Also at ]Physical Science and Engineering Division, Diablo Valley College,
Pleasant Hill, CA 94523, USA

# Generalized Lie Symmetries and Almost Regular Lagrangians: A Link Between
Symmetry and Dynamics

Achilles D. Speliotopoulos  Department of Physics, University of California,
Berkeley, CA 94720 USA  [<EMAIL_ADDRESS>
###### Abstract

The generalized Lie symmetries of almost regular Lagrangians are studied, and
their impact on the evolution of dynamical systems is determined. It is found
that if the action has a generalized Lie symmetry, then the Lagrangian is
necessarily singular; the converse is not true, as we show with a specific
example. It is also found that the generalized Lie symmetry of the action is a
Lie subgroup of the generalized Lie symmetry of the Euler-Lagrange equations
of motion. The converse is once again not true, and there are systems for
which the Euler-Lagrange equations of motion have a generalized Lie symmetry
while the action does not, as we once again show through a specific example.
Most importantly, it is shown that each generalized Lie symmetry of the action
contributes one arbitrary function to the evolution of the dynamical system.
The number of such symmetries gives a lower bound to the dimensionality of the
family of curves emanating from any set of allowed initial data in the
Lagrangian phase space. Moreover, if second- or higher-order Lagrangian
constraints are introduced during the application of the Lagrangian constraint
algorithm, these additional constraints could not have been due to the
generalized Lie symmetry of the action.

##  I Introduction

The symmetries of the Euler-Lagrange equations of motion were recently used to
study the constrained dynamics of singular Lagrangians [1]. The focus was on
almost regular Lagrangians [2, 3, 4, 5], and it was found that for these
Lagrangians the Euler-Lagrange equations of motion admit a generalized Lie
symmetry (also known as a local gauge symmetry). The generators
$\mathcal{S}\hbox{ym}$ of this symmetry group
$\hbox{Gr}_{\mathcal{S}\hbox{ym}}$ were determined in the Lagrangian phase
space approach to Lagrangian mechanics, and were found to lie in the kernel of
the Lagrangian two-form $\mathbf{\Omega}_{L}$. While it is well-known that the
solutions $\mathbf{X}_{E}$ of the energy equation,

$0=\mathbf{d}E-i_{\mathbf{X}_{E}}\mathbf{\Omega}_{L},$ (1)

is not unique for almost regular Lagrangians, it was shown in [1] that the
action of $\mathcal{S}\hbox{ym}$ on a general solution to this equation—and in
particular, on the second-order, Lagrangian vector field (SOLVF)—will result
in a vector field that is no longer a solution of Eq. $(\ref{EnergyE})$. Thus,
not all solutions of the energy equation have
$\hbox{Gr}_{\mathcal{S}\hbox{ym}}$ as a symmetry group. It is, however,
possible to construct solutions to Eq. $(\ref{EnergyE})$ for whom
$\mathcal{S}\hbox{ym}$ does generate a group of symmetry transformations [1].
These vector fields are called second-order, Euler-Lagrange vector fields
(SOELVFs). As the evolution of the dynamical system for singular Lagrangians
must lie on Lagrangian constraint surfaces [5], a Lagrangian constraint
algorithm for SOELVFs was also introduced in [1] to construct such solutions
to the energy equation. It was then shown that these SOELVFs, along with the
dynamical structures in the Lagrangian phase space needed to describe and
determine the motion of the dynamical system, are projectable to the
Hamiltonian phase space. In particular, the primary Hamiltonian constraints
can be constructed from vectors that lie in the kernel of
$\mathbf{\Omega}_{L}$, and the Lagrangian constraint algorithm for the SOELVF
is equivalent to the stability analysis of the total Hamiltonian (we follow
the terminology found in [6]; see also [7, 8, 9]) obtained using constrained
Hamiltonian mechanics. Importantly, the end result of this stability analysis
gives a Hamiltonian vector field that is the projection of the SOELVF obtained
from the Lagrangian constraint algorithm. The Lagrangian and Hamiltonian
formulations of mechanics for almost regular Lagrangians were thereby shown to
be equivalent.

While [1] focused on the generalized Lie symmetries of the Euler-Lagrange
equations of motion and whether the dynamical structures constructed in the
Lagrangian phase space are projectable to the Hamiltonian phase space, in this
paper the focus is on the symmetries of the action itself and the impact these
symmetries have on the evolution of dynamical systems. This impact is found to
be quite broad, surprisingly restrictive, and unexpectedly subtle. Indeed,
even the seemingly reasonable expectation that any generalized Lie symmetry of
the Euler-Lagrange equations of motion should be a reflection of the
symmetries of the action itself is not borne out.

We find that if the action has a generalized Lie symmetry, then its Lagrangian
is necessarily singular; the converse need not be true, as we show through a
specific example. We also find that the generators of the generalized Lie
symmetry of the action form a Lie sub-algebra of the generators of the
generalized Lie symmetry of the Euler-Lagrange equation of motion; once again,
the converse is not true. We give an example of a dynamical system for which
the Euler-Lagrange equations of motion has a generalized Lie symmetry, while
its action does not. Most importantly, for systems where the Lagrangian is
almost regular and for which the two-form $\mathbf{\Omega}_{L}$ has constant
rank, we show that each generalized Lie symmetry of the action contributes one
arbitrary constant to the SOELVF. The dimensionality of the space of solutions
to the energy equation that have $\hbox{Gr}_{\mathcal{S}\hbox{ym}}$ as a
symmetry group is thus at least as large as the number of generalized Lie
symmetries of the action. Moreover, if second- or higher-order Lagrangian
constraints are introduced during the application of the Lagrangian constraint
algorithm, these additional constraints cannot be due to the generalized Lie
symmetry of the action.

Symmetries of Lagrangian systems have been studied before. However, such
analyses have been focused on time-dependent Lagrangians [10, 11, 12, 13, 14,
15, 17, 16]; on systems of first-order evolution equations [18, 19, 20, 21,
22]; or on general solutions of Eq. $(\ref{EnergyE})$ [23] (see also [24]).
Importantly, the great majority of these studies have been done using first-
order prolongations on first-order jet bundles with a focus on the Lie
symmetries of first-order evolution equations. Our interest is in the
symmetries of the action, which naturally leads us to consider generalized Lie
symmetries and second-order prolongations. To our knowledge, such symmetry
analysis of the action has not been done before. (The framework for
$k^{\hbox{{th}}}$-order prolongations on $k^{\hbox{{th}}}$-order jet bundles
have been introduced before [17, 23, 16, 25, 26], but they were not applied to
the action or to the Euler-Lagrange equations of motion.)

The rest of the paper is arranged as follows. In Section II the conditions
under which the action for a dynamical system, and the conditions under which
the Euler-Lagrange equations of motion for this action, have a generalized Lie
symmetry are determined. To compare the conditions for each, the analysis for
the two are done separately, with each self-contained. In Section III
properties of the Lagrangian phase space are reviewed, and the notation used
here established. The generators of the generalized Lie symmetry group for the
Euler-Lagrange equations of motion were determined in [1], and a summary of
the results found therein that are needed here is given. In Section IV the
generators of the generalized Lie symmetry group for the action is found
within the Lagrangian phase space approach, and their relation to the
generators for the symmetry group of the Euler-Lagrange equations of motion is
determined. The impact of the symmetries of the action on the SOELVF is then
analyzed by applying the Lagrangian constraint algorithm introduced in [1] to
these SOELVF. The results obtained in this paper is then applied to three
different dynamical systems in Section V. In particular, an example of a
dynamical system that has no generalized Lie symmetries and yet is still
singular, and another example where the action has no symmetries and yet the
Euler-Lagrange equations of motion do, are given. Concluding remarks can be
found in Section VI.

##  II Generalized Lie symmetries and Lagrangian Mechanics

In this section we determine the conditions under which the action of a
dynamical system, and the conditions under which the Euler-Lagrange equations
of motion for this system, has a generalized Lie symmetry. While the
determination for both is done within Lagrangian mechanics, the analysis for
the action is completed separately from that of the equations of motion—with
each self-contained—so that the two conditions can be compared. We will later
show that every generator of the generalized Lie symmetry of the action is a
generator of a generalized Lie symmetry of the Euler-Lagrange equations of
motion. Interestingly, the converse is not true.

###  II.1 Symmetries of the Action

We begin with Lagrangian mechanics, and an analysis of the generalized Lie
symmetry [27] of the action

$S:=\int_{t_{1}}^{t_{2}}L\left(q(t),\dot{q}(t)\right)dt,$ (2)

for a dynamical system on a $D$-dimensional configuration space $\mathbb{Q}$.
Here, $L\left(q(t),\dot{q}(t)\right)$ is the Lagrangian along a path
$q(t)=\left(q^{1}(t),\dots,q^{D}(t)\right)$ on $\mathbb{Q}$ with end points
given by $Q_{1}:=q(t_{1}),Q_{2}:=q(t_{2})$. These points are chosen at the
same time the choice of $S$ is made, and are fixed.

As $L\left(q(t),\dot{q}(t)\right)$ depends on both the position $q(t)$ and the
velocity $\dot{q}(t)$ of the path, we consider a generalized Lie symmetry that
is generated by

$\mathbf{g}_{L}:=\rho_{L}(q,\dot{q})\cdot\frac{\mathbf{\partial}\>\>\,}{\mathbf{\partial}q},$
(3)

where $\rho_{L}(q,\dot{q})$ does not depend explicitly on time. Evolution
along the path gives the total time derivative

$\frac{\mathbf{d}\>\>\>}{\mathbf{d}t}:=\dot{q}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q}+\ddot{q}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}\dot{q}}.$
(4)

This in turn gives $\dot{\rho}_{L}:=\mathbf{d}\rho_{L}/\mathbf{d}t$, and the
second-order prolongation vector [27],

$\hbox{{pr
}}\mathbf{g}_{L}:=\rho_{L}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q}+\dot{\rho}_{L}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}\dot{q}}+\ddot{\rho}_{L}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}\ddot{q}},$
(5)

on the second-order jet space $\mathbb{M}^{(2)}=\\{(t,q,\dot{q},\ddot{q})\\}$
where this $\hbox{{pr }}\mathbf{g}_{L}\in\mathbf{T}\mathbb{M}^{(2)}$.

Under this generalized Lie symmetry, the action varies by

$\delta S=\int_{t_{1}}^{t_{2}}\hbox{{pr
}}\mathbf{g}_{L}\Big{[}L(q(t),\dot{q}(t))\Big{]}dt,$ (6)

with the requirement that
$\rho_{L}(q(t_{1}),\dot{q}(t_{1}))=0=\rho_{L}(q(t_{2}),\dot{q}(t_{2}))$. Then
after an integration by parts,

$\delta S=\int_{t_{1}}^{t_{2}}\rho_{L}\cdot\left[\frac{\partial L}{\partial
q}-\frac{d\>\>\>}{dt}\left(\frac{\partial
L}{\partial\dot{q}}\right)\right]dt.$ (7)

It is important to realize that the action may be evaluated along any path on
$\mathbb{Q}$. As such, if $\mathbf{g}_{L}$ generates a symmetry of the action,
then Eq. $(\ref{Ae1})$ must vanish for all paths $q(t)$ on $\mathbb{Q}$, and
not just for those that minimize the action.

To make connection with the Lagrangian phase space approach used in the rest
of the paper, we make use of

$E\left(q,\dot{q}\right):=\dot{q}^{a}\frac{\partial
L\left(q,\dot{q}\right)}{\partial\dot{q}^{a}}-L\left(q,\dot{q}\right),$ (8)

along with

$M_{ab}\left(q,\dot{q}\right):=\frac{\partial^{2}L\left(q,\dot{q}\right)}{\partial\dot{q}^{a}\partial\dot{q}^{b}},\quad\hbox{and
}\quad
F_{ab}\left(q,\dot{q}\right):=\frac{\partial^{2}L\left(q,\dot{q}\right)}{\partial\dot{q}^{a}\partial
q^{b}}-\frac{\partial^{2}L\left(q,\dot{q}\right)}{\partial\dot{q}^{b}\partial
q^{a}},$ (9)

to express Eq. $(\ref{Ae1})$ as

$\delta S=-\int_{t_{1}}^{t_{2}}\rho^{a}_{L}\left(\frac{\partial E}{\partial
q^{a}}+F_{ab}(q,\dot{q})\dot{q}^{b}+M_{ab}(q,\dot{q})\ddot{q}^{b}\right)dt.$
(10)

Here, Latin indices run from $1$ to $D$, and Einstein’s summation convention
is used. We then arrive at our first result.

###### Lemma 1

An action $S$ of a dynamical system has a generalized Lie symmetry generated
by $\mathbf{g}_{L}$ if and only if there exists a $\rho_{L}\in\hbox{ker
}M_{ab}$ such that

$0=\rho_{L}^{a}(q,\dot{q})\left(\frac{\partial E}{\partial
q^{a}}+F_{ab}(q,\dot{q})\dot{q}^{b}\right),$ (11)

on $\mathbf{T}\mathbb{Q}$.

Proof. If $\mathbf{g}_{L}$ generates a generalized Lie symmetry of $S$, then
Eq. $(\ref{e2})$ must vanish for all paths on $\mathbb{Q}$. For an arbitrary
path on $\mathbb{Q}$ the curvature of the path $\ddot{q}$ will not depend on
either the $q(t)$ or the $\dot{q}(t)$ for the path, however. As such, for
$\delta S=0$, it must be that $\rho^{a}_{L}M_{ab}\ddot{q}^{b}=0$ for any
choice of $\ddot{q}$, and thus $\rho_{L}^{a}\in\hbox{ker }M_{ab}$. The
remaining terms in Eq. $(\ref{e2})$ gives the condition Eq. $(\ref{e3})$.

The set of all vector fields $\mathbf{g}_{L}$ that satisfy Lemma 1 is denoted
by $\mathfrak{g}_{L}$, while $\hbox{{pr }}\mathfrak{g}_{L}:=\\{\hbox{{pr
}}\mathbf{g}_{L}\ |\ \ \mathbf{g}_{L}\in\mathfrak{g}_{L}\\}$ is the set of
their prolongations. This $\hbox{{pr }}\mathfrak{g}_{L}$ is involutive [27],
and the conditions under which $\hbox{{pr }}\mathfrak{g}_{L}$ generates a
generalized Lie symmetry group are given in [27].

We see from Lemma 1 that if the action has a generalized Lie symmetry, then
the Lagrangian is necessarily singular, and as such the Lagrangian two-form
$\mathbf{\Omega}_{L}$ will not have maximum rank. It is also important to note
that while equations of the form Eq. $(\ref{e3})$ often appear in the
Lagrangian phase space description of mechanics [1], they appear as Lagrangian
constraints, conditions that must be imposed for evolution under the Euler-
Lagrange equations to be well defined. Here, Eq. $(\ref{e3})$ is not a
constraint. Rather, because the action must have this symmetry for all
possible paths on $\mathbb{Q}$, and since the set of all possible paths cover
$\mathbb{Q}$, Eq. $(\ref{e3})$ is a condition on $\rho_{L}$ that must be
satisfied identically on all of $\mathbf{T}\mathbb{Q}$—and thus, on the
Lagrangian phase space—for $\mathbf{g}_{L}$ to be a generator of the symmetry
group. We will see that not all the vectors in $\hbox{ker }M_{ab}$ satisfy the
identity Eq. $(\ref{e3})$, however, and thus not all of these vectors will
generate a generalized Lie symmetry of the action.

###  II.2 Symmetries of the Euler-Lagrange Equations of Motion

While in Section II.1 the focus was on arbitrary paths on the configuration
space $\mathbb{Q}$ and the symmetries of the action, in this section the focus
is on the trajectories that minimizes the action and the generalized Lie
symmetries of them. These trajectories are solutions of the Euler-Lagrange
equations of motion, and for almost regular Lagrangians such solutions form a
family of curves. It is, in fact, the presence of this family of curves that
gives rise to the generalized Lie symmetry. The treatment here follows closely
to that given in [1].

