infinitesimal generators of the
transformation (2.23)-(2.24).

In general, a function
$H\left(t,\dot{x}^{k},\ddot{x}^{k},...,x^{\left(n\right)k}\right)=0$ is
invariant under the transformation (2.23)-(2.24) if and only if

$X^{\left[n\right]}\left(H\right)=\lambda H~{},~{}modH=0.$ (2.28)

where $\lambda$ is a function to be determined [4].

Below we give an example in which the Lie symmetries are calculated using the
symmetry condition (2.27).

###### Example 2.3.2

Find the Lie symmetries of the ODE $\ddot{x}=0.$

Solution: Condition (2.27) gives $\eta_{\left[2\right]}=0.~{}$From (2.16) the
following condition is found

$\eta_{,tt}+2\left(\eta_{,tx}-\xi_{,tt}\right)\dot{x}+\left(\eta_{,xx}-2\xi_{,tx}\right)\dot{x}^{2}-\dot{x}^{3}\xi_{,xx}=0$
(2.29)

since $\ddot{x}=0$. Functions $\xi,\eta$ are dependent only on the variables
$\left\\{t,x\right\\}$, hence equation (2.29) is a polynomial of $\dot{x}.$
This polynomial must vanish identically hence the coefficients of all powers
of $\dot{x}$ must vanish. Therefore, we have the following determining
equations

$\displaystyle\left(\dot{x}\right)^{0}$ $\displaystyle:$
$\displaystyle\eta_{,tt}=0$ $\displaystyle\left(\dot{x}\right)^{1}$
$\displaystyle:$ $\displaystyle\eta_{,xy}-\xi_{,tt}=0$
$\displaystyle\left(\dot{x}\right)^{2}$ $\displaystyle:$
$\displaystyle\eta_{,xx}-2\xi_{,tx}=0$ $\displaystyle\left(\dot{x}\right)^{3}$
$\displaystyle:$ $\displaystyle\xi_{,xx}=0.$

whosw solution is

$\displaystyle\xi\left(t,x\right)$ $\displaystyle=$ $\displaystyle
a_{1}+a_{2}t+a_{3}t^{2}+a_{4}x+a_{5}tx$ $\displaystyle\eta\left(t,x\right)$
$\displaystyle=$ $\displaystyle a_{6}+a_{7}t+a_{8}x+a_{3}tx+a_{5}x^{2}.$

We conclude that the second order ODE $\ddot{x}=0$ has eight Lie pont
symmetries generated by the generic vector field666As many as the unspecified
constants in the expression of the generic symmetric vector.

$X=\left(a_{1}+a_{2}t+a_{3}t^{2}+a_{4}x+a_{5}tx\right)\partial_{t}+\left(a_{6}+a_{7}t+a_{8}x+a_{3}tx+a_{5}x^{2}\right)\partial_{x}.$
(2.30)

These vectors are the generators of the projective algebra
$sl\left(3,R\right)$ of the 2-d Euclidian plane. Furthermore, this is the
maximum number of symmetries that a single second order ODE (in one variable!)
can have.

Lie symmetries can be used to find invariants or to reduce the order of an
ODE. Furthermore an ODE is characterized by the admitted algabra of Lie
symmetries. For second order ODEs we have the following theorem.

###### Theorem 2.3.3

If a second order ODE admits as Lie point symmetries the eight of
$sl\left(3,R\right)$, then, there exists a transformation which brings the
equation to the form $x^{\ast\prime\prime}=0$ and vice versa.

For instance the equations

$~{}\ddot{x}+\frac{1}{x}\dot{x}^{2}=0$ $\ddot{x}+3x\dot{x}+x^{3}=0$
$\ddot{x}+\omega_{0}^{2}x+\sin t=0$
$\left(4+x^{2}+t^{2}\right)\ddot{x}-2t\dot{x}^{3}+2x\dot{x}^{2}-2t\dot{x}+2x=0$

are equivalent to the equation of motion of a free particle [5, 6, 7], because
they are invariant under the action of the Lie algebra $sl\left(3,R\right).$
The transformation where the admitted $sl\left(3,R\right)$ algebra is written
in the form (2.30) is a coordinate transformation on $M:$
$\left(t,x\right)\rightarrow\left(\tau,y\right)$ which transforms the ODE to
the form $\frac{d^{2}y}{d\tau^{2}}=0.$ This procedure is called linearization
process (see [8, 9] and references therein).

Below we present two methods where the use of Lie symmetries reduces the order
of an ODE.

###### Canonical coordinates

###### Definition 2.3.4

Let $X~{}$be a vector field with coordinates
$X=\xi\left(t,x\right)\partial_{t}+\eta\left(t,x\right)\partial_{x}$. If under
the coordinate transformation
$\left\\{t,x\right\\}\rightarrow\left\\{r,s\right\\}$ holds that

$Xr=0~{}~{},~{}~{}Xs=1$ (2.31)

then, we say that $\\{r,s\\}$ are the canonical coordinates of $X,$ i.e.
$X=\partial_{s}$.

Canonical coordinates can be used to reduce by one the order of an ODE.
Consider the $nth$ order ODE $(n\succeq 2)$

$\frac{d^{n}s}{dr^{2}}=\bar{\omega}\left(r,s,\frac{ds}{dr},...,\frac{d^{n-1}s}{dr^{n-1}}\right).$
(2.32)

Let $X_{C}=\partial_{s}$ be a Lie symmetry of (2.32) written in canonical
coordinates. The nth prolongation of the symmetry vector is
$X_{C}^{\left[n\right]}=X_{C},~{}$so condition (2.27) sets the constraint

$\frac{\partial}{\partial
s}\bar{\omega}\left(r,s,\frac{ds}{dr},...,\frac{d^{n-1}s}{dr^{n-1}}\right)=0.$

This implies that the function $\bar{\omega}$ is independent of $s,$
consequently (2.32) can be written in the form

$\frac{d^{n-1}S}{dr^{n-1}}=\bar{\omega}\left(r,S,\frac{dS}{dr},...,\frac{d^{n-2}S}{dr}\right)$
(2.33)

which is a $\left(n-1\right)$ order ODE, where $S~{}$is defined by
$S=\frac{ds}{dr}$.

###### Example 2.3.5

Use the canonical coordinates of the Lie symmetry
$X=x\partial_{x}-t\partial_{t}$ to reduce the order of the ODE

$\ddot{x}+x\dot{x}=0$ (2.34)

Solution: The canonical coordinates of $X$ are

$x=e^{s}~{}~{},~{}~{}t=re^{-s}.$

For the first and the second derivative of $x\left(t\right)$ we compute

$\dot{x}=\frac{e^{2s}s^{\prime}}{1-rs^{\prime}}$
$\ddot{x}=\frac{e^{3s}}{\left(1-rs^{\prime}\right)^{3}}\left[\left(2s^{\prime
2}+s^{\prime\prime}\right)\left(1-rs^{\prime}\right)+s^{\prime}\left(s^{\prime}+rs^{\prime\prime}\right)\right]$

where $s^{\prime}=\frac{ds}{dr}\,$.

Replacing in 2.34, we find the reduced equation

$\left(2S^{2}+S^{\prime}\right)\left(1-rS\right)+S\left(S+rS^{\prime}\right)+S\left(1-rS\right)^{2}=0$
(2.35)

where we have set $S=s^{\prime}$. Equation (2.35) is an Abel equation of the
first kind [10] invariant under $X=\frac{d}{ds}.$

###### Invariants

As in the case of functions in $M~{}$(see section 2.2.2), condition (2.28) is
equivalent to the following Lagrange system

$\frac{dt}{\xi}=\frac{dx}{\eta}=\frac{d\dot{x}}{\eta_{\left[1\right]}}=...=\frac{dx^{\left(n\right)}}{\eta_{\left[n\right]}}.$
(2.36)

The Lagrange system (2.36) provides us with characteristic functions

$W^{\left[0\right]}\left(t,x\right),~{}W^{\left[1\right]i}\left(t,x,\dot{x}\right),~{}W^{\left[n\right]}\left(t,x,\dot{x},...,\dot{x}^{\left(n\right)}\right)$

where $W^{\left[n\right]}$ is called the nth order invariant of the Lie
symmetry vector. If (2.25) is a Lie symmetry for the ODE

$x^{\left(n\right)}=\omega\left(t,\dot{x},\ddot{x},...,x^{\left(n-1\right)}\right)$
(2.37)

it follows that (2.37) can be written as a function of the characteristic
functions $W^{\left[1\right]},...,W^{\left[n\right]},~{}$of (2.25).

Let $u=W^{\left[0\right]}~{},~{}v=W^{\left[1\right]},$ where
$W^{\left[0\right]},~{}W^{\left[1\right]}$ are the zero and the first order
invariants of a Lie symmetry repetitively. From the invariants $u,~{}v,we$
define the differential invariants

$\frac{dv}{du}~{},...,\frac{d^{n-1}v}{du^{n-1}}.$ (2.38)

where

$\frac{dv}{du}=\frac{\frac{\partial v}{\partial t}+\frac{\partial v}{\partial
x}\dot{x}+...+\frac{\partial v}{\partial\dot{x}}\ddot{x}}{\frac{\partial
v}{\partial t}+\frac{\partial v}{\partial x}\dot{x}}.$

The differential invariants are functions of $x^{\left(n\right)}$, hence, it
is feasible that (2.37) may be written in terms of the differential invariants
(2.38), i.e.

$\frac{d^{n-1}v}{du^{n-1}}=\Omega\left(u,v,\frac{dv}{du}~{},...\frac{d^{n-1}v}{du^{n-1}}\right)$
(2.39)

which is a $\left(n-1\right)$ order ODE.

###### Example 2.3.6

Use the first order invariants of the Lie symmetry
$X=x\partial_{x}-nt\partial_{t}$ to reduce the order of the Lane-Emden
equation

$\ddot{x}+\frac{2}{t}\dot{x}+x^{2n+1}=0$

where $n\neq-\frac{1}{2},0$. The Lane-Emden equation arises in the study of
equilibrium configurations of a spherical gas cloud acting under the mutual
attractions of its molecules and subject to the laws of thermodynamics [11,
12].

Solution: The first prolongation of $X$ is

$X^{\left[1\right]}=x\partial_{x}-nt\partial_{t}+\left(n+1\right)\dot{x}\partial_{\dot{x}}$

hence, the corresponding Lagrange system is

$\frac{dt}{-nt}=\frac{dx}{x}=\frac{d\dot{x}}{\left(n+1\right)\dot{x}}.$

The zero and the first order invariants are found to be

$u=xt^{\frac{1}{n}}~{}~{},~{}~{}v=\dot{x}t^{1+\frac{1}{n}}.$

The differential invariant is defined as

$\frac{dv}{du}=\frac{\left(1+\frac{1}{n}\right)\dot{x}+t\ddot{x}}{\frac{1}{n}t^{-1}x+\dot{x}}.$

Substituting in the Lane-Emden equation we obtain the first order ODE

$v^{\prime}\left(u+nv\right)-\left(1-n\right)v+nu^{2n+1}=0.$ (2.40)

where $v^{\prime}=\frac{dv}{du}$. Equation (2.40) is an Abel equation of the
second kind [10].

An interesting application of the Lie invariants to the classical Kepler
system can be found in [13] where the authors derive the Runge-Lenz vector
using the first order invariants of a point transformation.

It is possible an ODE to admit many symmetries which span a Lie algebra
$G_{m}$ of dimension $m>1$. The following theorem relates the Lie point
symmetries of the reduced and of the original equation.

###### Theorem 2.3.7

Consider an ODE which admits the Lie point symmetries $X_{1},~{}X_{2}$ which
are such that $\left[X_{1},X_{2}\right]=C_{12}^{1}X_{1}.~{}$Then, the
reduction by $X_{1}$ leads to a reduced equation which admits $X_{2}$ as a Lie
symmetry whereas reduction of the ODE by $X_{2}$ leads to a reduced equation,
which does not admit $X_{1}$ as a Lie symmetry.