For almost regular Lagrangians the solutions of the Euler-Lagrange equations
of motion

$M_{ab}(q,\dot{q})\ddot{q}^{b}=-\frac{\partial E}{\partial
q^{a}}-F_{ab}(q,\dot{q})\dot{q}^{b},$ (12)

are not unique. While for these Lagrangians the rank of
$M_{ab}\left(q,\dot{q}\right)=D-N_{0}$—with $N_{0}=\hbox{dim }\left(\hbox{ker
}M_{ab}(q,\dot{q})\right)$—is constant, this rank is not maximal, and thus Eq.
$(\ref{2ndEL1})$ does not have a unique solution for $\ddot{q}$. Instead, for
a chosen set of initial data
$\left(q_{0}=q(t_{0}),\dot{q}_{0}=\dot{q}(t_{0})\right)$, the solution to Eq.
$(\ref{2ndEL1})$ results in a family of solutions that evolve from this
$(q_{0},\dot{q}_{0})$. As with the paths in Section II.1, these solutions are
related to one another through a generalized Lie symmetry [27].

Following [27], the collection of functions

$\Delta_{a}(q,\dot{q},\ddot{q}):=\frac{\partial E(q,\dot{q})}{\partial
q^{a}}+F_{ab}(q,\dot{q})\dot{q}^{b}+M_{ab}(q,\dot{q})\ddot{q}^{b},$ (13)

defines a set of surfaces $\Delta_{a}(q,\dot{q},\ddot{q})=0$ on
$\mathbb{M}^{(2)}$, while the family of solutions to Eq. $(\ref{2ndEL1})$

$\mathcal{O}\left(q_{0},\dot{q}_{0}\right):=\big{\\{}q\left(t\right)\ |\ \
\Delta_{a}(q,\dot{q},\ddot{q})=0\hbox{ with }q\left(t_{0}\right)=q_{0},\
\dot{q}\left(t_{0}\right)=\dot{q}_{0}\big{\\}},$ (14)

that evolve from the same initial data $(q_{0},\dot{q}_{0})$ gives the
collection of trajectories that lie on these surfaces. Indeed, for any two
such solutions $q^{a}(t)$ and $Q^{a}(t)$ there exists a
$\mathfrak{z}(q,\dot{q})\in\hbox{ker }M_{ab}(q,\dot{q})$ such that
$\ddot{Q}^{a}-\ddot{q}^{a}=\mathfrak{z}^{a}$. Importantly, because
$\mathfrak{z}^{a}$ depends on both $q$ and $\dot{q}$, the symmetry group that
maps one member of $\mathcal{O}$ to another must be a generalized Lie
symmetry. We therefore take the generator of this symmetry group to be

$\mathbf{g}:=\rho(q,\dot{q})\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q},$
(15)

with the corresponding the second-order prolongation vector for $\mathbf{g}$
being,

$\hbox{{pr
}}\mathbf{g}:=\rho\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q}+\dot{\rho}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}\dot{q}}+\ddot{\rho}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}\ddot{q}},$
(16)

with this $\hbox{{pr }}\mathbf{g}\in\mathbf{T}\mathbb{M}^{(2)}$. As with the
above, the total time derivative is given by Eq. $(\ref{Dt})$, but unlike the
analysis in Section II.1, the evolution of the path—and indeed, for all the
trajectories in $\mathcal{O}(q_{0},\dot{q}_{0})$—here is given by the Euler-
Lagrange equations of motion.

The action of this prolongation on $\Delta_{a}$ on the $\Delta_{a}=0$ surface
gives,

$\hbox{{pr
}}\mathbf{g}\left[\Delta_{a}(q,\dot{q},\ddot{q})\right]=-\frac{\partial\ddot{q}^{b}}{\partial
q^{a}}M_{bc}(q,\dot{q})\rho^{c}+\frac{d\>\>\>}{dt}\left[F_{ab}(q,\dot{q})\rho^{b}+M_{ab}(q,\dot{q})\dot{\rho}^{b}\right].$
(17)

Since $N_{0}>0$, $\ddot{q}$ is not unique on this surface, and yet
$\mathbf{g}$ must generate the same symmetry group for all the trajectories in
$\mathcal{O}(q_{0},\dot{q}_{0})$. Necessarily, $\rho(q,\dot{q})\in\hbox{ker
}M_{ab}(q,\dot{q})$. It then follows that $\hbox{{pr
g}}[\Delta_{a}(q,\dot{q},\ddot{q})]=0$ if and only if (iff) there are
constants $b_{a}$ such that $b_{a}=F_{ab}\rho^{b}+M_{ab}\dot{\rho}^{b}$. The
solutions in $\mathcal{O}(q_{0},\dot{q}_{0})$ all have the same initial data,
however, and thus necessarily
$\rho(q_{0},\dot{q}_{0})=0=\dot{\rho}(q_{0},\dot{q}_{0})$. We conclude that
$b_{a}=0$. The following result, first proved in [1], then follows.

###### Lemma 2

If $\mathbf{g}$ is a generalized infinitesimal symmetry of $\Delta_{a}$, then
$\rho^{a}(q,\dot{q})\in\hbox{ker }M_{ab}(q,\dot{q})$, and
$\dot{\rho}^{a}(q,\dot{q})$ is a solution of

$0=F_{ab}(q,\dot{q})\rho^{b}(q,\dot{q})+M_{ab}(q,\dot{q})\dot{\rho}^{b}(q,\dot{q}).$
(18)

As before, we denote the set of all vector fields $\mathbf{g}$ that satisfy
Lemma 2 by $\mathfrak{g}$, while $\hbox{{pr }}\mathfrak{g}:=\\{\hbox{{pr
}}\mathbf{g}\ |\ \ \mathbf{g}\in\mathfrak{g}\\}$ is the set of their
prolongations. Once again $\hbox{{pr }}\mathfrak{g}$ is involutive, and the
conditions under which $\hbox{{pr }}\mathfrak{g}$ generates a generalized Lie
symmetry group are given in [27]. Note, however, that while $\rho=0$ and
$\dot{\rho}=\mathfrak{z}$ for any $\mathfrak{z}\in\hbox{ker
}M_{ab}(q,\dot{q})$ is a solution of Eq. $(\ref{sol})$, we require that
$\dot{\rho}=\mathbf{d}\rho/\mathbf{d}t$; these solutions cannot be generators
of the generalized Lie symmetry. Next, if $\dot{\rho}$ is a solution of Eq.
$(\ref{sol})$, then $\dot{\rho}^{a}+\mathfrak{z}$ is a solution of Eq.
$(\ref{sol})$ as well, and thus these solutions are not unique. This, along
with the previous observation, leads us to generators that are constructed
from equivalence classes of prolongations. Finally, Eq. $(\ref{delta})$ gives
for any $\mathfrak{z}\in\hbox{ker }M_{ab}(q,\dot{q})$,

$0=\mathfrak{z}^{a}\left(\frac{\partial E}{\partial
q^{a}}+F_{ab}(q,\dot{q})\dot{q}^{b}\right),$ (19)

on the solution surface $\Delta_{a}(q,\dot{q},\ddot{q})=0$. If Eq. (19) does
not hold identically, it must be imposed, leading to Lagrangian constraints
[5]. More importantly, because each $q(t)\in\mathcal{O}(\mathfrak{u}_{0})$
must lie on the Lagrangian constraint submanifold, any symmetry transformation
of $q(t)$ generated by $\mathbf{pr}\>\>\mathbf{g}$ must give a path $Q(t)$
that also lies on the constraint submanifold.

Not all vectors in $\mathbf{pr}\>\mathfrak{g}$ will be generators of the
generalized Lie symmetry group for $\mathcal{O}(\mathfrak{u}_{0})$.
Determining which of these vectors are, and the relationship between the
generators of symmetries of the Euler-Lagrange equations of motion and those
of the action, is best done within the Lagrangian phase space framework. To
accomplish this, we will need the following generalization of Lemma 2.

Consider the vector

$\mathbf{k}:=c\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q}+\dot{c}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}\dot{q}},$
(20)

with a $c\in\hbox{ker }M_{ab}(q,\dot{q})$ along with the quantity

$l_{a}:=F_{ab}c^{b}(q,\dot{q})+M_{ab}\dot{c}^{b}(q,\dot{q}).$ (21)

After an integration by parts,

$\displaystyle l_{a}$ $\displaystyle=$ $\displaystyle
c^{b}(q,\dot{q})\left\\{F_{ab}(q,\dot{q})-\frac{\mathbf{d}\>\>\,}{\mathbf{d}t}\frac{\mathbf{\partial}^{2}L}{\mathbf{\partial}\dot{q}^{a}\mathbf{\partial}\dot{q}^{b}}\right\\},$
$\displaystyle=$ $\displaystyle
c^{b}(q,\dot{q})\left\\{F_{ab}(q,\dot{q})-\left[\frac{\mathbf{d}\>\>\,}{\mathbf{d}t},\frac{\partial\>\>\,}{\mathbf{\partial}\dot{q}^{a}}\right]\frac{\mathbf{\partial}L}{\mathbf{\partial}\dot{q}^{b}}-\frac{\mathbf{\partial}\>\>\,}{\mathbf{\partial}\dot{q}^{a}}\left(\frac{\mathbf{d}\>\>\,}{\mathbf{d}t}\frac{\mathbf{\partial}L}{\mathbf{\partial}\dot{q}^{b}}\right)\right\\}.$

Using Eq. $(\ref{Dt})$ we have

$\left[\frac{\mathbf{d}\>\>\,}{\mathbf{d}t},\frac{\mathbf{\partial}\>\>\,}{\mathbf{\partial}\dot{q}^{a}}\right]\frac{\partial
L}{\partial\dot{q}^{b}}=-\frac{\mathbf{\partial}^{2}L}{\mathbf{\partial}q^{a}\mathbf{\partial}\dot{q}^{b}}-\frac{\mathbf{\partial}\ddot{q}^{c}}{\mathbf{\partial}\dot{q}^{a}}\frac{\mathbf{\partial}^{2}L}{\mathbf{\partial}\dot{q}^{c}\mathbf{\partial}\dot{q}^{b}}.$
(22)

As $q(t)$ is a solution of the Euler-Lagrange equations of motion, we find
that

$l_{a}=c^{b}(q,\dot{q})\left\\{F_{ab}(q,\dot{q})+\frac{\mathbf{\partial}^{2}L}{\mathbf{\partial}q^{a}\mathbf{\partial}\dot{q}^{b}}-\frac{\mathbf{\partial}^{2}L}{\mathbf{\partial}\dot{q}^{a}\mathbf{\partial}q_{b}}+\frac{\mathbf{\partial}\ddot{q}^{c}}{\mathbf{\partial}\dot{q}^{a}}\frac{\mathbf{\partial}^{2}L}{\mathbf{\partial}\dot{q}^{c}\mathbf{\partial}\dot{q}^{b}}\right\\}.$
(23)

This last expression vanishes after the definition of $F_{ab}(q,\dot{q})$ is
used along with the requirement that $c\in\hbox{ker }M_{ab}(q,\dot{q})$. We
then have the following result.

###### Lemma 3

For any vector

$\mathbf{k}=c\cdot\frac{{\partial}\>\>\>}{{\partial}q}+\dot{c}\cdot\frac{{\partial}\>\>\>}{{\partial}\dot{q}},$
(24)

such that $c\in\hbox{ker }M_{ab}$,

$0=F_{ab}c^{b}(q,\dot{q})+M_{ab}\dot{c}^{b}(q,\dot{q}).$ (25)

.

##  III Generators of the Generalized Lie Symmetry for the Euler-Lagrange
Equations of Motion

The generators of the generalized Lie symmetry for both the Euler-Lagrange
equations of motion and the action are best found using the Lagrangian phase
space approach to mechanics. This phase space and its concomitant mathematical
structure provide the tools needed to determine both the generators of the
symmetry and the solutions to the energy equation on which they act. For the
Euler-Lagrange equations of motion this determination was done in [1]. In this
section we will review the Lagrangian phase space approach, establish the
notation used in this paper, and summarize the results obtained in [1] that
are needed here. (We will also take the opportunity to correct typographical
errors made in [1].) Proofs of the majority of the assertions listed in this
section will not be given; the reader is instead referred to [1] where the
proofs and the context of their development can be found.

###  III.1 The Lagrangian Phase space

For a configuration space $\mathbb{Q}$ the Lagrangian phase space
$\mathbb{P}_{L}$ is the tangent space $\mathbb{P}_{L}=\mathbf{T}\mathbb{Q}$,
with the coordinates on $\mathbb{P}_{L}$ denoted as
$\mathfrak{u}=(q^{1},\dots,q^{D},v^{1},\dots v^{D})$. Integral flows on
$\mathbb{P}_{L}$, $t\in[t_{0},\infty)\to\mathfrak{u}(t)\in\mathbb{P}_{L}$
[28], for a set of initial data $\mathfrak{u}_{0}=(q_{0},v_{0})$ are given as
solutions to

$\frac{d\mathfrak{u}}{dt}:=\mathbf{X}(\mathfrak{u}),$ (26)

where $\mathbf{X}$ is a smooth vector field in
$\mathbf{T}\mathbb{P}_{L}=\mathbf{T}(\mathbf{T}\mathbb{Q})$. The two tangent
spaces $\mathbf{T}\mathbb{Q}$ and $\mathbf{T}\mathbb{P}_{L}$ have the bundle
projections: $\tau_{\mathbb{Q}}:\mathbf{T}\mathbb{Q}\to\mathbb{Q}$ and
$\tau_{\mathbf{T}\mathbb{Q}}:\mathbf{T}(\mathbf{T}\mathbb{Q})\to\mathbf{T}\mathbb{Q}$.
They can be used to construct two other projection maps:
$\tau_{\mathbb{Q}}\circ\tau_{\mathbf{T}\mathbb{Q}}:\mathbf{T}(\mathbf{T}\mathbb{Q})\to\mathbb{Q}$
and the prolongation of $\tau_{\mathbf{T}\mathbb{Q}}$ to
$\mathbf{T}(\mathbf{T}\mathbb{Q})$ (see [3] and [28]). This prolongation is
the map
$\mathbf{T}\tau_{\mathbb{Q}}:\mathbf{T}(\mathbf{T}\mathbb{Q})\to\mathbf{T}\mathbb{Q}$,
and is defined by requiring that the two maps
$\tau_{\mathbb{Q}}\circ\tau_{\mathbf{T}\mathbb{Q}}$ and
$\tau_{\mathbb{Q}}\circ\mathbf{T}\tau_{\mathbb{Q}}$ map any point in
$\mathbf{T}(\mathbf{T}\mathbb{Q})$ to the same point in $\mathbb{Q}$. The
vertical subbundle $[\mathbf{T}\mathbb{P}_{L}]^{v}$ of
$\mathbf{T}(\mathbf{T}\mathbb{Q})$ is
$[\mathbf{T}\mathbb{P}_{L}]^{v}=\hbox{ker }\mathbf{T}\tau_{\mathbb{Q}}$ [3]; a
$\mathbf{X}^{v}\in[\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}]^{v}$ above a point
$\mathfrak{u}\in\mathbb{P}_{L}$ is called a vertical vector field. The
horizontal subbundle $[\mathbf{T}\mathbb{P}_{L}]^{q}$ of
$\mathbf{T}(\mathbf{T}\mathbb{Q})$ is
$[\mathbf{T}\mathbb{P}_{L}]^{q}=\hbox{Image }\mathbf{T}\tau_{\mathbb{Q}}$; a
$\mathbf{X}^{q}\in[\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}]^{q}$ is called a
horizontal vector field. Consequently, each
$\mathbf{X}\in\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}$ consists of a
$\mathbf{X}^{q}\in\left[\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{q}$
and a
$\mathbf{X}^{v}\in\left[\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{v}$
with $\mathbf{X}=\mathbf{X}^{q}+\mathbf{X}^{v}$. In terms of local
coordinates,

$\mathbf{X}^{q}:=X^{qa}\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q^{a}},\quad\hbox{and}\quad\mathbf{X}^{v}:=X^{va}\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v^{a}}.$
(27)