In case the generators $X_{1},X_{2}$ form an Abelian Lie algebra, i.e.
$\left[X_{1},X_{2}\right]=0,$ then, the reduction preserves the Lie
symmetries. Theorem 2.3.7 is important because it gives a hint as to which
generator the reduction of an ODE must start in order to continue the
reduction process with the reduced equation.

Since the reduced equation is different from the original equation, it is
possible the reduced equation to admit extra Lie symmetries, which are not Lie
symmetries of the original equation. These new Lie symmetries have been named
Type II hidden symmetries. Type II hidden symmetries can be used to continue
the reduction process and - if it is feasible - to find a solution of the
original equation [14].

There are other methods to apply Lie symmetries; a quite intriguing
application of Lie symmetries is to produce integrals or Lagrangian functions
for a system of ODEs by the method of Jacobi’s last multiplier (see [15, 16,
17] and references therein).

####  2.4 Lie symmetries of PDEs

A partial differential equation (PDE) is a function
$H=H\left(x^{i},u^{A},u_{,i}^{A},u_{,ij}^{A},..,u_{ij...i_{n}}^{A}\right)$ in
the jet space $\bar{B}_{\bar{M}},~{}$where $x^{i}$ are the independent
variables and $u^{A}$ are the dependent variables. As in the case of ODEs we
define the invariance of a PDE under the infinitesimal point transformation

$\displaystyle\bar{x}^{i}$ $\displaystyle=$ $\displaystyle
x^{i}+\varepsilon\xi^{i}\left(x^{k},u^{B}\right)$ (2.41)
$\displaystyle\bar{u}^{A}$ $\displaystyle=$
$\displaystyle\bar{u}^{A}+\varepsilon\eta^{A}\left(x^{k},u^{B}\right)$ (2.42)

with generator $X$ as follows.

###### Definition 2.4.1

Let

$X=\xi^{i}\left(x^{k},u^{B}\right)\partial_{t}+\eta^{A}\left(x^{k},u^{B}\right)\partial_{B}$
(2.43)

be the generator of the infinitesimal point transformation (2.41)-(2.42) and

$X^{\left[n\right]}=X+\eta_{\left[i\right]}^{A}\partial_{\dot{x}^{i}}+...+\eta_{\left[ij...i_{n}\right]}^{A}\partial_{u_{ij...i_{n}}}$

be the nth prolongation vector, where $\eta_{\left[ij...i_{n}\right]}^{A}$ is
given from (2.19). Then, the transformation (2.41)-(2.42) leaves invariant the
PDE

$H\left(x^{i},u^{A},u_{,i}^{A},u_{,ij}^{A},..\right)=0$ (2.44)

if there exist a function $\lambda$ where the following condition holds,

$X^{\left[n\right]}\left(H\right)=\lambda H~{}~{},~{}~{}modH=0.$ (2.45)

The infinitesimal generator $X$ is called a Lie point symmetry of the PDE
(2.44).

In case the PDE (2.44) can we written in solved form, i.e.

$u_{ij...i_{n}}^{A}=h^{A}\left(x^{i},u^{B},u_{,i}^{B},u_{,ij}^{B},..,u_{ij...i_{n-1}}^{B}\right)$

then the Lie condition (2.45) is equivalent to the following system of
equations

$\eta_{\left[ij...i_{n}\right]}^{A}=X^{\left[n-1\right]}h^{A}\left(x^{i},u^{B},u_{,i}^{B},u_{,ij}^{B},..,u_{ij...i_{n-1}}^{B}\right).$
(2.46)

Consequently the family of all solutions of (2.44) is invariant under
(2.41)-(2.42) if condition (2.45) holds. In the example below we compute the
Lie symmetries of the homogeneous heat equation.

###### Example 2.4.2

Find the Lie symmetries of the 1+1 heat equation

$H\left(u_{,t},u_{,xx}\right):u_{,xx}-u_{,t}=0.$ (2.47)

Solution: Let

$X=\xi^{t}\left(t,x,u\right)\partial_{t}+\xi^{x}\left(t,x,u\right)\partial_{x}+\eta\left(t,x,u\right)\partial_{u}$

be a Lie symmetry of (2.47). Then condition (2.46) gives

$\eta_{\left[xx\right]}-\eta_{\left[t\right]}=0.$

Substituting $\eta_{[t]},~{}\eta_{\left[xx\right]}$ from (2.20),(2.21) and
collecting terms of derivatives of $u\left(t,x\right)$ we find the following
determining equations

$\xi_{,u}^{t}=0~{}~{},~{}~{}\xi_{,x}^{t}=0~{}~{}~{},~{}~{}\xi_{,u}^{x}=0~{}~{},~{}~{}\xi_{,ttt}^{t}=0$
$\xi_{,t}^{x}=-2\eta_{,xu}~{}~{},~{}~{}\xi_{,x}^{x}=\frac{1}{2}\xi_{,t}^{t}~{}~{},~{}~{}\eta_{,uu}=0$
$\eta_{,xx}=\eta_{,t}~{}~{},~{}~{}4\eta_{,tu}+\xi_{,tt}=0.$

The solution of the system of equations is

$\displaystyle\xi^{t}\left(t,x,u\right)$ $\displaystyle=$ $\displaystyle
a_{1}t^{2}+2a_{2}t+a_{3}$ $\displaystyle\xi^{x}\left(t,x,u\right)$
$\displaystyle=$ $\displaystyle a_{1}tx+a_{2}x+a_{4}t+a_{5}$
$\displaystyle\eta\left(t,x,u\right)$ $\displaystyle=$
$\displaystyle-\frac{a_{1}}{2}\left(t+\frac{1}{2}x^{2}\right)u-\frac{a_{4}}{2}xu+a_{6}u+b\left(t,x\right)$

where $b\left(t,x\right)$ is a solution of the heat equation. The generic
vector of the infinitesimal transformation which leaves invariant the heat
equation is

$\displaystyle X$ $\displaystyle=$
$\displaystyle\left(a_{1}t^{2}+2a_{2}t+a_{3}\right)\partial_{t}+\left(a_{1}tx+a_{2}x+a_{4}t+a_{5}\right)\partial_{x}+$
$\displaystyle+\left[-\frac{a_{1}}{2}\left(t+\frac{1}{2}x^{2}\right)u-\frac{a_{4}}{2}xu+a_{6}u+b\left(t,x\right)\right]\partial_{u}.$

###### Invariant solutions

In section 2.3.2, it has been explained how to use the canonical coordinates
to reduce the order of an ODE. The infinitesimal generator
$X_{3}=\partial_{t}$ is a Lie symmetry of the heat equation (2.47), written in
canonical coordinates. Nevertheless (2.47) cannot be reduced as it happened
with the case of ODEs. However, one can calculate from the Lie symmetry
$X_{3}$ the zero order invariants, which are $\ y=x~{},~{}w=u$. We select $y$
to be the independent variable and $w=w\left(y\right).$ Substituting the
invariants in (2.47) we get the ODE $w_{yy}=0.$ Then, the following solution
can be found easily

$u\left(x\right)=c_{1}x+c_{2}$ (2.48)

which is independent on the variable $t.$ That is, the use of the Lie symmetry
$X_{3}~{}$ reduces by one the number of independent variables but not the
order of the PDE. The solution (2.48) is called an invariant solution.

###### Definition 2.4.3

The function $u=U\left(x^{i}\right)$ is an invariant solution corresponding to
the infinitesimal generator (2.43), if and only if $U\left(x^{i}\right)~{}$is
an invariant of the infinitesimal generator, i.e. $X\left(U\right)=0$ and
solves the PDE (2.44).

By definition, a Lie symmetry maps a solution onto a solution. The action of
the point transformation with infinitesimal generator $X_{3}$ on a solution
$u$ of (2.47) is
$u\left(\bar{t},\bar{x}\right)=u\left(t+\varepsilon,x\right)$, that is,
$X_{3}\bar{u}=0$ or $\bar{u}_{,t}=0$, hence the use of Lie point symmetries
gives a constraint equation. However, if a solution of a PDE is already known
we can apply the point transformation to obtain a family of solutions. This
new solutions will depend on at most as many new parameters as there are in
the symmetry transformation used.

###### Theorem 2.4.4

Assume that the PDE (2.44) admits a Lie point symmetry and let
$u=u\left(x^{i}\right)$ be a solution of (2.44) which is not invariant under
this Lie symmetry. Then, under the transformation (2.41)-(2.42) the solution
$u\left(x^{i}\right)$ defines a family of solutions of the PDE.

For instance, the point transformation corresponding to the Lie symmetry

$X_{1}=t^{2}\partial_{t}+tx\partial_{x}-\frac{1}{2}\left(t+\frac{1}{2}x^{2}\right)u\partial_{u}$

of the heat equation (2.47) is

$\bar{t}=t\left(1-\varepsilon
t\right)^{-1}~{}~{},~{}~{}\bar{x}=x\left(1-\varepsilon t\right)^{-1}$
$\bar{u}=u\sqrt{1-\varepsilon t}e^{-\frac{\varepsilon
x^{2}}{4\left(1-\varepsilon t\right)}}.$

Then, if $\bar{u}_{c}=\bar{u}_{c}\left(\bar{t},\bar{x}\right)$ is a solution
of (2.47), which is not an invariant solution, the function

$u_{c}\left(x,t\right)=\frac{1}{\sqrt{1-\varepsilon t}}e^{\frac{\varepsilon
x^{2}}{4\left(1-\varepsilon t\right)}}\bar{u}_{c}\left(\frac{x}{1-\varepsilon
t},\frac{t}{1-\varepsilon t}\right)$

is also a solution of (2.47).

####  2.5 Lie Bäcklund symmetries

So far we have considered point transformations which depend on the variables
of the base manifold only. However there exist point transformations who are
defined in the jet space and depend also on the derivatives.

The infinitesimal transformation

$\displaystyle\bar{x}^{i}$ $\displaystyle=$ $\displaystyle
x^{i}+\varepsilon\xi^{i}\left(x^{i},u,u_{,i},u_{,ij}...\right)$ (2.49)
$\displaystyle\bar{u}$ $\displaystyle=$ $\displaystyle
u+\varepsilon\eta\left(x^{i},u,u_{,i},u_{,ij}...\right)$ (2.50)

is called Lie Bäcklund transformation. This point transformation does not
concern the present work but for the completeness we we give the basic
definitions.

###### Definition 2.5.1

The generator
$X=\xi\left(x^{i},u,u_{,i},u_{,ij}...\right)\partial_{i}+\eta\left(x^{i},u,u_{,i},u_{,ij}...\right)\partial_{u}$
generates a Lie Bäcklund symmetry for the DE
$H\left(x^{i},u,u_{,i},u_{,ij}...\right)=0,$ if and only if there exist a
function $\lambda\left(x^{i},u,u_{,i},u_{,ij}...\right)$ such as the following
condition holds.

$\left[X,H\right]=\lambda H~{}\ ,~{}modH=0.$

It follows from the above definition that a Lie Bäcklund symmetry preserves
the set of solutions $u$ of the DE, i.e. $X\left(u\right)=Cu,$ where $C$ is a
constant. A special class of Lie Bäcklund symmetries are the contact
symmetries. Contact symmetry is a Lie Bäcklund symmetry where the generator
depends only on the first derivative of $u_{,i}$, i.e. the generator of
transformation (2.49)-(2.49) has the form
$X=\xi\left(x^{i},u,u_{,i}\right)\partial_{i}+\eta\left(x^{i},u,u_{,i}\right)\partial_{u}$.