Of special interest is the second order Lagrangian vector field
$\mathbf{X}_{L}$. This vector field is the particular solution of Eq.
$(\ref{EnergyE})$ for which $\mathbf{T}\tau_{\mathbb{Q}}\circ\mathbf{X}_{L}$
is the identity on $\mathbf{T}\mathbb{Q}$ (see [28]). In terms of local
coordinates

$\mathbf{X}_{L}=v^{a}\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q^{a}}+X^{va}\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v^{a}}.$
(28)

The space of one-forms on $\mathbb{P}_{L}$ is the cotangent space
$\mathbf{T}^{*}\mathbb{P}_{L}$. For a one-form
$\mathbf{\alpha}\in\mathbf{T}^{*}_{\mathfrak{u}}\mathbb{P}_{L}$, and a vector
field $\mathbf{X}\in\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}$, the dual
prolongation map $\mathbf{T}^{*}\tau_{\mathbb{Q}}$ is defined as

$\langle\mathbf{\alpha}|\mathbf{T}\tau_{\mathbb{Q}}\mathbf{X}\rangle=\langle\mathbf{T}^{*}\tau_{\mathbb{Q}}\mathbf{\alpha}|\mathbf{X}\rangle,$
(29)

after a useful adaptation of Dirac’s bra and ket notation. In addition, for a
general $k$-form $\mathbf{\omega}$ in the $k$-form bundle
$\mathbf{\Lambda}^{k}\left(\mathbb{P}_{L}\right)$,

$\mathbf{\omega}\left(x\right):\mathbf{Y}_{1}\otimes\cdots\otimes\mathbf{Y}_{k}\rightarrow\left\langle\left.\mathbf{\omega}\left(x\right)\right|\mathbf{Y}_{1}\otimes\cdots\otimes\mathbf{Y}_{k}\right\rangle\in\mathbb{R},$
(30)

with $\mathbf{Y}_{j}\in\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}$ for
$j=1,\dots,k$. The vertical one-form subbundle
$[\mathbf{T}^{*}\mathbb{P}_{L}]^{v}$ of $\mathbf{T}^{*}\mathbb{P}_{L}$ is
$[\mathbf{T}^{*}\mathbb{P}_{L}]^{v}:=\hbox{ker
}\mathbf{T}^{*}\tau_{\mathbb{Q}}$; a
$\mathbf{\alpha}_{v}\in[\mathbf{T}^{*}_{\mathfrak{u}}\mathbb{P}_{L}]^{v}$ is
called a vertical one-form. The horizontal one-form subbundle
$[\mathbf{T}^{*}\mathbb{P}_{L}]^{q}$ of $\mathbf{T}^{*}\mathbb{P}_{L}$ is
$[\mathbf{T}^{*}\mathbb{P}_{L}]^{q}=\hbox{Image
}\mathbf{T}^{*}\mathbb{P}_{L}$; a
$\mathbf{\alpha}_{q}\in[\mathbf{T}^{*}_{\mathfrak{u}}\mathbb{Q}]^{q}$ is
called a horizontal one-form. Each one-form
$\mathbf{\varphi}\in\mathbf{T}^{*}_{\mathfrak{u}}\mathbb{P}_{L}$ consists of a
$\mathbf{\varphi}_{q}\in\left[\mathbf{T}_{\mathfrak{u}}^{*}\mathbb{P}_{L}\right]^{q}$
and a
$\mathbf{\varphi}_{v}\in\left[\mathbf{T}_{\mathfrak{u}}^{*}\mathbb{P}_{L}\right]^{q}$
such that $\mathbf{\varphi}=\mathbf{\varphi}_{q}+\mathbf{\varphi}_{v}$. In
terms of local coordinates $\mathbf{\varphi}_{q}:=\varphi_{qa}\
\mathbf{d}q^{a}$ and $\mathbf{\varphi}_{v}:=\varphi_{va}\mathbf{d}v^{a}$.

Following [3, 4], the Lagrangian two-form is defined as
$\mathbf{\Omega}_{L}:=-\mathbf{d}\mathbf{d}_{J}L$, where $\mathbf{d}_{J}$ is
the vertical derivative (see [3]). This two-form can be expressed as
$\mathbf{\Omega}_{L}:=\mathbf{\Omega}_{F}+\mathbf{\Omega}_{M}$ such that for
any $\mathbf{X},\mathbf{Y}\in\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}$.

$\mathbf{\Omega}_{F}(\mathbf{X},\mathbf{Y}):=\mathbf{\Omega}_{L}(\mathbf{T}\tau_{\mathbb{Q}}\mathbf{X},\mathbf{T}\tau_{\mathbb{Q}}\mathbf{Y}),$
(31)

and is thus the horizontal two-form of $\mathbf{\Omega}_{L}$. As
$\mathbf{\Omega}_{M}(\mathbf{X},\mathbf{Y})=\mathbf{\Omega}_{L}(\mathbf{X},\mathbf{Y})-\mathbf{\Omega}_{F}(\mathbf{X},\mathbf{Y})$,
$\mathbf{\Omega}_{M}$ is then a mixed two-form of $\mathbf{\Omega}_{L}$. In
terms of local coordinates,

$\mathbf{\Omega}_{L}=-\mathbf{d{\theta}}_{L},\quad\hbox{where}\quad\mathbf{\theta}_{L}:=\frac{\partial
L}{\partial v^{a}}\mathbf{d}q^{a},$ (32)

while

$\mathbf{\Omega}_{F}:=\frac{1}{2}F_{ab}\mathbf{d}q^{a}\wedge\mathbf{d}q^{b},\
\hbox{and}\ \mathbf{\Omega}_{M}:=M_{ab}\mathbf{d}q^{a}\wedge\mathbf{d}v^{b}.$
(33)

For regular Lagrangians $\mathbf{X}_{L}$ is the unique solution of Eq.
$(\ref{EnergyE})$. For almost regular Lagrangians, on the other hand, this
solution is not unique, but instead depends on

$\ker\>\mathbf{\Omega}_{L}\left(\mathfrak{u}\right):=\left\\{\mathbf{K}\in\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\
|\ \ i_{\mathbf{K}}\mathbf{\Omega}_{L}=0\right\\}.$ (34)

From Section II we expect this kernel to play a role in determining the
generators of the generalized Lie symmetry of both the Euler-Lagrange
equations of motion and the action. Indeed, consider the natural isomorphism
$iso:(t,q,\dot{q},\ddot{q})\in\mathbb{M}^{(2)}\to(t,q,v,X^{va}_{L})$ defined
in [1], and the prolongation $\hbox{{pr} }\mathbf{g}$ of a generator
$\mathbf{g}\in\mathbf{\mathfrak{g}}$ of a generalized Lie symmetry of the
Euler-Lagrange equations of motion. This $\hbox{{pr }}\mathbf{g}$ contains the
vector

$\mathbf{k}=\rho\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q}+\dot{\rho}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}\dot{q}}.$
(35)

The collection of all such vectors has been shown to be involutive (see [1]).
The isomorphism maps $iso:\mathbf{k}\to\mathbf{k}^{\prime}$ where

$\mathbf{k}^{\prime}=\rho\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q}+\dot{\rho}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v}.$
(36)

Then $\mathbf{k}^{\prime}\in\mathbf{T}\mathbb{P}_{L}$, and from Lemma 2,
$\mathbf{k}^{\prime}\in\hbox{ker }\mathbf{\Omega}_{L}(\mathfrak{u})$ as well.
A similar result holds for the generators in $\mathbf{\mathfrak{g}}_{L}$ after
Lemma 1 and Lemma 3 are used.

The two-form $\mathbf{\Omega}_{L}$ gives the lowering map
$\mathbf{\Omega}_{L}^{\flat}:\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\rightarrow\mathbf{T}_{\mathfrak{u}}^{\ast}\mathbb{P}_{L}$,
with $\Omega_{L}^{\flat}\mathbf{X}:=i_{\mathbf{X}}\mathbf{\Omega}_{L}$. This
map consists of
$\Omega_{L}^{\flat}=\Omega_{F}^{\flat}+\Omega_{M}^{v\flat}+\Omega_{M}^{q\flat}$,
with
$\Omega_{F}^{\flat}:\mathbf{X}\in\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\to\left[\mathbf{T}^{*}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{q}$;
$\Omega_{M}^{q\flat}:\mathbf{X}\in\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\to\left[\mathbf{T}^{*}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{q}$;
and
$\Omega_{M}^{v\flat}:\mathbf{X}\in\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\to\left[\mathbf{T}^{*}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{v}$.
In terms of local coordinates,
$\Omega_{F}^{\flat}\mathbf{X}=F_{ab}X^{qa}\mathbf{d}q^{b}$,
$\Omega_{M}^{q\flat}\mathbf{X}=-M_{ab}X^{va}{}\mathbf{d}q^{b}$, and
$\Omega_{M}^{v\flat}\mathbf{X}=M_{ab}X^{qa}\mathbf{d}v^{b}$.

For almost regular Lagrangians
$\ker\>\Omega_{M}^{v\flat}=\mathcal{C}\oplus\left[\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{v}$
while
$\ker\>\Omega_{M}^{q\flat}=\left[\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{q}\oplus\mathcal{G}$.
Here

$\mathcal{C}:=\left\\{\mathbf{C}\in[\mathbf{T}_{q}\mathbb{P}_{L}]^{q}\ |\
i_{\mathbf{C}}\mathbf{\Omega}_{M}=0\right\\},$ (37)

and

$\mathcal{G}:=\left\\{\mathbf{G}\in[\mathbf{T}_{q}\mathbb{P}_{L}]^{v}\ |\
i_{\mathbf{G}}\mathbf{\Omega}_{M}=0\right\\}.$ (38)

As $M_{ab}(\mathfrak{u})$ has constant rank on $\mathbb{P}_{L}$, there exists
a basis,

$\Big{\\{}\mathbf{\mathfrak{z}}_{\left(n\right)}\left(\mathfrak{u}\right)=\left(\mathfrak{z}_{\left(n\right)}^{1}\left(\mathfrak{u}\right),\ldots,\mathfrak{z}_{\left(n\right)}^{D}\left(\mathfrak{u}\right)\right)|\
M_{ab}\left(\mathfrak{u}\right)\mathfrak{z}_{\left(n\right)}^{b}\left(\mathfrak{u}\right)=0,n=1,\ldots,N_{0}\Big{\\}},$
(39)

for $\ker M_{ab}\left(\mathfrak{u}\right)$ at each
$\mathfrak{u}\in\mathbb{P}_{L}$. Spans of both

$\displaystyle\mathcal{C}$ $\displaystyle=$ $\displaystyle\hbox{span
}\left\\{\mathbf{U}^{q}_{(n)}=\mathbf{\mathfrak{z}}_{(n)}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q},n=1,\dots,N_{0}\right\\},\hbox{and}\>$
$\displaystyle\mathcal{G}$ $\displaystyle=$ $\displaystyle\hbox{span
}\left\\{\mathbf{U}^{v}_{(n)}=\mathbf{\mathfrak{z}}_{(n)}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v},n=1,\dots,N_{0}\right\\},$

can then be constructed. Importantly, $\mathcal{G}$ is involutive [5], and
when the rank of $\mathbf{\Omega}_{L}(\mathfrak{u})$ is constant on
$\mathbb{P}_{L}$, $\hbox{ker }\mathbf{\Omega}_{L}(\mathfrak{u})$ is involutive
as well.

Corresponding to $\mathbf{U}^{q}_{(n)}$ and $\mathbf{U}^{v}_{(n)}$ we have the
one-forms $\mathbf{\Theta}^{(m)}_{q}$ and $\mathbf{\Theta}^{(m)}_{v}$ where
$\langle\mathbf{\Theta}^{(m)}_{q}|\mathbf{U}^{q}_{(n)}\rangle=\delta_{(n)}^{(m)}$
and
$\langle\mathbf{\Theta}^{(m)}_{v}|\mathbf{U}^{v}_{(n)}\rangle=\delta_{(n)}^{(m)}$.
Then
$\left[\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{q}=\mathcal{C}\oplus\mathcal{C}_{\perp}$
and
$\left[\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{v}=\mathcal{G}\oplus\mathcal{G}_{\perp}$,
where

$\displaystyle\mathcal{C}_{\perp}:=\bigg{\\{}\mathbf{X}\in\left[\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{q}\
$ $\displaystyle|$ $\displaystyle\
\left\langle\left.\mathbf{\Theta}_{q}^{(n)}\right|\mathbf{X}\right\rangle=0,\>\>n=1,\dots,N_{0}\
\bigg{\\}},\hbox{and}$
$\displaystyle\mathcal{G}_{\perp}:=\bigg{\\{}\mathbf{X}\in\left[\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{v}\
$ $\displaystyle|$ $\displaystyle\
\left\langle\left.\mathbf{\Theta}_{v}^{(n)}\right|\mathbf{X}\right\rangle=0,\>\>n=1,\dots,N_{0}\
\bigg{\\}}.$

The vectors that lie in $\hbox{ker }\mathbf{\Omega}_{L}(\mathfrak{u})$ can be
determined by using the reduced matrix
${F}_{nm}:=\mathfrak{z}_{\left(n\right)}^{a}F_{ab}\mathfrak{z}_{\left(m\right)}^{b}$
to define

$\overline{\mathcal{C}}:=\left\\{\overline{\mathbf{C}}\in\mathcal{C}\
\bigg{|}\
\sum_{m=1}^{N_{0}}\bar{F}_{nm}\overline{C}^{\left(m\right)}=0\right\\}\subset\mathcal{C}.$
(40)

Then,

###### Theorem 4

The vectors $\mathbf{K=K}^{q}+\mathbf{K}^{v}\in\ker\mathbf{\Omega}_{L}$ are
given by,

$\mathbf{K}^{q}=\overline{\mathbf{C}}\mathbf{,\ \
K}^{v}=\mathbf{G+}\widehat{\mathbf{C}},$ (41)

where, $\overline{\mathbf{C}}\in\overline{\mathcal{C}}$,
$\mathbf{G}\in\mathcal{G}$, and $\widehat{\mathbf{C}}\in\mathcal{G}_{\perp}$
is the unique solution of $M_{ab}\widehat{C}^{b}=-F_{ab}\overline{C}^{b}$.

We found in [1] that
$\dim\>\left(\ker\mathbf{\Omega}_{L}\left(\mathfrak{u}\right)\right)=N_{0}+\bar{D},$
where $\ \bar{D}:=\dim\>\overline{\mathcal{C}}\leq N_{0}$ (see [1] for proof).
However, the results of Lemma 3 show that we can construct from any vector
$\mathbf{U}^{q}\in\mathcal{C}$ a vector that lies in the $\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})$, and as dim $(\mathcal{C})=N_{0}$, it
follows that dim $(\hbox{ker }\mathbf{\Omega}_{L}(\mathfrak{u}))=2N_{0}$.

###  III.2 First-order Lagrangian constraints

For singular Lagrangians solutions of the energy equation $\mathbf{X}_{E}$ are
not unique. It is well known that they also do not, in general, exist
throughout $\mathbb{P}_{L}$, but are instead confined to a submanifold of the
space given by Lagrangian constraints.

With $\mathbf{X}_{E}=\mathbf{X}^{q}_{E}+\mathbf{X}^{v}_{E}$, it is convenient
to use the one form $\mathbf{\Psi}$

$\Omega_{M}^{q\flat}\mathbf{X}_{E}^{v}=\mathbf{\Psi}.$ (42)

constructed from the energy equation. The first-order constraint functions are
then
$\gamma_{n}^{\left[1\right]}:=\left\langle\left.\mathbf{\Psi}\right|\mathbf{U}_{\left(n\right)}^{q}\right\rangle=0$
for $n=1,\ldots,N_{0}$. In terms of local coordinates,

$\gamma_{n}^{\left[1\right]}=U_{\left(n\right)}^{qa}\left(\frac{\partial
E}{\partial q^{a}}+F_{ab}v^{b}\right).$ (43)

They may also be expressed [3, 4] as
$\gamma^{[1]}_{n}=\langle\mathbf{d}E|\mathbf{P}_{(n)}\rangle=\mathbf{P}_{(n)}E$
for any basis $\\{\mathbf{P}_{(n)}\\}$ of ker
$\mathbf{\Omega}_{L}(\mathfrak{u})$ for which
$\langle\mathbf{\Theta}^{(m)}_{q}|\mathbf{P}_{(n)}\rangle=\delta^{(m)}_{(n)}$.
In general, $\gamma_{n}^{\left[1\right]}\neq 0$ on $\mathbb{P}_{L}$. Instead,
the condition $\gamma_{n}^{\left[1\right]}=0$ must be imposed, and this in
turn defines a set of submanifolds of $\mathbb{P}_{L}$ given by the collection
$\hbox{C}_{L}^{\left[1\right]}:=\left\\{\gamma_{1}^{\left[1\right]},\ldots,\gamma_{N_{0}}^{\left[1\right]}\right\\}$.
The collection of these surfaces,
$\mathbb{P}_{L}^{\left[1\right]}:=\left\\{\mathfrak{u}\in\mathbb{P}_{L}\ |\ \
\gamma_{n}^{\left[1\right]}\left(\mathfrak{u}\right)=0\,,n=1,\ldots,N_{0}\
\right\\}$ is called the first-order Lagrangian constraint submanifold, and
has $\dim\mathbb{P}_{L}^{\left[1\right]}=2D-I_{\left[1\right]}$. Here
$I_{\left[1\right]}$ is the number of independent functions in
$\hbox{C}_{L}^{\left[1\right]}$ with $I_{\left[1\right]}=\hbox{rank
}\left\\{\mathbf{d}\gamma_{n}^{\left[1\right]}\right\\}\leq N_{0}$.

The constraint one-form

$\mathbf{\beta}[\mathbf{X}_{E}]:=\mathbf{d}E-i_{\mathbf{X}_{E}}\mathbf{\Omega}_{L},$
(44)

was introduced in [1] with the condition $\mathbf{\beta}[\mathbf{X}_{E}]=0$
giving both the solution of the energy equation and the submanifold
$\mathbb{P}_{L}^{[1]}$. As
$\langle\mathbf{\beta}|\mathbf{U}^{q}_{(n)}\rangle=\gamma_{n}^{[1]}$, this
$\mathbf{\beta}[\mathbf{X}_{E}]$ can also be expressed as

$\mathbf{\beta}[\mathbf{X}_{E}]=\sum_{n=1}^{N_{0}}\gamma_{n}^{[1]}\mathbf{\Theta}^{(n)}_{q}.$
(45)

###  III.3 The Generalized Lie Symmetry Group for the Euler-Lagrange Equations
of motion

The generalized Lie symmetry group for $\mathcal{O}(\mathfrak{u}_{0})$ is
determined using

$\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}:=\\{\mathbf{P}\in\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})\ |\
[\mathbf{G},\mathbf{P}]\in\left[\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{v}\>\>\forall\>\>\mathbf{G}\in\mathcal{G}\\},$
(46)

along with the following collection of functions on $\mathbb{P}_{L}$,

$\overline{\mathcal{F}}:=\\{f\in C^{\infty}\hbox{on }\mathbb{P}_{L}\ |\ \
\mathbf{G}f=0\>\>\forall\>\>\mathbf{G}\in\mathcal{G}\\}.$ (47)

This $\overline{\hbox{ker }\mathbf{\Omega}_{L}(\mathfrak{u})}$ is also
involutive.

The following results were proved in [1].

###### Lemma 5

Let $\mathbf{X}\in\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}$ and
$\mathbf{G}\in\mathcal{G}$ such that $[\mathbf{G},\mathbf{X}]\in\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})$. Then
$[\mathbf{G},\mathbf{X}]\in\left[\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{v}$
iff $[\mathbf{G},\mathbf{X}]\in\mathcal{G}$.

It then follows that $[\mathbf{G},\mathbf{P}]\in\mathcal{G}$ for all
$\mathbf{P}\in\overline{\hbox{ker }\mathbf{\Omega}_{L}(\mathfrak{u})}$. As
$\mathcal{G}$ is involutive and as $\mathcal{G}\subset\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})$, $\mathcal{G}\subset\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}$ as well, and thus $\mathcal{G}$ is an
ideal of $\overline{\hbox{ker }\mathbf{\Omega}_{L}(\mathfrak{u})}$.

###### Lemma 6

There exists a choice of basis for ker $\mathbf{\Omega}_{L}(\mathfrak{u})$
that is also a basis of $\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}$.

As $\mathcal{G}$ is an ideal of $\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}$, we may define for any
$\mathbf{P}_{1},\mathbf{P}_{2}\in\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}$ the equivalence relation:
$\mathbf{P}_{1}\sim\mathbf{P}_{2}$ iff
$\mathbf{P}_{1}-\mathbf{P}_{2}\in\mathcal{G}$. The equivalence class,

$\left[\mathbf{P}\right]:=\\{\mathbf{Y}\in\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}\ |\ \mathbf{Y}\sim\mathbf{P}\\},$ (48)

can be constructed along with the quotient space $\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}/\mathcal{G}$. (For the sake of notational
clarity we will suppress the square brackets for equivalence classes when
there is no risk of confusion.) This space is a collection of vectors that lie
in the kernel of $\mathbf{\Omega}_{L}$, but with the vectors in $\mathcal{G}$
removed; $\overline{\hbox{ker }\mathbf{\Omega}_{L}(\mathfrak{u})}/\mathcal{G}$
thereby addresses the first two observations listed at the end of Section
II.2.

We now turn our attention to the third observation. Because the integral flow
$\mathfrak{u}_{\mathbf{X}}(t)$ of any solution $\mathbf{X}$ of the energy
equation must lie on $\mathbb{P}_{L}^{[1]}$, a symmetry transformation of
$\mathfrak{u}_{\mathbf{X}}(t)$ must result in an integral flow
$\mathfrak{u}_{\mathbf{Y}}(t)$ of another solution $\mathbf{Y}$ of the energy
equation, which must also lie on $\mathbb{P}_{L}^{[1]}$. Implementing this
condition is done through $\mathbf{\beta}[\mathbf{X}_{E}]$.

As
$\langle\mathbf{\beta}[\mathbf{X}_{E}]|\mathbf{G}\rangle=\langle\mathbf{d}E|\mathbf{G}\rangle=\mathbf{G}E=0$
for all $\mathbf{G}\in\mathcal{G}$ on $\mathbb{P}_{L}^{[1]}$, the Lie
derivative $\mathfrak{L}_{\mathbf{G}}$ of $\mathbf{\beta}$ along $\mathbf{G}$
is,

$\mathfrak{L}_{\mathbf{G}}\mathbf{\beta}[\mathbf{X}_{E}]=\sum_{n=1}^{N_{0}}\left(\mathbf{G}\gamma_{n}^{[1]}\right)\mathbf{\Theta}^{(n)}_{q}.$
(49)

Given a $\mathbf{P}_{(n)}\in\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}$ such that
$\mathbf{P}_{(n)}=\mathbf{U}_{(n)}^{q}+\widehat{\mathbf{U}}_{(n)}+\mathbf{G}^{\prime}$
with $\mathbf{G}^{\prime}\in\mathcal{G}$,
$\mathbf{G}\gamma_{n}^{[1]}=[\mathbf{G},\mathbf{P}_{(n)}]E+\mathbf{P}_{(n)}\mathbf{G}E$.
But $\mathcal{G}$ is an ideal of $\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}$, and thus $\mathbf{G}\gamma_{n}^{[1]}=0$
on the first-order constraint manifold. It follows that
$\mathfrak{L}_{\mathbf{G}}\mathbf{\beta}=0$ on $\mathbb{P}^{[1]}_{L}$. The
collection of vectors,

$\mathcal{S}\hbox{ym}:=\big{\\{}\mathbf{P}\in\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}/\mathcal{G}\ |\ \
\mathfrak{L}_{\mathbf{P}}\mathbf{\beta}[\mathbf{X}_{E}]=\mathbf{d}\langle\mathbf{\beta}[\mathbf{X}_{E}]|\mathbf{P}\rangle\hbox{
on }\mathbb{P}_{L}^{[1]}\big{\\}},$ (50)

is therefore well defined, and is involutive. It follows that
$\mathbf{P}\in\mathcal{S}\hbox{ym}$ iff
$\langle\mathbf{d}\mathbf{\beta}[\mathbf{X}_{E}]|\mathbf{P}\otimes\mathbf{X}\rangle=0$
for all $\mathbf{X}\in\mathbf{T}\mathcal{P}_{L}$. We are then able to
construct from each $\mathbf{P}\in\mathcal{S}\hbox{ym}$ a one-parameter
subgroup $\mathbf{\sigma}_{\mathbf{P}}(\epsilon,x)$ defined as the solution to

$\frac{d\mathbf{\sigma}_{\mathbf{P}}}{d\epsilon}:=\mathbf{P}\left(\mathbf{\sigma}_{\mathbf{P}}\right),$
(51)

where $\sigma_{\mathbf{P}}(0,\mathfrak{u})=\mathfrak{u}$ for
$\mathfrak{u}\in\mathbb{P}_{L}$. The collection of such subgroups with give
the Lie group $\hbox{Gr}_{\mathcal{S}\hbox{ym}}$.

###  III.4 Euler-Lagrange Solutions of the Energy Equation

We denote the set of general solutions to the energy equation as

$\mathcal{S}\hbox{ol}:=\\{\mathbf{X}_{E}\in\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\
|\ \ i_{\mathbf{X}_{E}}\mathbf{\Omega}_{L}=\mathbf{d}E\hbox{ on
}\mathbb{P}_{L}^{[1]}\\}.$ (52)

If $\mathfrak{u}(t)$ is the integral flow of a vector in
$\mathcal{S}\hbox{ol}$ whose projection onto $\mathbb{Q}$ corresponds to a
solution of the Euler-Lagrange equations of motion, then
$\hbox{Gr}_{\mathcal{S}\hbox{ym}}$ must map one of such flows into another
one. However, while
$\mathfrak{L}_{\mathbf{G}}\mathbf{X}_{L}=[\mathbf{G},\mathbf{X}_{L}]\in\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})$, in general
$\mathfrak{L}_{\mathbf{G}}\mathbf{X}_{L}\notin\mathcal{G}$. The action of
$\sigma_{\mathbf{P}}$ on the flow $\mathfrak{u}_{\mathbf{X}_{L}}$ will in
general result in a flow $\mathfrak{u}_{\mathbf{Y}}$ generated by a
$\mathbf{Y}$ that is not a SOLVF. It need not even be a solution of the energy
equation. By necessity, general solutions of the energy equation must be
considered, leading us to consider the collection of solutions

$\overline{\mathcal{S}\hbox{ol}}:=\\{\overline{\mathbf{X}}_{EL}\in\mathcal{S}\hbox{ol}\
|\
[\mathbf{G},\overline{\mathbf{X}}_{EL}]\in\left[\mathbf{T}_{\mathfrak{u}}\mathbb{P}_{L}\right]^{v}\>\>\>\>\forall\mathbf{G}\in\mathcal{G}\\}.$
(53)

This collection generates the family of integral flows

$\mathcal{O}_{EL}(\mathfrak{u}_{0}):=\bigg{\\{}\mathfrak{u}(t)\ \bigg{|}\
\frac{d\mathfrak{u}}{dt}=\overline{\mathbf{X}}_{EL}(\mathfrak{u}),\overline{\mathbf{X}}_{EL}\in\overline{\mathcal{S}\hbox{ol}},\>\>\hbox{and
}\mathfrak{u}(t_{0})=\mathfrak{u}_{0}\bigg{\\}}.$ (54)

Importantly, if $\mathbf{P}\in\mathcal{S}\hbox{ym}$, then

$i_{[\mathbf{X}_{E},\mathbf{P}]}\mathbf{\Omega}_{L}=i_{\mathbf{P}}\mathbf{d}\mathbf{\beta}[\mathbf{X}_{E}]=0.$
(55)

As such, we find that

###### Lemma 7

$[\overline{\mathbf{X}}_{EL},\mathbf{P}]\in\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}$ for all
$\mathbf{P}\in\mathcal{S}\hbox{ym}$.

It then follows that

###### Theorem 8

$\hbox{Gr}_{\mathcal{S}\hbox{ym}}$ forms a group of symmetry transformations
of $\mathcal{O}_{EL}(\mathfrak{u}_{0})$.

Proof of both assertions can be found in [1].

The generators of the generalized Lie symmetry for
$\mathcal{O}_{EL}(\mathfrak{u}_{0})$ are thus given by $\mathcal{S}\hbox{ym}$.
The corresponding solutions to the Euler-Lagrange equations that have this
symmetry are given by $\overline{\mathcal{S}\hbox{ol}}$, and a vector
$\overline{\mathbf{X}}_{EL}\in\overline{\mathcal{S}\hbox{ol}}$ is called a
second-order, Euler-Lagrange vector field (SOELVF). It has the general form,

$\overline{\mathbf{X}}_{EL}=\overline{\mathbf{X}}_{L}+\sum_{m=1}^{N_{0}}u^{m}(\mathfrak{u})\left[\mathbf{P}_{(m)}\right],$
(56)

where $u^{m}(\mathfrak{u})\in\overline{\mathcal{F}}$ and
$\\{[\mathbf{P}_{(n)}],n=1,\dots,N_{0}\\}$ is a choice of basis for
$\overline{\hbox{ker }\mathbf{\Omega}_{L}(\mathfrak{u})}/\mathcal{G}$. The
vector field $\overline{\mathbf{X}}_{L}$ is constructed from the second order
Lagrangian vector field $\mathbf{X}_{L}$ and vectors in $\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})$ by requiring
$\overline{\mathbf{X}}_{L}\in\overline{\mathcal{S}\hbox{ol}}$. This
construction is described in [1]; we will only need the existence of such a
vector field in this paper.

##  IV Generalized Lie Symmetries of the Action and its Impact on Dynamics

We now turn our attention to the generators of the generalized Lie symmetry of
the action, and the impact this symmetry has on the evolution of dynamical
systems.

###  IV.1 The Generalized Lie Symmetry of the Action

In determining the conditions (as listed in Lemma 1) under which the action
admits a generalized Lie symmetry, the understanding that the action must have
this symmetry for all possible paths on $\mathbb{Q}$ played an essential role.
By necessity, these conditions could only be placed on $\rho_{L}$, and not on
$\dot{\rho}_{L}$; unlike $\rho_{L}$, $\dot{\rho}_{L}$ depends explicitly on
the evolution of a particular path, while the symmetry must hold for all
paths. We note, however, that the family $\mathcal{O}_{EL}$ of trajectories
determined by the Euler-Lagrange equations of motion also consists of paths on
$\mathbb{Q}$, and as such the generalized Lie symmetry of the action is also a
symmetry of $\mathcal{O}_{EL}$. Importantly, how these trajectories evolve
with time is known, and as such, the $\dot{\rho}_{L}$ for a given $\rho_{L}$
is also known for these trajectories. With this understanding, and after
comparing Lemma 1 and the results of Lemma 3 with Lemma 2, we conclude that
the generators of the generalized Lie symmetry of the action must also be
generators of the generalized Lie symmetry of the Euler-Lagrange equations of
motion. This leads us to consider the following collection of vectors.

$\mathcal{S}\hbox{ym}\mathcal{L}=\\{\mathbf{P}\in\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}/\mathcal{G}\ |\
\gamma_{\mathbf{P}}^{[1]}=\langle\mathbf{\beta}|\mathbf{P}\rangle=0\hbox{ on
}\mathbb{P}_{L}\\}.$ (57)

We will also need $N_{\mathcal{S}\hbox{ym}\mathcal{L}}=\hbox{dim
}(\mathcal{S}\hbox{ym}\mathcal{L})$ in the following.

###### Lemma 9

$\mathcal{S}\hbox{ym}\mathcal{L}\subset\mathcal{S}\hbox{ym}$.