###### Proposition 2.5.2

The generator $\bar{X}=\left(\eta-\xi^{k}u_{,k}\right)\partial_{u}$ is the
canonical form of the operator $X=\xi\partial_{i}+\eta\partial_{u}.$

The operator
$D_{i}=\partial_{i}+u_{,i}\partial_{u}+u_{,ij}\partial_{u_{i}}+...$ is always
a Lie Bäcklund symmetry (the trivial one, since $D_{i}$ in the canonical form
is $\bar{D}_{i}=\left(u_{,i}-u_{,i}\right)\partial_{u}+...=0$), hence
$f^{i}\left(x^{i},u,u_{,i},u_{,ij}...\right)D_{i}$ is also a Lie Bäcklund
symmetry. If $X=\xi\partial_{i}+\eta\partial_{u}$ is a Lie Bäcklund symmetry,
then the generator $\bar{X},~{}$

$\bar{X}=X-fD_{i}=\left(\xi^{k}-f^{k}\right)\partial_{k}+\left(\eta-f^{k}u_{,k}\right)\partial_{u}+...$

is also a Lie Bäcklund symmetry. Since $f^{k}$ is an arbitrary function we set
$f^{k}=\xi^{k}$ and obtain

$\bar{X}=\left(\eta-\xi^{k}u_{,k}\right)\partial_{u}.$

We can always absorb the term $\xi^{k}u_{k}$ inside the $\eta$ and have the
final result that
$\bar{X}=Z\left(x^{i},u,u_{,i},u_{,ij}...\right)\partial_{u}$ is the generator
of a Lie Bäcklund symmetry.

For a PDE the generator of a Lie point symmetry is

$X=\xi\left(x^{i},u\right)\partial_{i}+\eta\left(x^{i},u\right)\partial_{u}$

then in this case the vector $\bar{X}$ can be written as

$\bar{X}=\left(\eta\left(x^{i},u\right)-\xi^{k}\left(x^{i},u\right)u_{,k}\right)\partial_{u}$

which is linear in the first derivative; that is, for PDEs, Lie symmetries are
equivalent to contact symmetries that are linear in the first derivative
$u_{k}$ (_that property does not hold for ODEs_). For example, the Lie point
symmetry $X=\frac{\partial}{\partial x}$ of a PDE is equivalent to the Lie
Bäcklund symmetry $\bar{X}=-u_{,x}\frac{\partial}{\partial u}$.

####  2.6 Noether point symmetries

In the following sections we consider the Lie symmetries and the conservation
laws of systems admitting a Lagrangian function, i.e. systems whose equations
of motion follow from a variational principle (e.g. Hamilton principle).

#####  2.6.1 Noether symmetries of ODEs

In Analytical Mechanics the Lagrangian $L=L\left(t,x^{k},\dot{x}^{k}\right)$
is a function describing the dynamics of a system. The equations of motion of
the dynamical system follow from the action of the Euler Lagrange vector
$E_{i}$ on the Lagragian $L$, i.e.

$E_{i}\left(L\right)=0.$ (2.51)

where the Euler Lagrange vector

$E_{i}=\frac{d}{dt}\frac{\partial}{\partial\dot{x}^{i}}-\frac{\partial}{\partial
x^{i}}$ (2.52)

If the Lagrangian is invariant under the action of the transformation
(2.23)-(2.24), then, it is easy to observe that the Euler Lagrange equations
(2.51) are invariant under the transformation (2.23)-(2.24). E. Noether proved
that if the action of a one parameter point transformation leaves invariant
the Euler Lagrange equations (2.51) then there exists a conserved quantity
corresponding to the point transformation.

###### Theorem 2.6.1

Let

$X=\xi\left(t,x^{k}\right)\partial_{t}+\eta^{i}\left(t,x^{k}\right)\partial_{i}$
(2.53)

be the infinitesimal generator of transformation (2.23)-(2.24) and

$L=L\left(t,x^{k},\dot{x}^{k}\right)$ (2.54)

be a Lagrangian describing the dynamical system (2.51). The action of the
transformation (2.23)-(2.24) on (2.54)777For the application of Noether
theorem in higher order Lagrangians see [18] leaves the Euler Lagrange
equations (2.51) invariant, if and only if there exist a function
$f=f\left(t,x^{k}\right)$ such that the following condition holds

$X^{\left[1\right]}L+L\frac{d\xi}{dt}=\frac{df}{dt}$ (2.55)

where $X^{\left[1\right]}$ is the first prolongation of (2.53).

If the generator (2.53) satisfies (2.55), the generator (2.53) is a Noether
symmetry of the dynamical system described by the Lagrangian (2.54). Noether
symmetries form a Lie algebra called the Noether algebra. If the dynamical
system (2.51) admits Lie symmetries which span a Lie algebra $G_{m}$ of
dimension $m\succeq 1$ then the Noether symmetries of (2.51) form a Lie
algebra $G_{h}~{},~{}h\geq 0,$ which is a subalgebra of $G_{m},$
$G_{h}\subseteq G_{m}$.

###### Theorem 2.6.2

For any Noether point symmetry (2.53) of the Lagrangian (2.54) there
corresponds a function $\phi\left(t,x^{k},\dot{x}^{k}\right)$

$\phi=\xi\left(\dot{x}^{i}\frac{\partial
L}{\partial\dot{x}^{i}}-L\right)-\eta^{i}\frac{\partial L}{\partial x^{i}}+f$
(2.56)

which is a first integral i.e. $\frac{d\phi}{dt}=0$ of the equations of
motion. The function (2.56) is called a Noether integral (first integral) of
(2.51).

Since a Noether symmetry leaves the DEs (2.51) invariant, it is also a Lie
symmetry of (2.51) and from (2.26) we can say that (2.56) satisfies the
condition $X\left(\phi\right)=0$, that is, Noether integrals are invariant
functions of the Noether symmetry vector $X.~{}$

The existence and the number of Noether symmetries characterize a dynamical
system. If a dynamical system (2.51) on $n$ degrees of freedom admits (at
least) $n$ linear independent first integrals888Not necessarilly Noether point
integrals. $\phi_{J},~{}J=1...n,$ which are in involution, i.e.

$\left\\{\phi_{J},\phi_{K}\right\\}=0$ (2.57)

where $\left\\{~{},~{}\right\\}$ is the Poisson bracket, then the solution of
the dynamical system can be obtained by quadratures.

We calculate the Noether symmetries of the free particle and of the harmonic
oscillator.

###### Example 2.6.3

Find the Noether symmetries of the Lagrangian $L=\frac{1}{2}\dot{x}^{2}.$

Solution: The Lagrangian corresponds to the equation of motion of the free
particle $\ddot{x}=0$. In example 2.3.2, the Lie symmetries of the free
particle were calculated. The generic Lie symmetry vector is applied to
condition (2.55) and it is found that the generic Noether symmetry of the free
particle is

$X=\left(a_{1}+2a_{4}t+a_{5}t^{2}\right)\partial_{t}+\left(a_{2}+a_{3}t+a_{4}x+a_{5}tx\right)\partial_{x}.$
(2.58)

From the generic vector (2.58) and (2.56), it is found that the corresponding
Noether integral is

$\phi=a_{1}\dot{x}^{2}+a_{2}\dot{x}+a_{3}\left(t\dot{x}-x\right)+a_{4}\left(t\dot{x}^{2}-x\dot{x}\right)+a_{5}\left(t^{2}\dot{x}^{2}-2tx\dot{x}+x^{2}\right)$

###### Example 2.6.4

Find the Noether symmetries of the one dimensional harmonic oscillator with
Lagrangian

$L=\frac{1}{2}\dot{x}^{2}-\frac{1}{2}x^{2}.$ (2.59)

Solution: To derive the Noether symmetries of (2.59) we apply theorem 2.6.1.
Let $X=\xi\left(t,x\right)\partial_{t}+\eta\left(t,x\right)\partial_{x}$ be
the infinitesimal generator. The first prolongation $X^{\left[1\right]}~{}$is

$X^{\left[1\right]}=\xi\partial_{t}+\eta\partial_{x}+\eta_{\left[1\right]}\partial_{\dot{x}}.$

where
$\eta_{\left[1\right]}=\left[\eta_{,t}+\dot{x}\left(\eta_{,x}-\xi_{,t}\right)-\dot{x}^{2}\xi_{,x}\right].~{}$For
the terms of (2.55) we have

$\frac{df}{dt}=f_{,t}+\dot{x}f_{,x}.$
$X^{\left[1\right]}L=\dot{x}\eta_{,t}+\dot{x}^{2}\left(\eta_{,x}-\xi_{,t}\right)-\dot{x}^{3}\xi_{,x}-x\eta$
$L\frac{d\xi}{dt}=\frac{1}{2}\dot{x}^{2}\xi_{,t}-\frac{1}{2}x^{2}\xi_{,t}+\frac{1}{2}\dot{x}^{3}\xi_{,x}-\frac{1}{2}x^{2}\dot{x}\xi_{,x}.$

Replacing in (2.55) we find:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\left[x\eta+\frac{1}{2}x^{2}\xi_{,t}+f_{,t}\right]+\dot{x}\left[\eta_{,t}-\frac{1}{2}x^{2}\xi_{,x}-f_{,x}\right]$
$\displaystyle+\dot{x}^{2}\left[\eta_{,x}-\frac{1}{2}\xi_{,t}\right]+\dot{x}^{3}\left[\xi_{,x}\right]$

The determining equations are

$\xi_{,x}=0~{}~{},~{}~{}\eta_{,x}-\frac{1}{2}\xi_{,t}=0$ (2.60)
$\eta_{,t}-\frac{1}{2}x^{2}\xi_{,x}-f_{,x}=0~{}~{},~{}~{}x\eta+\frac{1}{2}x^{2}\xi_{,t}+f_{,t}=0.$
(2.61)

The solution of the system of equations (2.60)-(2.61) gives the generic
Noether symmetry [19]

$\displaystyle X$ $\displaystyle=$ $\displaystyle\left(a_{1}+a_{4}\cos
2t+a_{5}\sin 2t\right)\partial_{t}+$ (2.62) $\displaystyle+\left(a_{2}\sin
t+a_{3}\cos t-a_{4}x\sin 2t+a_{5}x\cos 2t\right)\partial_{x}$

with the corresponding gauge function

$f\left(t,x\right)=a_{2}x\cos t-a_{3}\sin t-a_{3}x^{2}\cos 2t-a_{5}x^{2}\sin
2t.$

It should be underlined that the free particle and the harmonic oscillator
have the same number of Noether symmetries. It can be easily noticed that the
Lie algebras (2.58) and (2.62) are the same Lie algebra in different
representations.

#####  2.6.2 Noether point symmetries of PDEs

In case of PDEs

$H\left(x^{i},u,u_{i},u_{ij}\right)=0$ (2.63)

which arise from a variational principle the following theorem holds.

###### Theorem 2.6.5

The action of the transformation (2.41)-(2.42) on the Lagrangian

$L_{P}=L_{P}\left(x^{k},u,u_{k}\right)$ (2.64)

leaves (2.63) invariant if there exists a vector field
$F^{i}=F^{i}\left(x^{i},u\right)$ such that

$X^{\left[1\right]}L_{P}+L_{P}D_{i}\xi^{i}=D_{i}F^{i}.$

The generator of the point transformation (2.41)-(2.42) is called Noether
symmetry. The corresponding Noether flow is

$\Phi^{i}=\xi^{k}\left(u_{k}\frac{\partial L}{\partial
u_{i}}-L\right)-\eta\frac{\partial L}{\partial u_{i}}+F^{i}$ (2.65)

and satisfies the condition $D_{i}\Phi^{i}.$

The implementation of Noether flows in PDEs differs from that in the ODEs.
Conservation flow (2.65) is used to reduce the order of (2.63) by defining a
new dependent variable $v\left(x^{i}\right)$. It has been proved that the
solution of the system

$v_{,i}\left(x^{k}\right)=\Phi_{i}\left(x^{k},u,u_{k}\right)$ (2.66)

is also a solution of (2.63). Furthermore, it is feasible new symmetries to
arise from the system (2.66) which are not symmetries of (2.63). These new
symmetries have been called potential symmetries [20].

####  2.7 Collineations of Riemannian manifolds

Previously, we studied the case when a function is invariant under a point
transformation. In the following subsections we consider the cases in which a
function is invariant under a point transformation. In order to do this we
shall need the concept of the Lie derivative. In particular, the geometric
objects which will be considered the metric tensor $g_{ij}$ and the connection
coefficients $\Gamma_{jk}^{i}$ in a Riemannian space.