Proof. Let $\\{\mathbf{P}_{(l)},l=1,\dots,N_{0}\\}$ be a basis of
$\overline{\hbox{ker }\mathbf{\Omega}_{L}(\mathfrak{u})}/\mathcal{G}$ such
that $\mathbf{P}_{(l)}\in\mathcal{S}\hbox{ym}\mathcal{L}$ for
$l=1,\dots,N_{\mathcal{S}\hbox{ym}\mathcal{L}}$. We may choose the basis of
$\mathcal{C}$ such that
$\langle\mathbf{\Theta}^{(m)}_{q}|\mathbf{P}_{(l)}\rangle=\delta^{(m)}_{(l)}$.
Then for any $\mathbf{P}_{(n)}\in\mathcal{S}\hbox{ym}$, we see from Eq.
$(\ref{beta})$ that,

$\langle\mathbf{d}\mathbf{\beta}|\mathbf{P}_{(n)}\otimes\mathbf{Y}\rangle=\sum_{m=1}^{N_{0}}\left(\langle\mathbf{d}\gamma_{m}^{[1]}|\mathbf{P}_{(n)}\rangle\langle\mathbf{\Theta}^{(m)}_{q}|\mathbf{Y}\rangle-\langle\mathbf{d}\gamma_{m}^{[1]}|\mathbf{Y}\rangle\langle\mathbf{\Theta}^{(m)}_{q}|\mathbf{P}_{(n)}\rangle+\gamma_{m}^{[1]}\langle\mathbf{d}\mathbf{\Theta}^{(m)}_{q}|\mathbf{P}_{(n)}\otimes\mathbf{Y}\rangle\right),$
(58)

for any $\mathbf{Y}\in\mathbf{T}\mathbb{P}_{L}$. The last term vanishes on the
first-order constraint manifold $\mathbb{P}_{L}^{[1]}$, while for the second
term,
$\langle\mathbf{d}\gamma_{m}^{[1]}|\mathbf{Y}\rangle\langle\mathbf{\Theta}^{(m)}_{q}|\mathbf{P}_{(n)}\rangle=\langle\mathbf{d}\gamma_{n}^{[1]}|\mathbf{Y}\rangle\delta^{(m)}_{(n)}$.
But as $\mathbf{P}_{(n)}\in\mathcal{S}\hbox{ym}\mathcal{L}$,
$\gamma_{n}^{[1]}=0$ on $\mathbb{P}_{L}$, and this term vanishes as well.
Finally, for the first term,
$\langle\mathbf{d}\mathbf{\gamma}_{m}^{[1]}|\mathbf{P}_{(n)}\rangle=\mathbf{P}_{(n)}\mathbf{P}_{(m)}E=[\mathbf{P}_{(n)},\mathbf{P}_{(m)}]E+\mathbf{P}_{(m)}\mathbf{P}_{(n)}E$.
But $\gamma_{n}^{[1]}=\mathbf{P}_{(n)}E=0$ on $\mathbb{P}_{L}$, while
$\overline{\hbox{ker }\mathbf{\Omega}(\mathfrak{u})}$ is involutive. There
then exists a $\mathbf{P}_{(nm)}\in\overline{\hbox{ker
}\mathbf{\Omega}(\mathfrak{u})}$ such that
$\mathbf{P}_{(nm)}=[\mathbf{P}_{(n)},\mathbf{P}_{(m)}]$. As
$\mathbf{P}_{(nm)}E:=\gamma_{(nm)}^{[1]}$, this $\gamma_{(nm)}^{[1]}$ must be
a linear combination of first-order constraint functions, and they also vanish
on $\mathbb{P}_{L}^{[1]}$. It then follows that
$\langle\mathbf{d}\mathbf{\beta}|\mathbf{P}_{(n)}\otimes\mathbf{Y}\rangle=0$
on $\mathbb{P}_{L}^{[1]}$, and $\mathbf{P}_{(n)}\in\mathcal{S}\hbox{ym}$.

If $\mathbf{P}_{1},\mathbf{P}_{2}\in\mathcal{S}\hbox{ym}\mathcal{L}$, then
$\gamma^{[1]}_{[\mathbf{P}_{1},\mathbf{P}_{2}]}=\mathbf{P}_{1}\mathbf{P}_{2}E-\mathbf{P}_{2}\mathbf{P}_{1}E=\mathbf{P_{1}}\gamma_{\mathbf{P}_{2}}-\mathbf{P_{2}}\gamma_{\mathbf{P}_{1}}=0$,
and thus $\mathcal{S}\hbox{ym}\mathcal{L}$ is involutive. Then for each
$\mathbf{P}\in\mathcal{S}\hbox{ym}\mathcal{L}$ we once again have the one-
parameter subgroup
$\sigma^{\mathcal{S}\hbox{ym}\mathcal{L}}_{\mathbf{P}}(\epsilon,\mathfrak{u})$
define as the integral flow of

$\frac{\mathbf{d}\sigma^{\mathcal{S}\hbox{ym}\mathcal{L}}_{\mathbf{P}}}{\mathbf{d}\epsilon}:=\mathbf{P},$
(59)

with
$\sigma^{\mathcal{S}\hbox{ym}\mathcal{L}}_{\mathbf{P}}(0,\mathfrak{u})=\mathfrak{u}$
for $\mathfrak{u}\in\mathbb{P}_{L}$. The collection of such subgroups gives
the Lie group $\hbox{Gr}_{\mathcal{S}{ym}\mathcal{L}}$. As
$\mathcal{S}{ym}\mathcal{L}\subset\mathcal{S}{ym}$,
$\hbox{Gr}_{\mathcal{S}{ym}\mathcal{L}}$ is a Lie subgroup of
$\hbox{Gr}_{\mathcal{S}{ym}}$. It then follows from Theorem 8 that
$\hbox{Gr}_{\mathcal{S}\hbox{ym}\mathcal{L}}$ also forms a group of symmetry
transformations of $\mathcal{O}_{EL}(\mathfrak{u}_{0})$. As the family
$\mathcal{O}_{EL}(\mathfrak{u}_{0})$ of trajectories are paths on
$\mathbb{Q}$, and as the symmetry transformation of the action must be the
same for all paths on $\mathbb{Q}$, it also follows that,

###### Theorem 10

$\hbox{Gr}_{\mathcal{S}\hbox{ym}\mathcal{L}}$ forms the group of symmetry
transformations of the action $S$.

###  IV.2 Symmetries and Dynamics

While $\mathcal{O}_{EL}(\mathfrak{u}_{0})$ gives the family of integral flows
on which both $\hbox{Gr}_{\mathcal{S}\hbox{ym}}$ and
$\hbox{Gr}_{\mathcal{S}\hbox{ym}\mathcal{L}}$ act, a general flow in
$\mathcal{O}_{EL}(\mathfrak{u}_{0})$ need not be confined to
$\mathbb{P}_{L}^{\left[1\right]}$, and yet this is the submanifold on which
the solutions $\overline{\mathbf{X}}_{EL}\in mathcal{S}{\mathrm{ol}}$ of the
energy equations exist. In such cases it is necessary to jointly choose a
SOELVF $\overline{\mathbf{X}}_{EL}$ and a submanifold of
$\mathbb{P}_{L}^{\left[1\right]}$ on which the resultant flow
$\mathfrak{u}_{\overline{\mathbf{X}}_{EL}}$ will be confined. This is done
through the implementation of a constraint algorithm, one of which was
proposed in [1]. In that paper the product of this algorithm was the most that
could be said about the general structure of SOELVFs that have integral flow
fields which lie on $\mathbb{P}_{L}^{[1]}$. Here, with the results obtained in
Section IV.1, we can say much more, and we will see that the presence of a
generalized Lie symmetry of the action greatly restricts the structure of the
SOELVFs that such systems can have.

Following [1], we introduce for a
$\overline{\mathbf{X}}_{EL}\in\overline{\mathcal{S}\hbox{ol}}$ the notation

$\overline{\mathbf{X}}_{EL}^{[1]}:=\overline{\mathbf{X}}_{EL},\>\overline{\mathbf{X}}_{L}^{[1]}:=\overline{\mathbf{X}}_{L},\>\mathbf{P}^{[1]}_{(n)}:=\mathbf{P}_{(n)},\>u_{[1]}^{m}:=u^{m},\>N_{0}^{[1]}:=N_{0},$
(60)

when the constraint algorithm is implemented, with the superscript $[1]$
denoting the first iteration of this algorithm. (This notation is only used in
this section.) In addition, we choose
$\mathbf{P}^{[1]}_{(n)}\in\mathcal{S}\hbox{ym}\mathcal{L}$ for
$n=1,\dots,N_{\mathcal{S}\hbox{ym}\mathcal{L}}$.

For the integral flow field of $\overline{\mathbf{X}}_{EL}$ to lie on
$\mathbb{P}_{L}^{[1]}$,

$\mathfrak{L}_{\overline{\mathbf{X}}_{EL}}\mathbf{\beta}=0,$ (61)

which reduces to $\mathfrak{L}_{\overline{\mathbf{X}}_{EL}}\gamma_{n}^{[1]}=0$
on $\mathbb{P}_{L}^{[1]}$. This is called the constraint condition. As both
$u_{[1]}^{n},\gamma^{[1]}_{n}\in\overline{\mathcal{F}}$,
$\left[\mathbf{P}_{(n)}^{[1]}\right]\gamma^{[1]}_{m}=\mathbf{P}_{(n)}\gamma^{[1]}_{m}$,
and after making use of the general form of a SOELVF given in Eq.
$(\ref{EL})$, Eq. $(\ref{stable})$ reduces to

$\sum_{m=1}^{N_{0}}\Gamma^{[1]}_{nm}u^{m}_{[1]}=-\left\langle\mathbf{d}\gamma^{[1]}_{n}\Big{|}\overline{\mathbf{X}}^{[1]}_{L}\right\rangle,\>\hbox{with
}\Gamma^{[1]}_{nm}:=\left\langle\mathbf{d}\gamma^{[1]}_{n}\Big{|}\mathbf{P}^{[1]}_{(m)}\right\rangle.$
(62)

Since
$\left\langle\mathbf{d}\gamma^{[1]}_{n}\Big{|}\mathbf{P}^{[1]}_{(m)}\right\rangle=\mathbf{P}^{[1]}_{(m)}\mathbf{P}^{[1]}_{(n)}E=[\mathbf{P}^{[1]}_{(m)},\mathbf{P}^{[1]}_{(n)}]E+\Gamma^{[1]}_{mn}$.
But $\overline{\hbox{ker }\mathbf{\Omega}_{L}(\mathfrak{u})}$ is involutive,
and thus $[\mathbf{P}^{[1]}_{(m)},\mathbf{P}^{[1]}_{(n)}]E$ is a linear
combination of first-order Lagrangian constraints. As these constraints
vanishes on $\mathbb{P}_{L}^{[1]}$, $\Gamma^{[1]}_{nm}=\Gamma^{[1]}_{mn}$ on
the first-order constraint manifold.

Next, when $n=1,\dots,N_{\mathcal{S}\hbox{ym}\mathcal{L}}$,
$\mathbf{P}_{(n)}^{[1]}\in\mathcal{S}\hbox{ym}\mathcal{L}$, and
$\gamma_{n}^{[1]}=0$. Thus, $\Gamma^{[1]}_{nm}=0$ when
$n=1,\dots,N_{\mathcal{S}\hbox{ym}\mathcal{L}}$, and as $\Gamma^{[1]}_{nm}$ is
a symmetric matrix on $\mathbb{P}_{L}^{[1]}$, $\Gamma^{[1]}_{mn}=0$ for these
values of $n$ as well. Thus while $\Gamma^{[1]}_{nm}$ is a $N_{0}\times N_{0}$
matrix, the only nonzero components of this matrix lie in the
$\left(N_{0}-N_{\mathcal{S}\hbox{ym}\mathcal{L}}\right)\times\left(N_{0}-N_{\mathcal{S}\hbox{ym}\mathcal{L}}\right)$
submatrix
$\bar{\Gamma}^{[1]}_{\overline{n}\,\overline{m}}:=\left\langle\mathbf{d}\gamma^{[1]}_{\overline{n}+N_{\mathcal{S}\hbox{ym}\mathcal{L}}}\bigg{|}\mathbf{P}^{[1]}_{\overline{m}+N_{\mathcal{S}\hbox{ym}\mathcal{L}}}\right\rangle$
where
$\overline{n},\overline{m}=1,\dots,N_{0}-N_{\mathcal{S}\hbox{ym}\mathcal{L}}$.
As
$\left\langle\mathbf{d}\gamma^{[1]}_{n}\Big{|}\overline{\mathbf{X}}^{[1]}_{L}\right\rangle=0$
as well when $n=1,\dots,N_{\mathcal{S}\hbox{ym}\mathcal{L}}$, Eq.
$(\ref{first-order})$ reduces to

$\sum_{\overline{m}=1}^{N_{0}-N_{\mathcal{S}\hbox{ym}\mathcal{L}}}\bar{\Gamma}^{[1]}_{\overline{n}\bar{m}}u^{\overline{m}+N_{\mathcal{S}\hbox{ym}\mathcal{L}}}_{[1]}=-\left\langle\mathbf{d}\gamma^{[1]}_{\overline{n}+N_{\mathcal{S}\hbox{ym}\mathcal{L}}}\Big{|}\overline{\mathbf{X}}^{[1]}_{L}\right\rangle.$
(63)

It is then readily apparent that the $N_{\mathcal{S}\hbox{ym}\mathcal{L}}$
arbitrary functions $u^{m}_{[1]}$ for
$m=1,\dots,N_{\mathcal{S}\hbox{ym}\mathcal{L}}$ are not determined at this
iteration, while $r^{[1]}=\hbox{rank
}\bar{\Gamma}^{[1]}_{\overline{n}\,\overline{m}}$ of the $u^{m}_{[1]}$ for
$m>N_{\mathcal{S}\hbox{ym}\mathcal{L}}$ are. There are then
$N_{0}^{[2]}:=N_{0}^{[1]}-r^{[1]}$ second-order Lagrangian constraint
functions

$\gamma^{[2]}_{n_{[2]}}:=\left\langle\mathbf{d}\gamma^{[1]}_{n_{[2]}}\Big{|}\overline{\mathbf{X}}_{L}^{[1]}\right\rangle,n_{[2]}=1,\cdots,N_{0}^{[2]},$
(64)

with the conditions $\gamma^{[2]}_{n_{[2]}}=0$ imposed if necessary. In
general there will be $I_{[2]}:=\hbox{rank
}\left\\{\mathbf{d}\gamma^{[1]}_{n_{[1]}},\mathbf{d}\gamma^{[2]}_{n_{[2]}}\right\\}$
independent functions in
$\hbox{C}^{[2]}_{L}:=\hbox{C}^{[1]}\cup\left\\{\gamma^{[2]}_{n_{[2]}}\ |\
n_{[2]}=1,\dots,N_{0}^{[2]}\right\\}$, and $\mathbb{P}_{L}^{[1]}$ is reduced
to the second-order constraint submanifold,

$\mathbb{P}_{L}^{[2]}:=\left\\{\mathfrak{u}\in\mathbb{P}_{L}^{[1]}\ \Big{|}\
\gamma^{[2]}_{[n_{2}]}(\mathfrak{u})=0,n_{[2]}=1,\dots,N_{0}^{[2]}\right\\},$
(65)

where dim $\mathbb{P}^{[2]}_{L}=2D-I_{[2]}$. At this point, there are two
possibilities. If $I_{[2]}=I_{[1]}$ or $I_{[2]}=2D$, the iterative process
stops, and no new Lagrangian constraints are introduced. If not, the process
continues.

For the second iteration in the constraint algorithm, we choose a basis
$\left\\{\mathbf{P}_{(n)}^{[2]}\right\\}$ for $\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}/\mathcal{G}$ and the arbitrary functions
$\left\\{u_{[2]}^{m}\right\\}$ such that for $m=1,\dots,N_{0}^{[2]}$,
$u_{[2]}^{m}$ are linear combinations of $u_{[1]}^{m}$ that lie in the kernel
$\Gamma^{[1]}_{nm}$. We once again require that
$\mathbf{P}_{(n)}^{[2]}\in\mathcal{S}\hbox{ym}\mathcal{L}$ for
$n=1,\dots,N_{\mathcal{S}\hbox{ym}\mathcal{L}}$. Then

$\overline{\mathbf{X}}_{EL}^{[2]}=\overline{\mathbf{X}}_{L}^{[2]}+\sum_{m=1}^{N_{0}^{[2]}}u_{[2]}^{m}\left[\mathbf{P}_{(m)}^{[2]}\right],$
(66)

with

$\overline{\mathbf{X}}_{L}^{[2]}=\overline{\mathbf{X}}_{L}^{[1]}+\sum_{m=N_{0}^{[2]}+1}^{N_{0}^{[1]}}u^{m}_{[2]}\left[\mathbf{P}^{[2]}_{(m)}\right].$
(67)

Here, the functions $u^{m}_{[2]}$ for $m=N_{0}^{[2]}+1,\dots,N_{0}^{[1]}$ have
been determined through the constraint analysis of $\gamma^{[1]}_{n}$.

As shown in [1], $\mathbf{G}u_{[1]}^{m}=0$. Similarly,
$\mathbf{G}\gamma^{[2]}_{n}=\mathfrak{L}_{[\mathbf{G},\overline{\mathbf{X}}_{EL}]}\mathbf{d}\gamma^{[2]}_{n}=0$.
Clearly $\gamma^{[2]}_{n}\in\overline{\mathcal{F}}$ and we may require
$u^{m}_{[2]}\in\overline{\mathcal{F}}$ as well. It then follows that
$\left[\mathbf{P}^{[2]}_{(n)}\right]\gamma^{[2]}_{m}=\mathbf{P}^{[2]}_{(n)}\gamma^{[2]}_{m}$,
and imposing Eq. $(\ref{stable})$ on $\gamma^{[2]}_{n}$, gives

$\sum_{m=1}^{N_{0}^{[2]}}\Gamma^{[2]}_{nm}u^{m}_{[2]}=-\left\langle\mathbf{d}\gamma^{[2]}_{n}\Big{|}\overline{\mathbf{X}}^{[2]}_{L}\right\rangle,\>\hbox{where
}\Gamma^{[2]}_{nm}:=\left\langle\mathbf{d}\gamma^{[2]}_{n}\Big{|}\mathbf{P}^{[2]}_{(m)}\right\rangle,\>\>n=1,\dots,N_{0}^{[2]}.$
(68)

Once again, $\Gamma^{[2]}_{nm}=\Gamma^{[2]}_{mn}$, but now on the constraint
manifold $\mathbb{P}^{[2]}_{L}$. Moreover, since
$\gamma^{[2]}_{n}=\gamma^{[1]}_{n}=0$ for
$n=1,\dots,N_{\mathcal{S}\hbox{ym}\mathcal{L}}$,
$\Gamma^{[2]}_{nm}=0=\Gamma^{[2]}_{mn}$, and
$\left\langle\mathbf{d}\gamma_{n}^{[2]}\Big{|}\overline{\mathbf{X}}^{[2]}_{L}\right\rangle=0$.
There is once again a reduction of Eq. $(\ref{second-order})$, and we are left
with

$\sum_{\overline{m}=1}^{N_{0}^{[2]}-N_{\mathcal{S}\hbox{ym}\mathcal{L}}}\overline{\Gamma}^{[2]}_{\overline{n}\,\overline{m}}u^{\overline{m}+N_{\mathcal{S}\hbox{ym}\mathcal{L}}}_{[2]}=-\langle\mathbf{d}\gamma^{[2]}_{\overline{n}+N_{\mathcal{S}\hbox{ym}\mathcal{L}}}|\overline{\mathbf{X}}^{[2]}_{L}\rangle.$
(69)

where
$\bar{\Gamma}^{[2]}_{\overline{n}\,\overline{m}}:=\left\langle\mathbf{d}\gamma^{[2]}_{\overline{n}+N_{\mathcal{S}\hbox{ym}\mathcal{L}}}\bigg{|}\mathbf{P}^{[2]}_{(\overline{m}+N_{\mathcal{S}\hbox{ym}\mathcal{L}})}\right\rangle$.
As before, the $N_{\mathcal{S}\hbox{ym}\mathcal{L}}$ arbitrary functions
$u^{m}_{[2]}$ are not determined, while $r^{[2]}:=\hbox{rank
}\bar{\Gamma}^{[2]}_{\overline{n}\,\overline{m}}$ of the remaining
$u^{m}_{[2]}$ for $m>N_{\mathcal{S}\hbox{ym}\mathcal{L}}$ are. There are now
$N^{[3]}_{0}=N_{0}^{[2]}-r^{[2]}$ third-order Lagrangian constraint functions,

$\gamma^{[3]}_{n_{[3]}}=\left\langle\mathbf{d}\gamma^{[2]}_{n_{[3]}}\Big{|}\overline{\mathbf{X}}^{[2]}_{L}\right\rangle,\>n_{[3]}=1,\dots,N_{0}^{[3]},$
(70)

with the conditions $\gamma^{[3]}_{n_{[3]}}=0$ imposed if necessary. With

$I_{[3]}:=\hbox{rank
}\left\\{\mathbf{d}\gamma_{n_{[1]}}^{[1]},\mathbf{d}\gamma_{n_{[2]}}^{[2]},\mathbf{d}\gamma_{n_{[3]}}^{[3]}\right\\},$
(71)

independent functions in
$\hbox{C}^{[3]}_{L}:=\hbox{C}_{L}^{[2]}\cup\left\\{\gamma_{n_{[3]}}^{[3]},n_{[3]}=1,\dots,N_{0}^{[3]}\right\\}$,
we now have the third-order constraint submanifold,

$\mathbb{P}_{L}^{[3]}:=\left\\{\mathfrak{u}\in\mathbb{P}^{[2]}_{L}\ \Big{|}\
\gamma^{[3]}_{n_{[3]}}(\mathfrak{u})=0,n_{[3]}=1,\dots,N_{0}^{[3]}\right\\}.$
(72)

Once again, the process stops when $I_{[3]}=I_{[2]}$ or $I_{[3]}=2D$. However,
if $I_{[2]}<I_{[3]}<2D$, the process continues until at the $n_{F}$-iteration
when either $I_{[n_{F}]}=I_{[n_{F}]-1}$ or $I_{[n_{F}]}=2D$.

Following [1], the end result of this algorithm is

  1. 1.

A submanifold $\mathbb{P}^{[n_{F}]}_{L}\subset\mathbb{P}_{L}$ on which
dynamics takes place.

  2. 2.

A collection $\hbox{C}_{L}^{[n_{F}]}\subset\overline{\mathcal{F}}$ of
constraint functions of order $1$ to $n_{F}$.

  3. 3.

A second-order, Euler-Lagrange vector field

$\overline{\mathbf{X}}_{EL}^{[n_{F}]}=\overline{\mathbf{X}}_{L}^{[n_{F}]}+\sum_{m=1}^{N^{[n_{F}]}_{0}}u^{m}_{[n_{F}]}(\mathfrak{u})\left[\mathbf{P}_{(m)}^{[n_{F}]}\right],$
(73)

with $N_{0}^{[n_{F}]}\geq N_{\mathcal{S}\hbox{ym}\mathcal{L}}$ arbitrary
functions $u^{m}_{[n_{F}]}(\mathfrak{u})\in\overline{\mathcal{F}}$ for
$m=1,\dots,N_{0}^{[n_{F}]}$, and

$\overline{\mathbf{X}}_{L}^{[n_{F}]}=\overline{\mathbf{X}}_{L}^{[1]}+\sum_{m=N_{0}^{[n_{F}]}+1}^{N_{0}^{[1]}}u^{m}_{[n_{F}]}(\mathfrak{u})\left[\mathbf{P}_{(m)}^{[n_{F}]}\right],$
(74)

where the $N_{0}^{[1]}-N_{0}^{[n_{F}]}$ functions
$u^{m}_{[n_{F}]}(\mathfrak{u})\in\overline{\mathcal{F}}$,
$m=N_{0}^{[n_{F}]}+1,\dots,N_{0}^{[1]}$, have been uniquely determined through
the constraint algorithm.

We assume that the rank of $\Gamma^{[l]}_{nm}$ is constant on $\mathbb{P}_{L}$
for each $l=1,\dots,n_{F}$, and that $\mathbb{P}^{[n_{F}]}_{L}$ is non-empty.

The end result of the constraint algorithm
$\overline{\mathbf{X}}_{EL}^{[n_{F}]}$ is still a SOELVF, and we define the
collection of such vector fields as

$\overline{\mathcal{S}\hbox{ol}}_{\mathbb{P}^{[n_{F}]}_{L}}:=\\{\overline{\mathbf{X}}_{EL}\in\overline{\mathcal{S}\hbox{ol}}\>\>|\>\mathfrak{L}_{\overline{\mathbf{X}}_{EL}}\mathbf{\beta}=0\\}.$
(75)

Importantly, $\hbox{dim
}\overline{\mathcal{S}\hbox{ol}}_{\mathbb{P}^{[n_{F}]}_{L}}\geq
N_{\mathcal{S}\hbox{ym}\mathcal{L}}$.

##  V The Generalized Lie Symmetries of Three Dynamical Systems

Three examples of dynamical systems with almost regular Lagrangians were
introduced in [1]. In that paper the focus of these examples was on the
explicit construction of the dynamical structures needed to describe and
predict motion in the Lagrangian phase space, and to show that these
structures are projectable to the Hamiltonian phase space. We return to these
examples here, but with the focus now being on the generalized Lie symmetries
of each, and the application of the results we have found in this paper. In
particular, we are in interested in the dimensionality of the symmetry groups
for each of the systems as compared to the dimensionality of
$\overline{\mathcal{S}\hbox{ol}}_{\mathbb{P}^{[n_{F}]}_{L}}$ of each. A
summary of our results can be found in Table 1

###  V.1 A Lagrangian With and Without a Generalized Lie Symmetry

Whether the action

$S_{1}:=\int\left[\frac{1}{2}m\left(\frac{d\widehat{q}}{dt}\right)^{2}-V(q^{a})\right]dt,$
(76)

with $|q|=\sqrt{q^{a}q_{a}}$ and $\widehat{q}^{a}:=q^{a}/|q|$, $a=1,\dots,D$,
has a generalized Lie symmetry depends on the choice of potential $V(q)$. With
one choice both the Lagrangian and the Euler-Lagrange equations of motion have
a generalized gauge symmetry; with a second choice the equations of motion has
a generalized Lie symmetry while the Lagrangian does not; and with a third
choice neither the action nor the equations of motion have a symmetry.
Irrespective of the choice of $V(q)$, however, $L$ is singular, demonstrating
that while all actions with a generalized Lie symmetry have a singular
Lagrangian, not all singular Lagrangians have a generalized Lie symmetry.

Defining $\Pi_{ab}(q):=\delta_{ab}-\widehat{q}_{a}\widehat{q}_{b}$, we find

$\mathbf{\Omega}_{M}=\frac{m}{|q|^{2}}\Pi_{ab}(q)\mathbf{d}q^{a}\wedge\mathbf{d}v^{b},\qquad\mathbf{\Omega}_{F}=\frac{m}{|q|^{3}}\left(\widehat{q}\cdot\mathbf{d}q\right)\wedge\left(v\cdot\Pi(q)\cdot\mathbf{d}q\right).$
(77)

Then $\mathcal{C}$ and $\mathcal{G}$ are spanned by
$\mathbf{U}^{q}_{(1)}=\widehat{q}\cdot\mathbf{\partial}/\mathbf{\partial}q$
and
$\mathbf{U}^{v}_{(1)}=\widehat{q}\cdot\mathbf{\partial}/\mathbf{\partial}v$,
respectively, while $\overline{\hbox{ker }\mathbf{\Omega}_{L}(\mathfrak{u})}$
is spanned by $\mathbf{U}^{v}_{(1)}$ and

$P_{(1)}=\widehat{q}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q}+\frac{1}{|q|}v\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v}.$
(78)

That $\hbox{dim }(\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}/\mathcal{G})=1$ then follows.

The energy is

$E=\frac{1}{2}\frac{m}{|q|^{2}}v\cdot\Pi(q)\cdot v+V(q),$ (79)

and there is only one first-order Lagrangian constraint,

$\gamma^{[1]}=\mathbf{U}^{q}_{(1)}V,$ (80)

so that
$\mathbf{\beta}[\mathbf{X}_{EL}]=\gamma^{[1]}\mathbf{\Theta}^{(1)}_{q}$, where
$\mathbf{\Theta}^{(1)}_{q}=\widehat{q}\cdot\mathbf{d}q$. Using Eq.
$(\ref{g-con})$,

$\mathfrak{L}_{\mathbf{P}_{(1)}}\mathbf{\beta}=\mathbf{d}\left[\mathbf{U}^{q}_{(1)}V\right]-\frac{1}{|q|^{2}}\widehat{q}\cdot\frac{\partial\>\>\>}{\partial
q}\left(\Pi_{a}^{\>\>b}(q)\frac{\partial
V}{\partial\widehat{q}^{b}}\right)\mathbf{d}q^{a}.$ (81)

Whether or not $\mathcal{S}\hbox{ym}$ or $\mathcal{S}\hbox{ym}\mathcal{L}$ is
empty therefore depends on the symmetries of $V(q)$, as we would expect.

It was found in [1] that

$\overline{\mathbf{X}}_{L}=v\cdot{\Pi(q)}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q}+\frac{(\widehat{q}\cdot
v)}{|q|}v\cdot\frac{\mathbf{\partial}\>\>\>}{\partial
v}-\frac{|q|^{2}}{m}\frac{\partial V}{\partial
q}\cdot\Pi(q)\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v},$ (82)

and a general SOELVF is given by
$\overline{\mathbf{X}}_{EL}=\overline{\mathbf{X}}_{L}+u(\mathfrak{u})\left[\mathbf{P}_{(1)}\right]$,
where $u(\mathfrak{u})\in\overline{\mathcal{F}}$. As the constraint algorithm
gives

$\mathfrak{L}_{\overline{\mathbf{X}}_{EL}}\gamma^{[1]}=v\cdot\Pi\cdot\frac{\partial\gamma^{[1]}}{\partial
q}+u(\mathfrak{u})\mathbf{U}^{q}_{(1)}\gamma^{[1]},$ (83)

whether or not $u(\mathfrak{u})$ (which in turn determines the dimensionality
of $\overline{\mathcal{S}\hbox{ol}}_{\mathbb{P}_{L}^{[n_{f}]}}$) is determined
by the constraint condition also depends on the symmetries of $V(q)$.

There are three cases to consider.

The symmetric potential

For $\mathbf{P}_{(1)}$ to generate a generalized Lie symmetry of the Euler-
Lagrange equations of motion,

$0=\frac{1}{|q|^{2}}\widehat{q}\cdot\frac{\partial\>\>\>}{\partial
q}\left(\Pi_{a}^{\>\>b}(q)\frac{\partial V}{\partial\widehat{q}^{b}}\right),$
(84)

and as such the potential must satisfy

$\frac{\partial V}{\partial\widehat{q}^{a}}=\frac{\partial
V_{AS}(\widehat{q}^{a})}{\partial\widehat{q}^{a}},$ (85)

where $V_{AS}$ is a function of $\widehat{q}^{a}$ only. It follows that
$\mathbf{P}_{(1)}$ generates a generalized Lie symmetry iff
$V(q^{a})=V_{Sph}(|q|)+V_{AS}(\widehat{q}^{a})$, where $V_{Sph}$ is a function
of $|q|$ only. For this potential, $\mathcal{S}\hbox{ym}$ is one-dimensional,
and is spanned by $\mathbf{P}_{(1)}$.

The constraint condition Eq. $(\ref{stab})$ for this potential reduces to

$0=u(\mathfrak{u})\frac{d^{2}V_{Sph}(q)}{d|q|^{2}},$ (86)

which must be satisfied on $\mathbb{P}_{L}^{[1]}$. There are two
possibilities.

Case 1: $\frac{d^{2}V_{Sph}}{d|q|^{2}}=0$.