###### Definition 2.7.1

Consider an n dimensional space $A_{n}$ of class $C^{p}$. An object is called
a geometric object of class $r~{}\left(r\leq p\right)$ if it has the following
properties .

i) In each coordinate system $\left\\{x^{i}\right\\},$ it has a well
determined set of components $\Omega^{a}\left(x^{k}\right)$.

ii) Under a coordinate transformation $x^{i^{\prime}}=J^{i}\left(x^{k}\right)$
the new components $\Omega^{a^{\prime}}$ of the object in the new coordinates
$\left\\{x^{i^{\prime}}\right\\}$ are represented as well determined functions
of class $r^{\prime}=p-r$ of the old components $\Omega^{a}$ in the old
coordinates $\left\\{x^{i}\right\\}$, of the functions $J^{i}$ and of their
sth derivatives $\left(1\leq s\leq p\right)$, that is, the new components
$\Omega^{i^{\prime}}$ of the object can be represented by equations of the
form

$\Omega^{a^{\prime}}=\Phi^{a}\left(\Omega^{k},x^{k},x^{k^{\prime}}\right).$
(2.67)

iii) The functions $\Phi^{a}$ have the group properties, that is, they satisfy
the following relations

$\Phi^{a}\left(\Phi\left(\Omega,x^{k},x^{k^{\prime}}\right),x^{k},x^{k^{\prime}}\right)=\Phi^{a}\left(\Omega,x^{k},x^{k^{\prime}}\right)$
(2.68)
$\Phi^{a}\left(F\left(\Omega,x^{k},x^{k^{\prime}}\right),x^{k^{\prime}},x^{k}\right)=\Omega^{a}$
(2.69)

The coordinate transformation law
$\Phi\left(\Omega,x^{k},x^{k^{\prime}}\right)$ characterizes the geometric
object. In case that the function $\Phi$ contains only $\Omega$ and the
partial derivatives of $J^{k}$ with respect to $x^{k},\,\ $the geometric
object is said to be a differential geometric object.

Furthermore, we say that a geometric object is linear if for the
transformation law $\Phi\left(\Omega,x^{k},x^{k^{\prime}}\right)$ it holds

$\Phi^{a}\left(\Omega,x^{k},x^{k^{\prime}}\right)=J_{b}^{a}\left(x^{k},x^{k^{\prime}}\right)\Omega^{b}+C\left(x^{k},x^{k^{\prime}}\right).$
(2.70)

When the transformation law is

$\Phi^{a}\left(\Omega,x^{k},x^{k^{\prime}}\right)=J_{b}^{a}\left(x^{k},x^{k^{\prime}}\right)\Omega^{b}$
(2.71)

we say that the geometric object is a linear homogeneous geometric object. One
important calss of linear homogeneous geometric objects are the tensors.

#####  2.7.1 Lie Derivative

Let $\Omega$ be a geometric object in $A_{n}$ with transformation law (2.67)
and consider the infinitesimal transformation

$\bar{x}^{i}=x^{i}+\varepsilon\xi^{i}\left(x^{k}\right).$ (2.72)

where $\xi=\xi^{i}\left(x^{k}\right)$ are the components of the infinitesimal
generator. From the transformation law (2.67), the geometric object in the
coordinate system $x^{i^{\prime}}=\bar{x}^{i}$ is
$\Phi\left(\Omega^{k},x^{k},x^{k^{\prime}}\right)$. The Lie derivative of the
geometric object $\Omega$ with respect to $\xi$ is defined as follows

$L_{X}\Omega=\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon}\left[\Phi\left(\Omega^{k},x^{k},x^{k^{\prime}}\right)-\Omega\right].$
(2.73)

In order for the Lie derivative of a geometric object to be again a geometric
object (not necessarily of the same type) it is necessary and sufficient that
the geometric object be linear [21].

By definition, the Lie derivative of a geometric object depends on the
transformation law. The transformation law (2.67) for functions is
$f^{\prime}\left(x^{i^{\prime}}\right)=f\left(x^{i^{\prime}}\right)$, hence,
under the point transformation (2.72) we have

$f^{\prime}\left(x^{i^{\prime}}\right)=f\left(x^{i}+\varepsilon\xi^{i}\right)=f\left(x^{i}\right)+\varepsilon
f_{,i}\xi^{i}+O\left(\varepsilon^{2}\right).$

From (2.73) the Lie derivative of functions $f$ along the vector field $\xi$
is computed to be

$L_{\xi}f=f_{,i}\xi^{i}=\xi\left(f\right).$

The transformation law for a tensor field $T$ of rank $\left(r,s\right)$ is

$T_{~{}~{}j_{i}^{\prime}...j_{s}^{\prime}}^{i_{1}^{\prime}...i_{r}^{\prime}}=J_{i_{1}}^{i_{1}^{\prime}}...J_{i_{r}}^{i_{r}^{\prime}}J_{j_{1}^{\prime}}^{j_{1}}...J_{j_{s}}^{j_{s}^{\prime}}T_{~{}~{}j_{i}...j_{s}}^{i_{1}...i_{r}}$

where $J$ is the Jacobian of the transformation. Thus, from (2.73), the Lie
derivative of $T$ with respect to $\xi$ is

$\displaystyle L_{\xi}T_{~{}~{}j_{i}...j_{s}}^{i_{1}...i_{r}}$
$\displaystyle=$
$\displaystyle\xi^{k}T_{~{}~{}j_{i}...j_{s,k}}^{i_{1}...i_{r}}-T_{~{}~{}j_{i}...j_{s}}^{m...i_{r}}\xi_{,m}^{i_{1}}-T_{~{}~{}j_{i}...j_{s}}^{i_{1}m...i_{r}}\xi_{m}^{i_{2}}+...$
(2.74)
$\displaystyle...+T_{~{}~{}m...j_{s}}^{i_{1}...i_{r}}\xi_{,j_{1}}^{m}+T_{~{}~{}j_{i}m...j_{s}}^{i_{1}...i_{r}}\xi_{j_{2}}^{m}+....$

In case of vector fields $X$ and 1-forms $\omega$ expression (2.74) gives

$L_{\xi}X=\xi^{k}X_{,k}^{i}-X^{k}\xi_{,k}^{i}=\left[\xi,X\right]$ (2.75)
$L_{\xi}\omega=\xi^{k}\omega_{i,k}-\omega_{k}\xi_{,i}^{k}.$ (2.76)

For a second order tensor $T_{ij}$, expression (2.74) becomes

$L_{\xi}T_{ij}=T_{ij,k}\xi^{k}+T_{ik}\xi_{,j}^{k}+T_{kj}\xi_{,i}^{k}.$ (2.77)

The connection coefficients $\Gamma_{jk}^{i}$ are linear differential
geometric objects with transformation law

$\Gamma_{j^{\prime}k^{\prime}}^{i^{\prime}}=J_{i}^{i^{\prime}}J_{j^{\prime}}^{j}J_{k^{\prime}}^{k}\Gamma_{jk}^{i}+\frac{\partial
x^{i^{\prime}}}{\partial x^{r}}\frac{\partial^{2}x^{r}}{\partial
x^{j^{\prime}}\partial x^{k^{\prime}}}.$ (2.78)

Connection coefficients have different transformation law from tensor fields,
that is, the Lie derivative $L_{\xi}\Gamma_{jk}^{i}$ will be different from
that of tensors. Applying (2.73),we find that the Lie derivative
$L_{\xi}\Gamma_{jk}^{i}$ is expressed as follows

$L_{\xi}\Gamma_{jk}^{i}=\xi_{,jk}^{i}+\Gamma_{jk,r}^{i}\xi^{r}-\xi_{,r}^{i}\Gamma_{jk}^{r}+\xi_{,j}^{s}\Gamma_{sk}^{i}+\xi_{,k}^{s}\Gamma_{js}^{i}.$
(2.79)

In case of symmetric connection $\Gamma_{jk}^{i}=\Gamma_{kj}^{i},$ condition
(2.79) can be written in the equivalent form

$L_{\xi}\Gamma_{jk}^{i}=\xi_{;jk}^{i}-R_{jkl}^{i}\xi^{l}$ (2.80)

where $R_{jkl}^{i}$ is the curvature tensor and the semicolon $";"$ means
covariant derivative.

###### Collineations

In section 2.2.2 we gave the conditions under which a function is invariant
under a one parameter point transformation. Similarly for linear (homogeneous)
differential geometric objects there is the following definition.

###### Definition 2.7.2

A linear differential geometric object $~{}\Omega\left(x^{i}\right)$ is
invariant under a one parameter point transformation

$\bar{x}^{i}=\bar{x}^{i}\left(x^{k},\varepsilon\right)$ (2.81)

if and only if
$~{}\bar{\Omega}\left(\bar{x}^{i}\right)=\Omega\left(x^{i}\right)$ at all
points where the one parameter point transformation acts. Equivalently, the
Lie derivative of the geometric object $\Omega$ with respect to the
infinitesimal generator of (2.81) vanishes, that is, $L_{\xi}\Omega=0.$

A direct result which arises from the definition of the Lie derivative and the
transformation law of linear differential geometric objects is that if a
linear differential geometric object $\Omega$ is invariant under the
transformation (2.81), then there exist a coordinate system with respect to
which the components of $\Omega$ are independent of one of the coordinates.

One can generalize the concept of symmetry in the sense that one does not
require the Lie derivative to be equal to zero, but to another tensor. That
is, the Lie derivative of the linear differential geometric object $\Omega$
with respect to the infinitesimal generator $\xi$ is

$L_{\xi}\Omega=\Psi$

where $\Psi$ has the same numbers of components and symmetries of the indices
with $\Omega.$ In this case the infinitesimal generator $\xi$ is said to be a
collineation of $\Omega,$ the type of collineations being defined by $\Psi$.
Collineations are a powerful tool in the study of the geometric properties of
Riemannian manifolds.

In Riemannian geometry, the geometric object $\Omega$ can be the metric tensor
$g_{ij}$ or any other geometric object defined from it (e.g. the connection
coefficients).

###### Definition 2.7.3

All collineations involving geometric objects $\Omega$ derived from the metric
$g_{ij}$ of a Riemannian space shall be called geometric collineations. In
particular, the collineation defined by the metric $L_{\xi}g_{ij}$ is called
the generic collineation because any other geometric collineation can be
written in terms of it. Furthermore, the geometric collineations can be
written in terms of the irreducible parts $\psi,H_{ij}$ as follows

$L_{\xi}g_{ij}=2\psi g_{ij}+2H_{ij}$ (2.82)

where the function $\psi$ is called the conformal factor and $H_{ij}$ is a
symmetric traceless tensor.

The role of the quantities $\psi,~{}H_{ij}$ is important because they can be
used as the variables in terms of which one can study any geometric
collineation. To do that, one has to express the Lie derivative of any metric
tensor in terms of the generic symmetry variables $\psi,~{}H_{ij}$ and their
derivatives.

In the following we shall be interested in geometric collineations,
particularly in the collineations of the metric and of the connection
coefficients of a Riemannian space.

#####  2.7.2 Motions of Riemannian spaces

Consider an n dimensional Riemannian space $V^{n}$ with line element

$ds^{2}=g_{ij}dx^{i}dx^{j}.$ (2.83)

where $g_{ij}$ is the metric tensor.

###### Definition 2.7.4

The point transformation (2.81) is called a motion of $V^{n},$ if and only if
the line element is invariant under the action of (2.81). Equivalently, the
Lie derivative of the metric tensor $g_{ij}$ with respect to the infinitesimal
generator $\xi$ of (2.81) vanishes, i.e.

$L_{\xi}g_{ij}=0.$ (2.84)

The point transformations (2.81) which are motions of $V^{n}$ form a group
named the group of motions. Since $g_{ij}$ is a metric, condition (2.84) can
be written in the equivalent form

$L_{\xi}g_{ij}=2\xi_{\left(i;j\right)}=0.$ (2.85)

This equation is known as Killing equation and $\xi$ is called an isometry or
Killing Vector (KV). The KVs of a metric form a Lie algebra, which is called
the Killing algebra.