Then $V_{Sph}(|q|)=a|q|+b$, but since

$\gamma^{[1]}=\frac{dV_{Sph}}{d|q|}=a,$ (87)

the condition $\gamma^{[1]}=0$ requires $a=0$. It then follows that
$\gamma^{[1]}=0$ on $\mathbb{P}_{L}$, and thus
$\mathcal{S}\hbox{ym}\mathcal{L}$ is one-dimensional; it also is spanned by
$\mathbf{P}_{(1)}$. The potential is then $V(q)=b+V_{AS}(\widehat{q}^{a})$,
and the Lagrangian is invariant under the transformation $q^{a}\to\alpha
q^{a}$, where $\alpha$ is an arbitrary, nonvanishing function on
$\mathbb{P}_{L}$. This Lagrangian therefore has a local conformal symmetry.
Importantly, the function $u(\mathfrak{u})$ is not determined, and thus the
dynamics of the particle is given only up to an arbitrary function. Then
$\hbox{dim }(\overline{\mathcal{S}\hbox{ol}}_{\mathbb{P}^{[n_{F}]}_{L}})=1$ as
well, and is also spanned by $\mathbf{P}_{(1)}$.

Case 2: $\frac{d^{2}V_{Sph}}{d|q|^{2}}\neq 0$.

In this case $u(\mathfrak{u})=0$, and the dynamics of the particle is
completely determined by its initial data;
$\overline{\mathcal{S}\hbox{ol}}_{\mathbb{P}^{[n_{F}]}_{L}}=\\{\overline{\mathbf{X}}_{L}\\}$.
The first-order Lagrangian constraint $\gamma^{[1]}$ does not vanish
automatically, but instead defines a surface on $\mathbf{P}_{L}$, and it
follows that $\mathcal{S}\hbox{ym}\mathcal{L}=\emptyset$. Indeed, the action’s
lack of a local gauge symmetry in this case can be seen explicitly.

Equation $(\ref{g-con})$ reduces to

$0=\widehat{q}\cdot\frac{\mathbf{\partial}V_{sph}}{\mathbf{\partial}q},$ (88)

and for dynamics to be possible the set of solutions

$\left\\{R_{i}\in\mathbb{R}\ \Bigg{|}\
\frac{dV_{Sph}}{d|q|}\Bigg{|}_{R_{i}}=0\right\\},$ (89)

must be non-empty. Dynamics are on the surfaces $|q|-R_{i}=0$ where the
potential reduces to $V(q)=V_{Sph}(R_{i})+V_{AS}(\widehat{q}^{a})$. This
reduced potential has the same symmetry as the potential
$V_{AS}(\widehat{q}^{a})$ in Case 1, and it is for this reason that the Euler-
Lagrange equations of motion have the same generalized Lie symmetry for the
two cases. This is explicitly shown in the appendix.

In Case 1 the action has a local conformal symmetry, while in Case 2 it does
not. (In [1] it was erroneously stated that in this case the action has a
global rotational symmetry.) The Lagrangian for the two cases do not have the
same invariances, resulting in one case dynamics that are determined only up
to an arbitrary $u(\mathfrak{u})$, and in the other case to a
$u(\mathfrak{u})=0$ and dynamics that are instead completely determined by the
choice of initial data.

The asymmetric potential

For a general $V$ the second term in Eq. $(\ref{e84})$ does not vanish,
$\mathbf{P}_{(1)}$ does not generate a symmetry of the equations of motion,
and $\mathcal{S}\hbox{ym}=\\{\emptyset\\}$. As before, $\gamma^{[1]}$ does not
vanish, and thus $\mathcal{S}\hbox{ym}\mathcal{L}=\\{\emptyset\\}$ as well.
Furthermore, as Eq.$(\ref{stab})$ results in

$\overline{\mathbf{X}}_{E}=\overline{\mathbf{X}}_{L}-\frac{v\cdot\Pi\cdot\frac{\partial\gamma^{[1]}}{\partial
q}}{q^{2}\mathbf{U}^{q}_{(1)}\gamma^{[1]}}[\mathbf{P}_{(1)}],$ (90)

the dynamics of the particle is uniquely determined by its initial data, and
$\overline{\mathcal{S}\hbox{ol}}_{\mathbb{P}^{[n_{F}]}_{L}}=\\{\overline{X}_{EL}\\}$
once again consists of a single point.

###  V.2 A Lagrangian with Local Conformal Symmetry

The action,

$S_{2}:=\int\Bigg{\\{}\frac{1}{2}m\left(\frac{d\widehat{q}_{1}}{dt}\right)^{2}+\frac{1}{2}m\left(\frac{d\widehat{q}_{2}}{dt}\right)^{2}+\frac{\lambda}{2}\left[\frac{q_{1}^{a}}{|q_{2}|}\frac{d\>\>}{dt}\left(\frac{q_{2a}}{|q_{1}|}\right)-\frac{q_{2}^{a}}{|q_{1}|}\frac{d\>\>}{dt}\left(\frac{q_{1a}}{|q_{2}|}\right)\right]\Bigg{\\}}dt,$
(91)

where $a=1,\dots,d$, $D=2d$, describes an interacting, two particle system
that is invariant under the local conformal transformation
$q_{1}^{a}\to\alpha(\mathfrak{u})q^{a}_{1}$ and
$q_{2}^{a}\to\alpha(\mathfrak{u})q^{a}_{2}$.

With

$\displaystyle\mathbf{\Omega}_{M}$ $\displaystyle=$
$\displaystyle\frac{m}{|q_{1}|^{2}}\Pi_{ab}(q_{1})\mathbf{d}q_{1}^{a}\wedge\mathbf{d}v_{1}^{b}+\frac{m}{|q_{2}|^{2}}\Pi_{ab}(q_{2})\mathbf{d}q_{2}^{a}\wedge\mathbf{d}v_{2}^{b},\hbox{
and}$ $\displaystyle\mathbf{\Omega}_{F}$ $\displaystyle=$
$\displaystyle\frac{m}{|q_{1}|^{3}}\left(\widehat{q}_{1}\cdot\mathbf{d}q_{1}\right)\wedge\left(v_{1}\cdot\Pi(q_{1})\cdot\mathbf{d}q_{1}\right)+\frac{m}{|q_{2}|^{3}}\left(\widehat{q}_{2}\cdot\mathbf{d}q_{2}\right)\wedge\left(v_{2}\cdot\Pi(q_{2})\cdot\mathbf{d}q_{2}\right)-$
$\displaystyle\frac{\lambda}{|q_{1}||q_{2}|}\left[\mathbf{d}q_{1}^{a}\wedge\left(\Pi(q_{2})\cdot\mathbf{d}q_{2}\right)_{a}+\left(\Pi(q_{1})\cdot\mathbf{d}q_{1}\right)_{a}\wedge\mathbf{d}q_{2}^{a}-\left(\Pi(q_{1})\cdot\mathbf{d}q_{1}\right)^{a}\wedge\left(\Pi(q_{2})\cdot\mathbf{d}q_{2}\right)_{a}\right]-$
$\displaystyle\frac{\lambda}{|q_{1}|^{2}}\left(\widehat{q}_{1}\cdot\mathbf{d}q_{1}\right)\wedge\left(\widehat{q}_{2}\cdot\Pi(q_{1})\cdot\mathbf{d}q_{1}\right)+\frac{\lambda}{|q_{2}|^{2}}\left(\widehat{q}_{2}\cdot\mathbf{d}q_{2}\right)\wedge\left(\widehat{q}_{1}\cdot\Pi(q_{2})\cdot\mathbf{d}q_{2}\right),$

$\mathcal{C}$ and $\mathcal{G}$ are two-dimensional, and are spanned by

$\mathbf{U}^{q}_{(1)}=\widehat{q}_{1}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q_{1}},\quad\mathbf{U}^{q}_{(2)}=\widehat{q}_{2}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q_{2}},\quad\hbox{and}\quad\mathbf{U}^{v}_{(1)}=\widehat{q}_{1}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v_{1}},\quad\mathbf{U}^{v}_{(2)}=\widehat{q}_{2}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v_{2}},$
(92)

respectively. The reduced $\bar{F}=0$, and $\overline{\hbox{ker
}\mathbf{\Omega}_{L}(\mathfrak{u})}$ is spanned by
$\mathbf{U}^{v}_{(1)},\mathbf{U}^{v}_{(2)}$,

$\displaystyle\mathbf{P}_{(+)}$ $\displaystyle=$ $\displaystyle
q_{1}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q_{1}}+q_{2}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q_{2}}+v_{1}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v_{1}}+v_{2}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v_{2}},$

and

$\displaystyle\mathbf{P}_{(-)}$ $\displaystyle=$ $\displaystyle
q_{1}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q_{1}}-q_{2}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q_{2}}+v_{1}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v_{1}}-v_{2}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v_{2}}-$
$\displaystyle
2\frac{\lambda}{m}\left[\frac{|q_{1}|}{|q_{2}|}q_{2}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v_{1}}+\frac{|q_{2}|}{|q_{1}|}q_{1}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v_{2}}\right].$

As such, $\hbox{dim }\overline{(\hbox{ker
}\mathbf{\Omega}_{L})}/\mathcal{G}=2$.

The energy is

$E=\frac{1}{2}\frac{m}{|q_{1}|^{2}}v_{1}\cdot\Pi(q_{1})\cdot
v_{1}+\frac{1}{2}\frac{m}{|q_{2}|^{2}}v_{2}\cdot\Pi(q_{2})\cdot v_{2}.$ (93)

We find that $\gamma^{[1]}_{(+)}=0$ while

$\gamma^{[1]}_{(-)}=-\frac{2\lambda}{|q_{1}||q_{2}|}\left(q_{2}\cdot\Pi(q_{1})\cdot
v_{1}+q_{1}\cdot\Pi(q_{2})\cdot v_{2}\right),$ (94)

giving,

$\displaystyle\mathbf{\beta}[\mathbf{X}_{EL}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\gamma^{[1]}_{(-)}\left(\frac{\mathbf{\Theta}_{q}^{(1)}}{|q_{1}|}-\frac{\mathbf{\Theta}_{q}^{(2)}}{|q_{2}|}\right).$

Then $\mathcal{S}\hbox{ym}\mathcal{L}$ is one-dimensional and spanned by
$\mathbf{P}_{(+)}$. As expected,
$\mathfrak{L}_{\mathbf{P}_{(+)}}\mathbf{\beta}=0$. Because

$\mathfrak{L}_{\mathbf{P}_{(-)}}\mathbf{\beta}=-\frac{4\lambda}{m}\left[1-(\widehat{q}_{1}\cdot\widehat{q}_{2})^{2}\right]\left(\frac{\mathbf{\Theta}_{q}^{(1)}}{|q_{1}|}-\frac{\mathbf{\Theta}_{q}^{(2)}}{|q_{2}|}\right),$
(95)

$\mathcal{S}\hbox{ym}$ is also one-dimensional, and is also spanned by
$\mathbf{P}_{(+)}$.

A general SOELVF is

$\overline{\mathbf{X}}_{EL}=\overline{\mathbf{X}}_{L}-\frac{m}{8\lambda^{2}}\frac{\overline{\mathbf{X}}_{L}\gamma^{[1]}_{(-)}}{\left[1-(\widehat{q}_{1}\cdot\widehat{q}_{2})\right]}\left[\mathbf{P}_{(-)}\right]+u^{(+)}(\mathfrak{u})\left[\mathbf{P}_{(+)}\right],$
(96)

where $u^{(+)}(\mathfrak{u})\in\overline{\mathcal{F}}$, and from [1],

$\displaystyle\overline{\mathbf{X}}_{L}$ $\displaystyle=$ $\displaystyle
v_{1}\cdot\Pi(q_{1})\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q_{1}}+v_{2}\cdot\Pi(q_{2})\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q_{2}}+$
$\displaystyle\left(\frac{\widehat{q}_{1}\cdot
v_{1}}{|q_{1}|}\right)v_{1}\cdot\Pi(q_{1})\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v_{1}}+\left(\frac{\widehat{q}_{2}\cdot
v_{2}}{|q_{2}|}\right)v_{2}\cdot\Pi(q_{2})\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v_{2}}+$
$\displaystyle\frac{\lambda}{m}\left(\frac{|q_{1}|}{|q_{2}|}v_{2}\cdot\Pi(q_{2})\cdot\Pi(q_{1})\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v_{1}}-\frac{|q_{2}|}{|q_{1}|}v_{1}\cdot\Pi(q_{1})\cdot\Pi(q_{2})\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v_{2}}\right),$

after the constraint algorithm is applied. Equation $(\ref{103})$ is a
consequence of the identity
$\langle\mathbf{d}\gamma^{[1]}_{(+)}|\overline{\mathbf{X}}_{L}\rangle=0$ and

$\displaystyle-\frac{1}{2\lambda}\langle\mathbf{d}\gamma^{[1]}_{(-)}|\overline{\mathbf{X}}_{L}\rangle$
$\displaystyle=$
$\displaystyle-2(\widehat{q}_{1}\cdot\widehat{q}_{2})\frac{E}{m}+\frac{2}{|q_{1}||q_{2}|}v_{1}\cdot\Pi(q_{1})\cdot\Pi(q_{2})\cdot
v_{2}-$
$\displaystyle\frac{\lambda}{m}(\widehat{q}_{1}\cdot\widehat{q}_{2})\left[v_{2}\cdot\Pi(q_{2})\cdot\widehat{q}_{1}-v_{1}\cdot\Pi(q_{1})\cdot\widehat{q}_{2}\right].$

We see that $\overline{\mathcal{S}\hbox{ol}}_{\mathbb{P}^{[n_{F}]}_{L}}$ is
also one-dimensional, and is also spanned by $\mathbf{P}_{(+)}$.

###  V.3 A Lagrangian with Local Conformal and Time-reparametization
Invariance

The action

$S_{3}:=sm\int\left[s\left(\frac{d\widehat{q}}{dt}\right)^{2}\right]^{1/2}dt,$
(97)

where $s=\pm 1$, is invariant under both the local conformal transformations,
$q^{a}\to\alpha(\mathfrak{u})q^{a}$, and the reparametization of time
$t\to\tau(t)$ where $\tau$ is a monotonically increasing function of $t$. Then

$\mathbf{\Omega}_{L}=\frac{m}{|q|}\frac{P_{ab}(u)}{\sqrt{sv\cdot\Pi(q)\cdot
v}}\mathbf{d}q^{a}\wedge\mathbf{d}v^{b},$ (98)

and $\mathbf{\Omega}_{F}=0$. Here, $a=1.\dots,D$,

$u_{a}=\frac{\Pi_{ab}(q)v^{b}}{\sqrt{sv\cdot\Pi(q)\cdot v}},$ (99)

so that $u^{2}=s$, while $P_{ab}(u)=\Pi_{ab}(q)-su_{a}u_{b}$. As such,
$\hbox{ker }\mathbf{\Omega}_{L}(\mathfrak{u})=$ ker
$\mathbf{\Omega}_{M}(\mathfrak{u})$. Both $\mathcal{C}$ and $\mathcal{G}$ are
two-dimensional, and are spanned by

$\mathbf{U}^{q}_{(1)}=\widehat{q}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q},\quad\mathbf{U}^{q}_{(2)}=u\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}q},\quad\hbox{and}\quad\mathbf{U}^{v}_{(1)}=\widehat{q}\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v},\quad\mathbf{U}^{v}_{(2)}=u\cdot\frac{\mathbf{\partial}\>\>\>}{\mathbf{\partial}v},$
(100)

respectively. It follows that $\hbox{dim }(\overline{\hbox{ker
}\mathbf{\Omega}_{L}}/\mathcal{G})=2$.

Because this system is fully constrained, $E=0$. As $\mathbf{\Omega}_{F}=0$ as
well, there are no Lagrangian constraints. It follows that
$\mathcal{S}\hbox{ym}\mathcal{L}$ is two dimensional and spanned by
$\mathbf{U}_{(1)}^{q}$ and $\mathbf{U}_{(2)}^{q}$. As $\mathbf{\beta}=0$ as
well, $\mathcal{S}\hbox{ym}$ is also two dimensional, and is also spanned by
$\mathbf{U}^{q}_{(1)}$ and $\mathbf{U}^{q}_{(2)}$.

We found in [1] that $\overline{\mathbf{X}}_{L}=0$. A general SOELVF is then
$\overline{\mathbf{X}}_{EL}=u^{1}(\mathfrak{u})\left[\mathbf{U}^{q}_{(1)}\right]+u^{2}(\mathfrak{u})\left[\mathbf{U}^{q}_{(2)}\right]$,
with $u^{n}(\mathfrak{u})\in\overline{\mathcal{F}}$ for $n=1,2$. It follows
that $\overline{\mathcal{S}\hbox{ol}}_{\mathbb{P}^{[n_{F}]}_{L}}$ is also two-
dimensional, and is spanned by $\mathbf{U}^{q}_{(1)}$ and
$\mathbf{U}^{q}_{(2)}$ as well.

Action | Potential | $\>\>\>\overline{\hbox{ker }\mathbf{\Omega}_{L}}/\mathcal{G}\>\>\>$ | $\>\>\>\mathcal{S}\hbox{ym}\>\>\>$ | $\>\>\>\mathcal{S}\hbox{ym}\mathcal{L}\>\>\>$ | $\>\>\>I_{\left[1\right]}\>\>\>$ | $\>\>\>\overline{\mathcal{S}\hbox{ol}}_{\mathbb{P}^{[n_{F}]}_{L}}\>\>\>$
---|---|---|---|---|---|---
| $V_{AS}(\hat{q}^{a})$ | 1 | 1 | 1 | 0 | 1
$S_{1}$ | $V_{sph}(|q|)+V_{AS}(\hat{q}^{a})$ | 1 | 1 | 0 | 1 | 0
| $V(q^{a})$ | 1 | 0 | 0 | 1 | 0
$S_{2}$ | $\frac{\lambda}{2}\left[\frac{q_{1}^{a}}{|q_{2}|}\frac{d\>\>}{dt}\left(\frac{q_{2a}}{|q_{1}|}\right)-\frac{q_{2}^{a}}{|q_{1}|}\frac{d\>\>}{dt}\left(\frac{q_{1a}}{|q_{2}|}\right)\right]$ | 2 | 1 | 1 | 1 | 1
$S_{3}$ | $0$ | 2 | 2 | 2 | 0 | 2
Table 1: A summary of the symmetries of the three examples considered in this
paper. With the exception of the $I_{[1]}$ column, the numerical entries are
the dimensionality of the vector spaces listed along the first row. Notice the
case where the Euler-Lagrange equations of motion has a generalized Lie
symmetry while the action itself does not. In all three examples, $\hbox{dim
}(\mathcal{S}\hbox{ym}\mathcal{L})=\hbox{dim
}(\overline{\mathcal{S}\hbox{ol}}_{\mathbb{P}_{L}^{[n_{f}]}})$.

##  VI Concluding Remarks

That each generalized Lie symmetry of the action contributes one arbitrary
function to the SOELVF for a dynamical system is known anecdotally, and is a
result expected on physical grounds. For almost regular Lagrangians, the
appearance in physics of a generalized Lie symmetry is due to a local gauge
symmetry of the dynamical system, and thus to the absence of a gauge—the
length of vectors for local conformal invariance, or a measure for time for
time-reparametization invariance—for some dynamical property of the system. As
the generalized Lie symmetries of the action for an almost regular Lagrangian
would have $N_{\mathcal{S}\hbox{ym}\mathcal{L}}$ of these gauge freedoms, it
is reasonable that the absence of these gauges will result in an equal number
of arbitrary functions in the SOELVF. An equal number of terms to fix these
gauges would then be needed to determine the dynamics of the system uniquely.
But while these expectations are reasonable, up to now they have been
fulfilled only on a case-by-case basis. This is in great part because the
analysis of dynamical systems with a local gauge symmetry has traditionally
been done using constrained Hamiltonian mechanics. Such analysis relies on the
canonical Hamiltonian, however, and the connection between the canonical
Hamiltonian and the symmetries of the Lagrangian is indirect at best, in
contrast to the Lagrangian approach followed here. Moreover, the process of
determining the total Hamiltonian for the system is often prescriptive, with
results that are specific to the system at hand. By focusing on the Lagrangian
and on the Lagrangian phase space, we have been able to show for all systems
with an almost regular Lagrangian that has a constant rank Lagrangian two-
form, a direct link between local gauge symmetries and its dynamics. In
particular, it establishes a link between the number of gauge symmetries of
the action and the number of arbitrary functions that naturally appear in the
evolution of such dynamical systems.

As $\gamma_{\mathbf{P}}^{[1]}=0$ for any choice of
$\mathbf{P}\in\mathcal{S}\hbox{ym}\mathcal{L}$, the vectors in
$\mathcal{S}\hbox{ym}\mathcal{L}$ do not contribute to the first-order
constraint manifold $\mathbb{P}_{L}^{[1]}$, and as such do not contribute to
the Lagrangian constraint algorithm at this order, or at any higher orders. It
is for this reason that the $N_{\mathcal{S}\hbox{ym}\mathcal{L}}$ arbitrary
functions $u^{m}_{[1]}$ are not determined by the algorithm, and why these
functions will still contribute to $\overline{\mathbf{X}}_{EL}$ even after the
algorithm has been completed. It also means that if second- and higher-order
Lagrangian constraints are introduced, they are accidental and cannot be due
to the local gauge symmetries of the action. Interestingly, we have yet to
find a dynamical system with a Lagrangian that is both almost-regular and has
a Lagrangian two-form with constant rank where second- or higher-order
Lagrangian constraints are introduced.

This impact of generalized Lie symmetries on the dynamics of particles
illustrates the inherent differences between the analysis of the symmetries of
regular Lagrangians and that of almost regular Lagrangians. For regular
Lagragians, the generator of the generalized Lie symmetry (at times referred
to as a global symmetry) gives rise to a prolongation vector, and the action
of this prolongation on the Lagrangian gives the variation of the action,
$\delta S$, under this symmetry. When the Euler-Lagrange equations of motion
are thenimposed, the conserved quantity for this symmetry along the path given
by the solution of these equations of motion is then obtained. While the
generator of the generalized Lie symmetry for the almost regular Lagrangian
$\mathbf{g}_{L}$ does give a prolongation vector pr $\mathbf{g}_{L}$ Eq.
$(\ref{prog})$, and while the action of pr $\mathbf{g}_{L}$ on $L$ does give
$\delta S$, imposing the Euler-Lagrange equations of motion on $\delta S$ in
Eq. $(\ref{Ae1})$ gives the vacuous statement $\delta S=0$. Instead, the
requirement that $\delta S=0$ for all paths on $\mathbb{Q}$ gives the
conditions that the generators of the symmetry must satisfy. This in turn
shows that the existence of these generators is due solely to the Lagrangian
being singular. These conditions then affect the dynamics of the system
through $\gamma_{\mathbf{P}}^{[1]}=0$, and in doing so, sets a lower bound to
the dimensionality of
$\overline{\mathcal{S}\hbox{ol}}_{\mathbb{P}_{L}^{[n_{f}]}}$.

We have found it quite difficult to construct more than one example of a
dynamical system that has an almost regular Lagrangian with both a generalized
Lie symmetry and a Lagrangian two-form with constant rank on $\mathbb{P}_{L}$.
We have, on the other hand, found it quite easy to construct examples of
dynamical systems that have an almost regular Lagrangian with a generalized
Lie symmetry and a Lagrangian two-form whose rank varies across
$\mathbb{P}_{L}$. Indeed, it is the latter case that is the more prevalent
one, and yet much of the results of this paper and a good portion of the
results of our previous one [1] relies on the condition that the rank of the
Lagrangian two-form be constant on $\mathbb{P}_{L}$. This is even more
concerning when we realize that these more prevalent systems are expected, by
their nature, to have much richer dynamics and mathematical structures
(indeed, we have found that such systems often require the introduction of
second- or higher-order Lagrangian constraints), and yet it is not known which
of the results that have been shown to hold for systems with constant rank
Lagrangian two-forms will still hold when the rank varies across
$\mathbb{P}_{L}$. Determining the generalized Lie symmetries of these systems;
showing that the passage from the Lagrangian to the Hamiltonian phase space is
possible; and finding the links between symmetry and dynamics is a necessity
for future research.

###### Acknowledgements.

This paper would not have been possible without the contributions by John
Garrison, who provided much of the underlying symmetry analysis of the action
used in Section II.1, and most of the essential mathematics in Section III.
Publication made possible in part by support from the Berkeley Research Impact
Initiative (BRII) sponsored by the UC Berkeley Library.

*

## Appendix A

The Euler-Lagrangian equations of motion for the action $S_{1}$ is

$0=\frac{m}{|q|^{3}}\Pi_{ab}(q)\ddot{q}^{b}-\frac{2m}{|q|^{3}}(\widehat{q}\cdot\dot{q})\Pi_{ab}(q)\dot{q}^{b}+\frac{\partial
V}{\partial q^{a}}.$ (101)

Contracting both sides of this equation with $\widehat{q}$ results in the
first-order Lagrangian constraint Eq. $(\ref{g-con})$, and it is clear that
dynamics is only possible on this constraint surface. Acting on Eq.
$(\ref{a1})$ with $\Pi_{ab}(q)$ gives

$0=\frac{m}{|q|^{3}}\Pi_{ab}(q)\ddot{q}^{b}-\frac{2m}{|q|^{3}}(\widehat{q}\cdot\dot{q})\Pi_{ab}(q)\dot{q}^{b}+\Pi_{a}^{b}(q)\frac{\partial
V}{\partial q^{b}},$ (102)

since $\Pi_{ac}(q)\Pi^{c}_{b}(q)=\Pi_{ab}(q)$. But in this case
$V(q^{a})=V_{sph}(|q|)+V_{AS}(\hat{q})$, and as

$\Pi_{a}^{b}(q)\frac{\partial V_{Sph}}{\partial
q^{b}}=\Pi_{a}^{b}(q)\frac{\partial|q|}{\partial
q^{b}}V^{\prime}_{Sph}(|q|)=0,$ (103)

while the identity

$\frac{\partial\hat{q}^{a}}{\partial q^{b}}=\Pi^{a}_{b}(q),$ (104)

ensures that

$\Pi_{a}^{b}(q)\frac{\partial V_{AS}}{\partial q^{b}}=\frac{\partial
V_{AS}}{\partial q^{a}},$ (105)

Eq. $(\ref{Ap})$ thereby reduces to the same equations of motion for the
system as found for Case 1. It is for this reason that the two cases have same
generalized Lie symmetry.

## References

  * [1] A. D. Speliotopoulos, Constrained dynamics: generalized Lie symmetries, singular Lagrangians, and the passage to Hamiltonian mechanics, J. Phys Commun., 4 065002 (2020).
  * [2] M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, Journal of Mathematical Physics, 19 2388–2399 (1978) 10.1063/1.523597.
  * [3] M. J. Gotay and J. M. Nester, Presymplectic lagrangian systems I: the constraint algorithm and the equivalence theorem, Annales de L’Institut Henri Poincare, Section A, 30(2) 129–142 (1979)
  * [4] M. J. Gotay and J. M. Nester, Presymplectic lagrangian systems II: the second-order problem, Annales de L’Institut Henri Poincare, Section A, 32(1) 1–13 (1980).
  * [5] J. F. Cariñena, Theory of singular Lagrangians, Fortschritte der Physik, 38(9) 641–679 (1990) 10.1002/prop.2190380902.
  * [6] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, (Princeton University Press, Princeton, New Jersey, 1992).
  * [7] P. A. M. Dirac, Generalized Hamiltonian dynamics, Canadian Journal of Mathematics, 2 129–148 (1950) 10.4153/CJM-1950-012-1.
  * [8] M. C. Muñoz-Lecanda, Hamiltonian systems with constraints: A geometric approach, International Journal of Theoretical Physics, 28(11) 1405–1417 (1989) 10.1007/BF00671858.
  * [9] L. Lusanna, Dirac-Bergmann constraints in physics: Singular Lagrangians, Hamiltonian constraints and the Second Noether Theorem, International Journal of Geometric Methods in Modern Physics, 15(10) 1830004 (2018), 10.1142/S0219887818300040.
  * [10] G. Prince, Toward a classification of dynamical symmetries in classical mechanics, Bulletin of the Australian Mathematical Society, 27 53–71 (1983) 10.1017/S0004972700011485.
  * [11] G. Prince, A complete classification of dynamical symmetries in classical mechanics, Bulletin of the Australian Mathematical Society, 32 299–308 (1985) 10.1017/S0004972700009977.
  * [12] M. Crampin, Tangent bundle geometry Lagrangian dynamics, Journal of Physics A: Mathematical and General Physics, 16 3755–3772 (1983) 10.1088/0305-4470/16/16/014.
  * [13] J. F. Cariñena, J. Fernández-Núñez and E. Martínez, A geometric approach to Noether’s Second Theorem in time-dependent Lagrangian mechanics, Letters in Mathematical Physics, 23 51–63 (1991) 10.1007/BF01811294.
  * [14] J. F. Cariñena and M. F. Rañada, Noether’s theorem for singular Lagrangians, Letters on Mathematical Physics, 15 305–311 (1988) 10.1007/BF00419588.
  * [15] J. F. Cariñena, E. Martínez and J. Fernández-Núñez, Noether’s theorem in time-dependent Lagrangian mechanics, Reports on Mathematical Physics, 31 189–203 (1992) 10.1016/0034-4877(92)90014-R.
  * [16] J. F. Cariñena, J. Fernández-Núñez and M. F. Rañada, Singular Lagragians affine in velocities, Journal of Physics A: Mathematical and General Physics, 36 3789–3807 (2003) 10.1088/0305-4470/36/13/311.
  * [17] J. F. Cariñena and J. Fernández-Núñez, Geometric theory of time-dependent singular Lagrangians, Fortschritte der Physik, 41(6) 517–552 (1993).
  * [18] J. F. Cariñnena and E. Martinez, in Summetries and Algebra Structures in Physics, Part 2: Integral Systems, Soli State Physics, and Theory of Phase Transitions, edited by V. V. Dodonov and V. I. Man’ko, (Nova Science Publishers, New York, 1991) Chap. Generalized Jacobi equation and inverse problem in classical mechanics, pp 84–98.
  * [19] G. Marmo, G. Mendella and W. M. Tulczyjew, Symmetries and constants of the motion for dynamics in implicit form, Annales de L’Institut Henri Poincare, Section A, 57(2) 147–166 (1992).
  * [20] X. Grácia and J. M. Pons, Symmetries and infinitesimal symmetries of singular differential equations, Journal of Physics A: Mathematical and General Physics, 35 5059–5077 (2002) 10.1088/0305-4470/35/24/306.
  * [21] X. Grácia and R. Martín, Geometric aspects of time-dependent singular differential equations, International Journal of Geometric Methods in Modern Physics, 2(4) 597–618 (2005) 10.1142/S0219887805000697.
  * [22] L. Popescu, Symmetries of second order differential equations on Lie algebroids, Journal of Geometry and Physics, 117 84–98 (2017) 10.1016/j.geomphys.2017.03.006.
  * [23] M. de León and D. M. de Diego, Symmetries and constants of the motion for singular Lagrangian systems, International Journal of Theoretical Physics, 35(5) 975–1011 (1996) 10.1007/BF02302383.
  * [24] N. Dimakis, P. A. Terzis and T. Christodoulakis , Contact symmetries of constrained quadratic Lagrangians, Journal of Physics: Conference Series, 670 1–6 (2016) 10.1088/1742-6596/670/1/012021.
  * [25] M. Popescu, Totally singular Lagrangians and affine Hamiltonians, Balkan Journal of Geometry and Its Applications, 14(1) 60–71 (2009)
  * [26] M. Popescu and P. Popescu, Totally singular Lagrangians and affine Hamiltonians of higher order, Balkan Journal of Geometry and Its Applications, 16(2) 122–132 (2011).
  * [27] P. J. Olver, Applications of Lie Groups to Differential Equations, (Springer-Verlag, New York, New York, 1993).
  * [28] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed, (Addison-Wesley, Reading, Massachusetts, 1978).