Motions are important in physics. For instance, the Euclidian space admits as
motions the group of translations and the group of rotations
$T\left(3\right)\otimes so\left(3\right)$ and this implies the conservation of
linear and angular momentum respectively. As a second example in Cosmology the
assumption that the universe is isotropic and homogeneous about all points
leads to the Friedmann Robertson Walker (FRW) spacetime.

The maximum dimension of the Killing algebra that $V^{n}$ can admit is given
in the following theorem.

###### Theorem 2.7.5

If an n dimensional Riemannian space $V^{n}$ admits a Killing algebra
$G_{KV},$ then, $0\leq\dim G_{KV}\leq\frac{1}{2}n\left(n+1\right).$

A Riemannian space which admits a Killing algebra of dimension
$\frac{1}{2}n\left(n+1\right)$ is called a maximally symmetric space. For
example, the Euclidian space $E^{3}$ and the Minkowski spacetime $M^{4}~{}$are
maximally symmetric spaces.

A special class of KVs are the gradient KVs. A KV $\xi$ is called gradient iff
$\xi_{i;j}=0$, that is,

$\xi_{\left(i;j\right)}=0~{}~{}\text{and~{}~{}}\xi_{\left[i;j\right]}=0.$

For every gradient KV $\xi,~{}$there exists a function $S$ so
$S_{,k}g^{ik}=\xi^{i}$ and $S_{;ij}=0.$

###### Theorem 2.7.6

If $V^{n}$ admits $p~{}$gradient KVs $\left(\text{where~{}}p\leq n\right)$
then, $V^{n}$ is called a $p~{}$decomposable space and in this case there
exists a coordinate system in which the line element (2.83) can be written in
the form

$ds^{2}=M_{\alpha\beta}dz^{\alpha}dz^{\beta}+h^{AB}\left(y^{A}\right)dx_{A}dx_{B}$

where $\alpha,\beta=1,2..,p$, $A,B=p+1,...n$ and
$M_{a\beta}=diag\left(c_{1},c_{2},...,c_{p}\right)$, $c_{1},c_{2},...,c_{p}$
are constants.

###### Example 2.7.7

Compute the KVs of the Euclidian sphere with line element

$ds^{2}=d\phi^{2}+\sin\phi^{2}d\theta^{2}.$ (2.86)

Solution: In order to find the KVs we solve the Killing equation (2.85). This
leads to the system of equations

$\xi_{,\theta}^{\theta}=0$
$2\xi_{,\phi}^{\phi}+2\xi^{\theta}\sin\theta\cos\theta=0$
$\xi_{,\phi}^{\theta}+\xi_{,\theta}^{\phi}-2\xi^{\phi}\cot\theta=0.$

whose solution are the elements of the $so\left(3\right)$ Lie algebra (see
example 2.2.8).

The 2D Euclidian sphere (2.86) admits a three dimensional Killing algebra,
hence, it is a maximally symmetric space. Moreover, all spaces of constant
curvature are maximally symmetric spaces.

###### Conformal motions

When the point transformation (2.81) does not change the angle between two
directions at a point, the transformation (2.81) is called a conformal motion.
Technically we have the following definition.

###### Definition 2.7.8

The infinitesimal generator $\xi$ of the point transformation (2.81) is called
Conformal Killing Vector (CKV) if the Lie derivative of the metric $g_{ij}$
with respect to $\xi$ is a multiple of $g_{ij}$. That is if,

$L_{\xi}g_{ij}=2\psi g_{ij}$ (2.87)

where $\psi=\frac{1}{n}\xi_{;k}^{k}$. In the case where $\psi_{;ij}=0,$ $\xi$
is a special CKV (sp.CKV)999For the conformal factor of a sp.CKV holds
$\psi_{;ij}=0,$ that is, $\psi_{,i}$ is a gradient KV. A Riemannian space
admits a sp.CKV if and only if admits a gradient KV and a gradient HV [22].
and if $\psi=$constant, $\xi$ is a Homothetic Killing Vector (HV).

The CKVs of a metric form a Lie algebra, which is called the conformal
algebra, $G_{CV}$. Obviously the KVs and the homothetic vector are elements of
the conformal algebra $G_{CV}$. If $G_{HV}$ is the algebra of HVs (including
the algebra $G_{KV}$ of KVs), then the following theorem applies.

###### Theorem 2.7.9

Let $V^{n}$ be an n dimensional Riemannian space, $n\geq 2,$ which admits a
conformal algebra $G_{CK}$, a homothetic algebra $G_{HV}$ and a Killing
algebra $G_{KV},$ then

i) $G_{KV}\subseteq G_{HV}\subseteq G_{CV}$.

ii) for arbitrary $n$, $G_{H-K}=G_{HV}-G_{HV}\cap G_{KV},$ then $0\leq\dim
G_{H-K}\leq 1$; that is, $V^{n}$ admits at most one HV.

iii) $V^{2}$ admits an infinite dimentional conformal algebra $G_{CV}$,

iv) for $n>2$, $0\leq\dim
G_{CV}\leq\frac{1}{2}\left(n+1\right)\left(n+2\right)\,.$

In the following by the term _proper conformal algebra_ we mean the algebra of
CKVs which are not HV/KVs. A generalization of theorem 2.7.6 for spaces which
admit a gradient CKV is the following.

###### Theorem 2.7.10

If $V^{n}$ admits a gradient CKV then there exists a coordinate system in
which the line element can be written as follows

$ds^{2}=f^{2}\left(x^{n}\right)\left[dx^{n}+h^{AB}\left(x^{A}\right)dx^{A}dx^{B}\right]$

where $A,B=1,2,...n-1$. In these coordinates the CKV is
$X_{C}=\partial_{x^{n}}$ with conformal factor $\psi=\frac{f_{,x^{n}}}{f}$. In
the case in which $f\left(x^{n}\right)=\exp\left(x^{n}\right),$ then $\psi=1$,
hence, $X_{C}$ becomes HV and if $f_{,x^{n}}=0$, $X_{C}$ becomes KV.

Two metrics $g_{ij},~{}\bar{g}_{ij}$ are conformally related if
$\bar{g}_{ij}=N^{2}g_{ij}$ where the function $N^{2}$ is the conformal factor.
If $\xi$ is a CKV of the metric $\bar{g}_{ij}$ so that
$L_{\xi}\bar{g}_{ij}=2\bar{\psi}\bar{g}_{ij},$ then $\xi$ is also a CKV of the
metric $g_{ij}$, that is $L_{\xi}g_{ij}=2\psi g_{ij}$ with conformal factor
$\psi\left(x^{k}\right)$

$\psi=\bar{\psi}N^{2}-NN_{,i}\xi^{i}.$ (2.88)

The last relation implies that two conformally related metrics have the same
conformal algebra, but with different subalgebras; that is, a KV for one may
be proper CKV for the other. This is an important observation which shall be
useful later. A special class of conformally related metrics are the
conformally flat metrics. A space $V^{n}$ is conformally flat if the metric
$g_{ij}$ of $V^{n}$ satisfies the relation $g_{ij}=N^{2}s_{ij}~{}$where
$s_{ij}\,$ is the metric of a flat space which has the same signature with
$g_{ij}$.

For conformally flat spaces the following proposition aplies.

###### Proposition 2.7.11

Let $V^{n}$ be an n-dimensional Riemannian space, $n\geq 2,$

i) If $V^{n},~{}n>2$ is conformally flat then $V^{n}$ admits a conformal
algebra of dimension $\frac{1}{2}\left(n+1\right)\left(n+2\right)~{}$.

ii) A three dimensional space is conformally flat if the Cotton-York tensor

$C^{ij}=2\varepsilon^{ikr}\left(R_{k}^{j}-\frac{1}{4}\delta_{k}^{j}\right)_{;r}\newline
$

vanishes.

iii) $V^{n},$ $n\geq 4\ $ is conformally flat if the Weyl tensor

$R_{ijkr}=C_{ijkr}+\frac{2}{n-2}\left(g_{i[k}R_{r]j}-g_{j[k}R_{r]i}\right)-\frac{2}{\left(n-1\right)\left(n-2\right)}Rg_{i[k}g_{r]j}$

vanishes.

iv) If $V^{n},$ $n>3$ is a maximally symmetric space then $V^{n}$ is
conformally flat.

v) All two dimensional spaces are conformally flat .

A result which will be used in subsequent sections is the following.

######  Example 2.7.12 (The conformal algebra of the flat space.)

Consider a flat space of dimension $n>2~{}$ with metric

$ds^{2}=\varepsilon dt^{2}+\delta_{AB}dy^{A}dy^{B}~{},~{}\varepsilon=\pm 1.$

The conformal algebra of the space consists of the following vectors
$n-$ gradient KVs,

$K_{G}^{1}=\partial_{t}~{},~{}K_{G}^{A}=\partial_{A}\newline $

$\frac{n\left(n-1\right)}{2}~{}$ non gradient KVs (rotations).

$X_{R}^{1A}=y^{A}\partial_{t}-\varepsilon t\partial_{A}$
$~{}X_{R}^{AB}=y^{B}\partial_{A}-y^{A}\partial_{B}$

one gradient HV

$H=t\partial_{t}+\sum\limits_{A}y^{A}\partial_{A}$

$n~{}$special CKVs

$X_{C}^{1}=\frac{1}{2}\left(t^{2}-\varepsilon\sum\limits_{A}\left(y^{A}\right)^{2}\right)\partial_{t}+t\sum\limits_{A}y^{A}\partial_{A}$
$X_{C}^{A}=ty^{A}\partial_{t}+\frac{1}{2}\left(\varepsilon
t^{2}+\left(y^{A}\right)^{2}-\sum\limits_{B\neq
A}\left(y^{B}\right)^{2}\right)\partial_{A}+y^{A}\sum\limits_{B\neq
A}y^{B}\partial_{B}$

where $y^{A}=1...n-1~{}$with conformal factor $\psi_{C}^{1}=t~{}$and
$\psi_{C}^{A}=y^{A}.$

For $n>2$ the flat space does not admit proper (non special) CKVs.

For $n=2$ the vector field

$X=\left[f_{1}\left(t+i\sqrt{\varepsilon}x\right)-f_{2}\left(t-i\sqrt{\varepsilon}x\right)+c_{0}\right]\partial_{t}+i\sqrt{\varepsilon}\left[f_{1}\left(t+i\sqrt{\varepsilon}x\right)+f_{2}\left(t-i\sqrt{\varepsilon}x\right)\right]\partial_{x}$

is the generic CKV, that is, it includes the KVs, the HV, and the CKVs.

#####  2.7.3 Symmetries of the connection

Let $\xi$ be the generator of an infinitesimal transformation of (2.81). In a
Riemannian space with metric $g_{ij}$ we have the identity

$L_{\xi}\Gamma_{jk}^{i}=g^{ir}\left[\left(L_{\xi}g_{rk}\right)_{;j}+\left(L_{\xi}g_{rj}\right)_{;k}-\left(L_{\xi}g_{jk}\right)_{;r}\right].$
(2.89)

If $\xi$ is a HV or KV then from (2.89) follows that $L_{\xi}\Gamma_{jk}^{i}$
vanishes, which implies that the $\Gamma_{jk}^{i}$ are invariant under the
action of transformation (2.81).

###### Definition 2.7.13

The infinitesimal generator $\xi$ of the point transformation (2.81) caries a
geodesic into a geodesic and also preserves the affine parameter iff the Lie
derivative of connection coefficients $\Gamma_{jk}^{i}$ with respect to $\xi$
vanishes, that is iff

$L_{\xi}\Gamma_{jk}^{i}=0$ (2.90)

The infinitesimal generator $\xi$ is called an Affine Killing vector or Affine
collineation (AC).

ACs of $V^{n}$ form a Lie algebra, which is called Affine algebra, $G_{AC}$.
Obviously the homothetic algebra101010Note that the proper CKVs do not satisfy
condition (2.90). $G_{HV},~{}$is a subalgebra of $G_{AC}$, $G_{HV}\subseteq
G_{AC}.~{}$We shall say that a spacetime admits proper ACs when $\dim
G_{HV}\prec\dim G_{AC}$.

For instance, in the case of flat space condition (2.90) becomes
$\xi_{,jk}^{i}=0,~{}$therefore, the general solution is
$\xi^{i}=A_{j}^{i}x^{j}+B^{i}~{}$where $A_{j}^{i},B^{i}$ are
$n\left(n+1\right)~{}$ constants. Therefore the flat space admits a
$n\left(n+1\right)$ dimensional Affine algebra (including the KVs and the HV).
We have the inverse result.

###### Theorem 2.7.14

If an n dimensional Riemannian space $V^{n}$ admits an Affine algebra $G_{AC}$
and $\dim G_{AC}=n\left(n+1\right)$, then $V^{n}$ is a flat space.

A generalization of affine symmetry which is of interest is the Projective
collineation.

###### Definition 2.7.15

The infinitesimal generator $\xi$ of the point transformation (2.81) is called
a Projective Collineation (PC) if there exists a function111111In general,
$\xi$ is PC if there exists a one form $\omega_{i}$ such
that$~{}L_{\xi}\Gamma_{jk}^{i}=2\omega_{(j}\delta_{k)}^{i}.$ In a Riemannian
space, the one form $\omega_{i}$ is necessarily closed, that is, there exist a
function $\phi$ such as $\omega_{i}=\phi_{,i}.$ $\phi\left(x^{k}\right)$ such
that

$L_{\xi}\Gamma_{jk}^{i}=\phi_{,j}\delta_{k}^{i}+\phi_{,k}\delta_{j}^{i}$
(2.91)

or equivalently

$\xi_{\left(i;j\right);k}=2g_{ij}\phi_{,k}+2g_{k(i}\phi_{,j).}$

The function $\phi$ is called the projective function. If the projective
function satisfies the condition $\phi_{;ij}=0,$ then we say that $\xi~{}$is a
special PC (sp.PC). Projective transformations transform the system of
geodesics (auto parallel curves) of $V^{n}$ into the same system but they do
not preserve the affine parameter.

The PCs of $V^{n}$ form a Lie algebra which is called Projective algebra,
$G_{PC}$. The affine algebra $G_{AC}$ is a subalgebra of $G_{PC}$,
$G_{AC}\subseteq G_{PC}.$

In flat space condition (2.91) gives the generic projective collineation

$\xi_{i}=A_{ij}x^{j}+\left(B_{j}x^{j}\right)x_{i}+C_{i}$

where $A_{ij},B_{j}$,$C_{i}$ are $n\left(n+2\right)$ constants.

###### Theorem 2.7.16

If an n dimensional Riemannian space $V^{n}~{}$admits a projective algebra
$G_{PC},$ then$~{}\dim G_{PC}\leq n(n+2$). In case $\dim
G_{PC}=n\left(n+2\right)$ then $V^{n}$ is a maximally symmetric space [23].

Some useful propositions for the existence of sp. PCs are the following [23,
24, 25].

###### Proposition 2.7.17

Let $V^{n}$ be an n dimensional Riemannian space, then

i) If $V^{n}$ admits a $p\leq n$ dimensional Lie algebra of sp.PCs then also
admits $p$ gradient KVs and a gradient HV and if $p=n$ the space is flat (the
reverse also holds true).

ii) A maximally symmetric space which admits a proper AC or a sp.PC is a flat
space.

A Riemannian space is possible to admit more general collineations, e.g.
Curvature collineations. A full classification of the collineations of a
Riemannian space (with definite or indefinite metric) can be found in [26]. A
summary of the above definitiosn is given in Table 2.1.

Table 2.1: Collineations of a Riemannian space  Collineation $\mathcal{L}_{\xi}\mathbf{A}=\mathbf{B}$ |  $\mathbf{A}$ | $\mathbf{B}$
---|---|---
Killing Vector (KV) | $g_{ij}$ | $0$
Homothetic vector (HV) | $g_{ij}$ | $2\psi g_{ij},~{}\psi_{,i}=0$
Conformal Killing vector (CKV) | $g_{ij}$ | $2\psi g_{ij},~{}\psi,_{i}\neq 0$
Affine Collineation (AC) | $\Gamma_{jk}^{i}$ | $0$
Proj. Collineations (PC) | $\Gamma_{jk}^{i}$ |  $2\phi_{(,j}\delta_{k)}^{i},$ $\phi,_{i}\neq 0$
Sp. Proj. collineation (sp.PC) | $\Gamma_{jk}^{i}$ |  $2\phi_{(,j}\delta_{k)}^{i},$ $\phi_{;jk}=0$

We note that the Lie symmetries $\left(sl\left(3,R\right)\right)$ of the free
particle form the projective algebra of the two dimensional Euclidian space.
Therefore, the natural question which arises is the following:

Is there any connection between the Lie symmetries of differential equations
of second order and collineations?

In the following chapters this will be confirmed and will be used to apply the
Lie symmetries of DEs in various areas of Geometry and Physics.

##  Part II Symmetries of ODEs

###  Chapter 3 Lie symmetries of geodesic equations

####  3.1 Introduction

In a Riemannian space the affinely parameterized geodesics are determined
uniquely by the metric. Therefore one should expect a close relation between
the geodesics as a set of homogeneous ordinary differential equations (ODEs)
linear in the highest order term and quadratic non-linear in first order
terms, and the metric as a second order symmetric tensor. A system of such
ODEs is characterized (perhaps not fully) by its Lie symmetries and a metric
by its collineations. Therefore it is reasonable to expect that the Lie
symmetries of the system of geodesics of a metric will be closely related with
the collineations of the metric. That such a relation exists it is easy to see
by the following simple example. Consider on the Euclidian plane a family of
straight lines parallel to the $x-$axis. These curves can be considered either
as the integral curves of the ODE $\frac{d^{2}y}{dx^{2}}=0$ or as the
geodesics of the Euclidian metric $dx^{2}+dy^{2}$. Subsequently consider a
symmetry operation defined by a reshuffling of these lines without preserving
necessarily their parametrization. According to the first interpretation this
symmetry operation is a Lie symmetry of the ODE $\ddot{y}=0$ and according to
the second interpretation it is a (special) projective symmetry of the
Euclidian two dimensional metric.

What has been said for a Riemannian space can be generalized to a space in
which there is only a linear connection. In this case the geodesics are called
autoparallels (or paths) and they comprise again a system of ODEs linear in
the highest order term and quadratic non-linear in the first order terms. In
this case one is looking for relations between the Lie symmetries of the
autoparallels and the projective or affine collineations of the connection.

The above matters have been discussed extensively in a series of interesting
papers. Classic is the work of Katzin and Levin [27, 28, 29]. Important
contributions have also been done by Aminova [30, 31, 32, 33, 34], Prince and
Crampin [35] and many others. More recent is the work of Feroze et al [36]. In
[36] they have considered the KVs of the metric and their relation to the Lie
symmetries of the system of affinely parameterized geodesics of maximally
symmetric spaces of low dimension. In the same paper a conjecture is made,
which essentially says that the maximally symmetric spaces of non-vanishing
curvature do not admit further Lie symmetries.

In the following we consider the set of autoparallels - not necessarily
affinely parameterized - of a symmetric connection. We find that the major
symmetry condition relates the Lie symmetries with the special projective
symmetries of the connection. A similar result has been obtained by Prince and
Crampin in [35] using the bundle formulation of second order ODEs.

Furthermore, because the geodesic equations follow from the variation of the
geodesic Lagrangian defined by the metric and due to the fact that the Noether
symmetries are a subgroup of the Lie group of Lie symmetries of these
equations, one should expect a relation / identification of the Noether
symmetries of this Lagrangian with the projective collineations of the metric
or with its degenerates. Recent work in this direction has been by Bokhari et
all [37, 38] in which the relation of the Noether symmetries with the KVs of
some special spacetimes is discussed.

In section 3.2 we derive the Lie symmetry conditions for a general system of
second order ODE polynomial in the first derivatives. In section 3.4 we apply
these conditions in the special case of Riemannian spaces and in Theorem 3.4.1
we give the Lie symmetry vectors in terms of the special projective
collineations of the metric and their degenerates. In section 3.4.1 we give
the second result of this work, that is Theorem 3.4.2, which relates the
Noether symmetries of the geodesic Lagrangian defined by the metric with the
homothetic algebra of the metric and comment on the results obtained so far in
the literature. Finally in section 3.5 we apply the results to various cases
and eventually we give the Lie symmetries, the Noether symmetries and the
associated conserved quantities of Einstein spaces, the Gödel spacetime, the
Taub spacetime and the Friedman Robertson Walker spacetimes.

####  3.2 The Lie symmetry conditions in an affine space

We consider the system of ODEs:

$\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x}^{k}+\sum\limits_{m=0}^{n}P_{j_{1}...j_{m}}^{i}\dot{x}^{j_{1}}\ldots\dot{x}^{j_{m}}=0$
(3.1)

where $\Gamma_{jk}^{i}$ are the connection coefficients of the space and
$P_{j_{1}...j_{m}}^{i}(t,x^{i})$ are smooth functions completely symmetric in
the lower indices and derive the Lie point symmetry conditions in geometric
form using the standard approach. Equation (3.1) is quite general and covers
most of the standard cases autonomous and non autonomous equations. For
instance for all $\mathbf{P}=0$ equation (3.1) becomes

$\ddot{x}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x}^{k}=0$ (3.2)

which are the geodesic equations. In case $P^{i}\mathbf{=}F^{i}$ and
$P_{j_{1}...j_{m}}^{i}=0$ equation (3.1) becomes

$\ddot{x}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x}^{k}+F^{i}\left(t,x^{i}\right)=0$
(3.3)

which are the equations of motions of a particle in a curved space under the
action of the force $\mathbf{F}$. Furthermore because the $\Gamma_{jk}^{i}$’s
are not assumed to be symmetric, the results are valid in a space with
torsion. Obviously they hold in a Riemannian space provided the connection
coefficients are given in terms of the Christofell symbols.

Following, the standard procedure (see e.g. [1, 3]) we find that the Lie
symmetry conditions for the values of $m\leq 4$ are111The detailed
calculations are given in Appendix 3.A.

$\displaystyle L_{\eta}P^{i}+2\xi,_{t}P^{i}+\xi
P^{i},_{t}+\eta^{i},_{tt}+\eta^{j},_{t}P_{.j}^{i}$ $\displaystyle=0$ (3.4)
$\displaystyle L_{\eta}P_{j}^{i}+\xi,_{t}P_{j}^{i}+\xi
P_{j}^{i},_{t}+\left(\xi,_{k}\delta_{j}^{i}+2\xi,_{j}\delta_{k}^{i}\right)P^{k}+2\eta^{i},_{t|j}-\xi,_{tt}\delta_{k}^{i}+2\eta^{k},_{t}P_{.jk}^{i}$
$\displaystyle=0$ (3.5) $\displaystyle
L_{\eta}P_{jk}^{i}+L_{\eta}\Gamma_{jk}^{i}+\left(\xi,_{d}\delta_{(k}^{i}+\xi,_{(k}\delta_{|d|}^{i}\right)P_{.j)}^{d}+\xi
P_{.kj,t}^{i}-2\xi,_{t(j}\delta_{k)}^{i}+3\eta^{d},_{t}P_{.dkj}^{i}$
$\displaystyle=0$ (3.6) $\displaystyle
L_{\eta}P_{.jkd}^{i}-\xi,_{t}P_{.jkd}^{i}+\xi,_{e}\delta_{(k}^{i}P_{.dj)}^{e}+\xi
P_{.jkd,t}^{i}+4\eta^{e},_{t}P_{.jkde}^{i}-\xi_{(,j|k}\delta_{d)}^{i}$
$\displaystyle=0~{}$ (3.7)

and for $\left(\dot{x}\right)^{l},l~{}\geq 4$

$\displaystyle
L_{\eta}P_{j_{1}...j_{m}}^{i}+P_{j_{1}...j_{m}~{},t}^{i}\xi+\left(2-m\right)\xi_{,t}P_{j_{1}...j_{m}}^{i}+$
$\displaystyle+\xi_{,r}\left(2-\left(m-1\right)\right)P_{j_{1}...j_{m-1}}^{i}\delta_{j_{m}}^{r}+\left(m+1\right)P_{j_{1}...j_{m+1}}^{i}\eta_{,t}^{j_{m+1}}+\xi_{,j}P_{j_{1}...j_{m-1}}^{j}\delta_{j_{m}}^{i}=0.$
(3.8)

From the above general relations it is possible to extract the Lie symmetry
conditions for the various values of functions $\mathbf{P}$. For example the
Lie symmetry conditions for the geodesic equations (3.2) are as follows.

$\displaystyle\eta^{i},_{tt}$ $\displaystyle=0$ (3.9) $\displaystyle
2\eta^{i},_{t|j}-\xi,_{tt}\delta_{k}^{i}$ $\displaystyle=0$ (3.10)
$\displaystyle L_{\eta}\Gamma_{jk}^{i}-2\xi,_{t(j}\delta_{k)}^{i}$
$\displaystyle=0$ (3.11) $\displaystyle\xi_{(,j|k}\delta_{d)}^{i}$
$\displaystyle=0.$ (3.12)

Moreover, the symmetry conditions for the system (3.3) are

$\displaystyle L_{\eta}F^{i}+2\xi,_{t}F^{i}+\xi F^{i},_{t}+\eta^{i},_{tt}$
$\displaystyle=0$ (3.13)
$\displaystyle\left(\xi,_{k}\delta_{j}^{i}+2\xi,_{j}\delta_{k}^{i}\right)F^{k}+2\eta^{i},_{t|j}-\xi,_{tt}\delta_{k}^{i}$
$\displaystyle=0$ (3.14) $\displaystyle
L_{\eta}\Gamma_{jk}^{i}-2\xi,_{t(j}\delta_{k)}^{i}$ $\displaystyle=0$ (3.15)
$\displaystyle\xi_{(,j|k}\delta_{d)}^{i}$ $\displaystyle=0.$ (3.16)

In the same manner we work for more terms of $\mathbf{P}.$ We note the
appearance of the term $L_{\eta}\Gamma_{jk}^{i}$ in these expressions.

####  3.3 Lie symmetries of autoparallel equation

Consider a $C^{p}$ manifold $M$ of dimension $n$, endowed with a
$\Gamma_{jk}^{i}$ symmetric222The coefficients $\Gamma_{jk}^{i}$ in general
are not symmetric in the lower indices. In the autoparallel equation (3.17)
the antisymmetric part of $\Gamma_{[jk]}^{i}$ (the torsion) does not play a
role. connection. In a local coordinate system $\\{x^{i}:i=1,\ldots,n\\}$ the
connection $\Gamma_{jk}^{i}\partial_{i}=\nabla_{j}\partial_{k}$ and the
autoparallels of the connection are defined by the requirement

$\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x}^{k}+\phi\left(t\right)\dot{x}^{i}=0$
(3.17)

where $t$ is a parameter along the paths. When $\phi\left(t\right)$ vanishes,
we say that the autoparallels are affinely parameterized and in this case $t$
is called an affine parameter, that is one has (3.2). Consider the
infinitesimal transformation

$\bar{t}=t+\varepsilon\xi\left(t,x^{k}\right)~{},~{}~{}\bar{x}^{i}=x^{i}+\varepsilon\eta^{i}\left(t,x^{k}\right)$
(3.18)

with infinitesimal generator

$X=\xi\left(t,x^{k}\right)\partial_{t}+\eta^{i}\left(t,x^{k}\right)\partial_{t}.$
(3.19)

The autoparallels (3.17) are invariant under transformation (3.18) if

$X^{\left[2\right]}\left(\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x}^{k}+\phi\left(t\right)\dot{x}^{i}\right)=0\;$
(3.20)

where $X^{\left[2\right]}$ is the second prolongation of (3.19). For
$P_{j_{1}...j_{m}}^{i}=0~{}$for $m\neq 0$ and
$P_{j_{1}}^{i}=\phi\left(t\right),~{}~{}\Gamma_{jk}^{i}\left(x^{k}\right)+P_{jk}^{i}\left(t,x^{k}\right)=\Gamma_{jk}^{i}\left(t,x^{k}\right)~{}$and
from conditions (3.4)-(3.8) we have the Lie symmetry conditions for the
autoparallel equations (3.17)

$\eta_{,tt}^{i}+\eta_{,t}^{i}\phi=0$ (3.21)
$\xi_{,tt}\delta_{j}^{i}-\xi\phi_{,t}\delta_{j}^{i}-2\left[\eta_{,tj}^{i}+\eta_{,t}^{k}\Gamma_{(kj)}^{i}\right]-\phi\xi_{,t}\delta_{j}^{i}=0$
(3.22)

$\mathcal{L}_{\mathbf{\eta}}\Gamma_{(jk)}^{i}=-2\phi\xi_{(,j}\delta_{k)}^{i}+\xi\Gamma_{(kj),t}^{i}+2\xi_{,t(j}\delta_{k)}^{i}$
(3.23) $\xi_{(,j|k}\delta_{d)}^{i}=0\;.$ (3.24)

Define the quantity

$\Phi=\xi_{,t}-\phi\xi.$ (3.25)

Then condition (3.23) is written $\left(\text{note that }\phi,_{i}=0\right)$:

$L_{\eta}\Gamma_{(jk)}^{i}=2\Phi_{(,j}\delta_{k)}^{i}-\xi\Gamma_{(kj),t}^{i}.$
(3.26)

If we consider the vector $\mathbb{\xi}\mathbf{=}\xi\partial_{t}$ (which does
not have components along $\partial_{i}$) we compute

$L_{\mathbf{\xi}}\Gamma_{(jk)}^{i}=\xi\Gamma_{(kj),t}^{i}$

hence (3.26) is written

$L_{X}\Gamma_{(jk)}^{i}=2\Phi_{(,j}\delta_{k)}^{i}.$ (3.27)

We observe that this condition is the condition for a projective collineation
of the connection $\Gamma_{(jk)}^{i}$ along the symmetry vector $X$ and with
projecting function $\Phi$. Concerning the other conditions we note that
(3.22) can be written in covariant form as follows:

$\Phi_{,t}\delta_{j}^{i}-2\eta_{,t\mid j}^{i}=0$ (3.28)

where $\eta_{,t\mid j}^{i}=\eta_{,tj}^{i}+\eta_{,t}^{k}\Gamma_{(kj)}^{i}$ is
the covariant derivative with respect to $\Gamma_{(kj)}^{i}$ of the vector
$\eta_{,t}^{i}.$ Condition (3.24) implies that $\xi,_{{i}}$ is a gradient KV
of the metric of the space $x^{i}$. Condition (3.21) is obviously in covariant
form with respect to the index $i.$

We arrive at the following conclusion.

a) The conditions for the Lie symmetries of the autoparallel equations (3.17)
are covariant equations because if we consider the connection in the augmented
$n+1$ space $\\{x^{i},t\\}$, all components of $\Gamma$ which contain an index
along the direction of $t$ vanish, therefore the partial derivatives with
respect to $t$ can be replaced with covariant derivatives with respect to $t$.

b) Equation (3.21) gives the functional dependence of $\eta^{i}$ on $t$ and
the non-affine parametrization function $\phi(t)$.

c) Equation (3.24) gives that the vector $\xi_{,i}$ is a gradient Killing
vector of the $n-$dimensional space $\\{x^{i}\\}$.

d) Equation (3.22) relates the functional dependence of $\eta^{i}$ and $\xi$
in terms of $t$.

e) Equation (3.23) is the most important equation for our purpose, because it
states that _the symmetry vector ( 3.18) is an affine collineation in the jet
space _$\\{t,x^{i}\\}$ because it preserves both the geodesics and their
affine parameter. In the space $\\{x^{i}\\}$ the vectors
$\eta^{i}\emph{(t,x)}\partial_{x^{i}}$ _are projective collineations_ because
they preserve the geodesics but not necessarily their parametrization.

In the following we restrict our considerations to the case of Riemannian
connections that is the $\Gamma_{jk}^{i}$ are symmetric and the covariant
derivative of the metric vanishes.

####  3.4 Lie and Noether symmetries of geodesic equations

We compute the Lie symmetry vectors of geodesics equations (3.2) with affine
parametrization; that is, we assume $\phi=0$ and $\Gamma_{jk,t}^{i}=0.$ The
later is a reasonable assumption because the $\Gamma_{jk}^{i\prime}$’s are
computed in terms of the metric which does not depend on the parameter $t$.
Under these assumptions the symmetry conditions for the geodesics equations
(3.2) are (3.9)-(3.12). We proceed with the solution of this system of
equations333See also [35] Table II..

Equation (3.9) implies

$\eta^{i}(t,x)=A^{i}(x)t+B^{i}(x)$ (3.29)

where $A^{i}(x),B^{i}(x)$ are arbitrary differentiable vector fields.

The solution of equation (3.12) is

$\xi(t,x)=C_{J}(t)S^{J}(x)+D(t)$ (3.30)

where $C_{J}(t),D(t)$ are arbitrary functions of the affine parameter $t$ and
$S^{J}(x)$ is a function whose gradient is a gradient KV, i.e.

$S^{J}(x)_{|(i,j)}=0.$ (3.31)

The index $J$ runs through the number of gradient KVs of the metric. Condition
(3.10) gives

$2A(x)_{|j}^{i}=\left[C_{J}(t),_{tt}S^{J}(x)+D(t),_{tt}\right]\delta_{j}^{i}.$
(3.32)

Because the left hand side is a function of $x$ only, we must have

$\displaystyle D(t),_{tt}$ $\displaystyle=M\Rightarrow
D(t)=\frac{1}{2}Mt^{2}+Kt+L\text{ \ where }M,K,L\text{ constants}$ (3.33)
$\displaystyle C_{J}(t),_{tt}$ $\displaystyle=G_{J}=\text{constant
}\Rightarrow C_{J}(t)=\frac{1}{2}G_{J}t^{2}+E_{J}t+F_{J}\text{\ where
}G_{J},E_{J},F_{J}\text{ constants.}$ (3.34)

Replacing in (3.32) we find

$2A(x)_{;j}^{i}=\left(G_{J}S^{J}(x)+M\right)\delta_{j}^{i}\Rightarrow
A(x)_{i;j}=\frac{1}{2}\left(G_{J}S^{J}(x)+M\right)g_{ij}$ (3.35)

where we have lowered the index because the connection is metric (i.e.
$g_{ij|k}=0)$. The last equation implies that the vector $A(x)^{i}$ is a
conformal Killing vector with conformal factor
$\psi=\frac{1}{2}(G_{J}S^{J}(x)+M)$. Because $A(x)_{[i;j]}=0$ this vector is a
gradient CKV.

We continue with condition (3.11) and replace $\eta^{i}(t,x)$ from (3.29)

$\displaystyle L_{A}\Gamma_{jk}^{i}t+L_{B}\Gamma_{jk}^{i}$
$\displaystyle=2\xi,_{t}{}_{(,j}\delta_{k)}^{i}=2\left[\left(G_{J}t+E_{J}\right)S^{J}(x)+Mt+K\right]_{|(j}\delta_{k)}^{i}=2\left(G_{J}t+E_{J}\right)S^{J}(x)_{|(j}\delta_{k)}^{i}\Rightarrow$
$\displaystyle L_{A}\Gamma_{jk}^{i}$
$\displaystyle=2G_{J}S^{J}(x)_{,(j}\delta_{k)}^{i}$ (3.36) $\displaystyle
L_{B}\Gamma_{jk}^{i}$ $\displaystyle=2E_{J}S^{J}(x)_{,(j}\delta_{k)}^{i}.$
(3.37)

The last two equations imply that the vectors $A^{i}(x),$ $B^{i}(x)$ are
_special projective collineations_ or _affine collineations_ of the metric -
or one of their specializations - with projective functions $G_{J}S^{J}(x)$
and $E_{J}S^{J}(x)$ or zero respectively. Note that relations (3.36), (3.37)
remain true if we add a KV to the vectors $A^{i}(x),$ $B^{i}(x)$ , therefore
these vectors are determined up to a KV444Because $A^{i}$ is a projective
collineation and a CKV it must be a HV..

It is well known that in a Riemannian space a CKV $K^{i}$ with conformal
factor $\psi(x^{i})$ satisfies the identity:

$L_{K}\Gamma_{(jk)}^{i}=g^{is}\left[\psi_{,j}g_{ks}+\psi_{,k}g_{js}-\psi_{,s}g_{jk}\right].$
(3.38)

Applying this identity to the CKV $A^{i}$ we find:

$G_{J}S^{J}(x)_{,k}=0\Rightarrow G_{J}S^{J}(x)=2\rho=\text{constant.}$ (3.39)

This implies that $A^{i}$_is a gradient HV (not necessarily proper) with
homothetic factor_ $\rho+\frac{1}{2}M$ . Furthermore (3.35) implies:

$\displaystyle 2A^{i}$ $\displaystyle=(2\rho+M)x^{i}+2L^{i}\Rightarrow$
$\displaystyle A^{i}$ $\displaystyle=(\rho+\frac{1}{2}M)x^{i}+L^{i}$ (3.40)

where $L^{i}$ is a non-gradient KV.

We continue with the special projective collineation vector $B^{i}$. For this
vector we use the property that for a symmetric connection the following
identity, holds

$\mathcal{L}_{\mathbf{B}}\Gamma_{(jk)}^{i}=B_{;jk}^{i}-R_{jkl}^{i}B^{l}.$

Replacing the left hand side from (3.37) we find

$B_{;jk}^{i}-R_{jkl}^{i}B^{l}=2E_{J}S^{J}(x)_{,(j}\delta_{k)}^{i}.$ (3.41)

Contracting the indices $i,j$ we find

$\left(B_{;i}^{i}-(n+1)E_{J}S^{J}(x)\right)_{;k}=0$ (3.42)

which implies

$B_{;i}^{i}=(n+1)E_{J}S^{J}(x)+2b$ (3.43)

where $b=$constant. In case this vector is an affine collineation then
$B_{;i}^{i}=2b.$ Using the above results we find for $\xi(t,x)$

$\displaystyle\xi(t,x)$ $\displaystyle=C_{J}(t)S^{J}+D(t)$
$\displaystyle=\left(\frac{1}{2}G_{J}t^{2}+E_{J}t+F_{J}\right)S^{J}+\frac{1}{2}Mt^{2}+Kt+L$
$\displaystyle=\frac{1}{2}(G_{J}S^{J}+M)t^{2}+(E_{J}S^{J}+K)t+F_{J}S^{J}+L$

We summarize the above results in the following Theorem.

###### Theorem 3.4.1

The Lie symmetry vector

$X=\xi(t,x)\partial_{t}+\eta^{i}(t,x)\partial_{x^{i}}$

of the equation of geodesics (3.2) in a Riemannian space is generated from the
elements of the special projective algebra as follows.

Case A. The metric admits gradient KVs. Then

a. The function

$\xi(t,x)=\frac{1}{2}\left(G_{J}S^{J}+M\right)t^{2}+\left[E_{J}S^{J}+K\right]t+F_{J}S^{J}+L,$
(3.44)

where $G_{J},M,b,K,F_{J}$ and $L$ are constants and the index $J\ $runs along
the number of gradient KVs

b. The vector

$\eta^{i}(t,x)=A^{i}(x)t+B^{i}(x)+D^{i}(x)$ (3.45)

where the vector $A^{i}(x)$ is a gradient HV with conformal factor
$\psi=\frac{1}{2}\left(G_{J}S^{J}+M\right)$ (if it exists), $D^{i}(x)$ is a
non-gradient KV of the metric and $B^{i}(x)$ is either a special projective
collineation with projection function $E_{J}S^{J}(x)$ or an AC and $E_{J}=0$
in (3.44).

Case B. The metric does not admit gradient KVs. Then

a. The function

$\xi(t,x)=\frac{1}{2}Mt^{2}+Kt+L$ (3.46)

b. The vector

$\eta^{i}(t,x)=A^{i}(x)t+B^{i}(x)+D^{i}(x),$ (3.47)

where $A^{i}(x)$ is a gradient HV with conformal factor $\psi=\frac{1}{2}M,$
$D^{i}(x)$ is a non-gradient KV of the metric and $B^{i}(x)$ is an AC. If in
addition the metric does not admit a gradient HV, then

$\displaystyle\xi(t)$ $\displaystyle=Kt+L$ (3.48) $\displaystyle\eta^{i}(x)$
$\displaystyle=B^{i}(x)+D^{i}(x).$ (3.49)

#####  3.4.1 Noether symmetries and conservation laws

In a Riemannian space the equations of geodesics (3.2) are produced from the
geodesic Lagrangian:

$L=\frac{1}{2}g_{ij}\dot{x}^{i}\dot{x}^{j}~{}.$ (3.50)

The infinitesimal generator

$X=\xi\left(t,x^{k}\right)\partial_{t}+\eta^{i}\left(t,x^{k}\right)\partial_{x^{i}}$
(3.51)

is a Noether symmetry of Lagrangian (3.50) if there exists a smooth function
$f(t,x^{i})$ such that

$X^{\left[1\right]}L+\frac{d\xi}{dt}L=\frac{df}{dt}$ (3.52)

where

$X^{\left[1\right]}=\xi\left(t,x^{k}\right)\partial_{t}+\eta^{i}\left(t,x^{k}\right)\partial_{x^{i}}+\left(\eta_{\left[1\right]}^{i}\right)\partial_{\dot{x}^{i}}$

is the first prolongation of $\mathbf{X}.$ We compute

$X^{\left[1\right]}L=\frac{1}{2}\left(\eta^{k}g_{ij,k}\dot{x}^{i}\dot{x}^{j}+2\frac{d\eta^{k}}{dt}g_{ik}\dot{x}^{i}-2\dot{x}^{i}\dot{x}^{j}\frac{d\xi}{dt}g_{ij}\right).$

Replacing the total derivatives in the rhs

$\displaystyle\frac{d\xi}{dt}$ $\displaystyle=\xi_{,t}+\dot{x}^{k}\xi_{,k}$
$\displaystyle\frac{d\eta^{i}}{dt}$
$\displaystyle=\eta_{,t}^{i}+\dot{x}^{k}\eta_{,k}^{i}$

we find that

$X^{\left[1\right]}L=\frac{1}{2}\left(\begin{array}[]{c}\eta^{k}g_{ij,k}\dot{x}^{i}\dot{x}^{j}+2\eta_{,t}^{i}g_{ij}\dot{x}^{j}+\eta_{,r}^{i}g_{ik}\dot{x}^{k}\dot{x}^{r}+\\\
+\eta_{,r}^{i}g_{kj}\dot{x}^{k}\dot{x}^{r}-2\xi_{,t}g_{ij}\dot{x}^{i}\dot{x}^{j}-2\xi_{,k}g_{ij}\dot{x}^{i}\dot{x}^{j}\dot{x}^{k}\end{array}\right).$

The term

$\frac{d\xi}{dt}L=\frac{1}{2}\left(\xi_{,t}+\dot{x}^{k}\xi_{,k}\right)g_{ij}\dot{x}^{i}\dot{x}^{j}.$

Finally the Noether symmetry condition (3.52) is

$-2f_{,t}+\left[2\eta_{,t}^{i}g_{ij}-2f_{,i}\right]\dot{x}^{j}-\xi_{,k}g_{ij}\dot{x}^{i}\dot{x}^{j}\dot{x}^{k}+\left[\eta^{k}g_{ij,k}+\eta_{,i}^{k}g_{ik}+\eta_{,i}^{k}g_{kj}-g_{ij}\xi_{,t}\right]\dot{x}^{i}\dot{x}^{j}=0.$

This relation is an identity hence the coefficient of each power of
$\dot{x}^{j}$ must vanish. This results in the equations:

$\displaystyle\xi_{,k}$ $\displaystyle=0$ (3.53) $\displaystyle
L_{\eta}g_{ij}$ $\displaystyle=2\left(\frac{1}{2}\xi_{,t}\right)g_{ij}$ (3.54)
$\displaystyle\eta_{,t}^{,i}g_{ij}$ $\displaystyle=f_{,i}$ (3.55)
$\displaystyle f_{,t}$ $\displaystyle=0$ (3.56)

Condition (3.53) gives $\xi_{,k}=0\Rightarrow\xi=\xi\left(t\right)$.

Condition (3.56) implies $f\left(x^{k}\right)$ and then condition (3.53) gives
that $\eta^{i}$ is of the form:

$\eta_{i}=f_{,i}t+K_{i}(x^{j}).$ (3.57)

Then from (3.54) follows that $\xi_{,t}$ must be at most linear in $t.$ Hence
$\xi(t)$ must be at most a function of $t^{2}.$ Furthermore from (3.54)
follows that $\eta^{i}$ is at most a CKV with conformal factor
$\psi_{H}=\frac{1}{2}(At+B),$ where $A,B$ are constants. We consider various
cases.

Case 1: Suppose $\xi=$constant=$C_{1}$. Then $\eta^{i}$ is a KV of the metric
which is independent of $t.$ This implies that either $f_{,i}=0$ and
$f=$constant $=A=0$ or that $f_{,i}$ is a gradient KV. In this case the
Noether symmetry vector is:

$X^{i}=C_{1}\partial_{t}+g^{ij}\left(f_{,j}t+K_{j}(x^{r})\right),$

where $K^{i}$ is a non-gradient KV of $g_{ij}.$

Case 2: Suppose $\xi=2t.$ Then $\eta^{i}$ is a HV of the metric $g_{ij}$ with
homothetic factor $1$. Then $\eta_{i}=H_{i}(x^{j})$ , $f_{,i}=0\Rightarrow
f=$constant$=0$ where $H^{i}$ is a HV of $g_{ij}$ with homothetic factor
$\psi,$ not necessarily a gradient HV. In this case the Noether symmetry
vector is:

$X^{i}=2\psi t\partial_{t}+H^{i}(x^{r}).$

Case 3: $\xi(t)=t^{2}.$ Then $\eta^{i}$ is a HV of the metric $g_{ij}$ (the
variable $t$ cancels) with homothetic factor $1.$ Again $f_{,i}$ is a gradient
HV with homothetic factor $\psi$ and the Noether symmetry vector is

$X^{i}=\psi t^{2}\partial_{t}+g^{ij}f_{,j}t.$

Therefore we have the result.

###### Theorem 3.4.2

The Noether Symmetries of the geodesic Lagrangian follow from the KVs and the
HV of the metric $g_{ij}$ as follows:

$\displaystyle X$ $\displaystyle=$ $\displaystyle\left(C_{3}\psi
t^{2}+2C_{2}\psi t+C_{1}\right)\partial_{t}+$ (3.58)
$\displaystyle+\left[C_{J}S^{J,i}+C_{I}KV^{Ii}+C_{IJ}tS^{J,i}+C_{2}H^{i}+C_{3}t(GHV)^{i}\right]\partial_{i}$

with corresponding gauge function

$f(x^{k})=C_{1}+C_{2}+C_{I}+C_{J}+\left[C_{IJ}S^{J}\right]+C_{3}\left[GHV\right],$
(3.59)

where $S^{J,i}$ are the $C_{J}$ gradient KVs, $KV^{Ii}$ are the $C_{I}$ non-
gradient KVs, $H^{i}$ is a HV not necessarily gradient and $(GHV)^{i}$ is the
gradient HV (if it exists) of the metric $g_{ij}$.

The importance of Theorems 3.4.1 and 3.4.2 are that one is able to compute the
Lie symmetries and the Noether symmetries of the geodesic equations in a