# REGULARITY PROPERTIES OF FIBER DERIVATIVES ASSOCIATED WITH HIGHER-ORDER
MECHANICAL SYSTEMS

Leonardo Colombo
Department of Mathematics, University of Michigan. Ann Arbor, MI 48109, USA
Pedro Daniel Prieto-Martínez
Departament de Matemàtiques. Building C3, North Campus UPC. Jordi Girona 1.
08034 Barcelona. Spain
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The aim of this work is to study fiber derivatives associated to Lagrangian
and Hamiltonian functions describing the dynamics of a higher-order autonomous
dynamical system. More precisely, given a function in $T^{*}T^{(k-1)}Q$, we
find necessary and sufficient conditions for such a function to describe the
dynamics of a $k$th-order autonomous dynamical system, thus being a $k$th-
order Hamiltonian function. Then, we give a suitable definition of
(hyper)regularity for these higher-order Hamiltonian functions in terms of
their fiber derivative. In addition, we also study an alternative
characterization of the dynamics in Lagrangian submanifolds in terms of the
solutions of the higher-order Euler-Lagrange equations.

Key words: Higher-order systems; Lagrangian and Hamiltonian mechanics;
Lagrangian submanifolds; Tulczyjew’s triple.

AMS s. c. (2010): 70H50, 53D05, 53D12

###### Contents

  1. 1 Introduction
  2. 2 Geometric background
    1. 2.1 Symplectic manifolds and Lagrangian submanifolds
    2. 2.2 Fiber derivative of a Hamiltonian function
    3. 2.3 Tulczyjew’s triple
    4. 2.4 Higher-order tangent bundles
  3. 3 Geometric description of higher-order dynamical systems
    1. 3.1 Higher-order Tulczyjew triple and dynamics generated by Lagrangian submanifolds
    2. 3.2 On the Legendre maps for higher-order dynamical systems
    3. 3.3 An example: the dynamics of the end of a javelin
  4. 4 On the definition and the regularity of a higher-order Hamiltonian function
    1. 4.1 Statement of the problems
    2. 4.2 A particular case: the Hamiltonian function associated to a (hyper)regular Lagrangian system
    3. 4.3 Higher-order Hamiltonian functions: definition and regularity

##  1 Introduction

Higher-order dynamical systems play a relevant role in certain branches of
theoretical physics, applied mathematics and numerical analysis. In
particular, they appear in theoretical physics, in the mathematical
description of relativistic particles with spin, string theories, Hilbert’s
Lagrangian for gravitation, Podolsky’s generalization of electromagnetism and
others, as well as in some problems of fluid mechanics and classical physics,
and in numerical models arising from the discretization of dynamical control
systems that preserve their inherent geometric structures. In these kinds of
systems, the dynamics have explicit dependence on accelerations or higher-
order derivatives of the generalized coordinates of position. The geometric
tools used to study those systems have been developed mainly by M. de León,
P.R. Rodrígues, D.J. Saunders and M. Crampin (among others) between the $70$’s
and $90$’s in [18, 51, 52] (see also [2, 9, 10, 11, 19, 20, 35, 39], and
references therein). These works are based in the ideas of the Lagrangian
formalism introduced by J. Klein at the beginning of the $60$’s in [33]. In
the aforementioned work, the Euler-Lagrange equations are obtained by J. Klein
in a purely geometric way using the canonical geometric structures of the
tangent and cotangent bundles, avoiding the use of variational calculus and
exploiting the geometry of these dynamical systems.

The interest in higher-order dynamical systems has been growing up from the
90’s due to the study of optimization and boundary value problems where the
cost function involves higher-order derivatives, which may be modeled as
variational problems with explicit dependence on higher-order derivatives of
the generalized coordinates of position. These “higher-order variational
problems” are of great interest for their useful applications in aeronautics,
robotics, computer-aided design, air traffic control, trajectory planning,
and, more generally, problems of interpolation and approximation of curves on
Riemannian manifolds. These kinds of problems have been studied in [4, 5, 7,
30, 37, 41, 44] and more recently, in [22, 23, 24, 43] the development of
variational principles which involve higher-order cost functions for
optimization problems on Lie groups and their application in template matching
for computational anatomy have been studied. These applications have produced
a great interest in the study and development of new modern geometric tools
and techniques to model properly higher-order variational problems, with the
additional goal of obtaining a deepest understanding of the intrinsic
properties of these problems. Some work in this line of research has been
carried out recently in the following references, [13, 12, 49, 50, 40, 45, 46,
6, 31, 32, 28, 56].

Let us recall that the dynamics for a $k$th-order dynamical system can be
obtained both by means of a Lagrangian function defined on a the $k$th-order
tangent bundle $T^{(k)}Q$ of the smooth manifold $Q$ that models the
configuration space of the system, or by means of a Hamiltonian function
defined on the cotangent bundle $T^{*}T^{(k-1)}Q$ (see [18] for details). The
relation between Lagrangian and Hamiltonian dynamics can be studied either
using the higher-order Legendre map [16, 17] or the Legendre-Ostrogradsky map
[18]. These two well-known transformations are both derived from the
Lagrangian function, and they provide a way to define a canonical Hamiltonian
function, and hence to give a Hamiltonian formulation of the system.
Nevertheless, for first-order dynamical systems it is known from [1] (Sections
§3.5 and §3.6) that, starting from a Hamiltonian function in the cotangent
bundle of the configuration space, it is possible to define a Lagrangian
function in the tangent bundle describing the dynamics of the system, or to
recover the starting Lagrangian if the Hamiltonian was defined using a
Legendre map. The fundamental tool to do so is the fiber derivative of the
Hamiltonian function. This same procedure can be carried out with a
Hamiltonian function defined on $T^{*}T^{(k-1)}Q$, and a “Lagrangian function”
can be defined in $TT^{(k-1)}Q$. This “Lagrangian function”, however, may not
be an actual Lagrangian function in the physical sense, since the generalized
coordinates in the base manifold $T^{(k-1)}Q$ are generally not related to the
fibered coordinates (the “velocities”). From the geometric point of view, this
pretended “Lagrangian function” may not be defined in the holonomic
submanifold $T^{(k)}Q\hookrightarrow TT^{(k-1)}Q$, which is the real domain
for a $k$th-order Lagrangian function. This problem arises from the fact that
the starting Hamiltonian function is not considered as a Hamiltonian function
for $k$th-order system, but just as a Hamiltonian function for a first-order
system defined on the cotangent bundle of a larger base manifold, since there
is not an enforced relation between the momenta.

The discussion in the previous paragraph gives rises to two natural questions.
First, what is a higher-order Hamiltonian function? And second, what does it
mean for such a function to be regular? Observe that these concepts are
clearly defined for first-order Lagrangian or Hamiltonian functions, and also
for a higher-order Lagrangian functions (see [15] for the constrained first
order case). In this work we pretend to give an answer to these questions.
Indeed, in Section 4 we propose a definition of both concepts, always taking
into account the particular case of Hamiltonian functions associated to a
regular Lagrangian system. Moreover, we extend in a nontrivial way some
results from [1], namely Propositions 3.6.7 and 3.6.8, and Theorem 3.6.9, to
higher-order autonomous dynamical systems by means of the answers to the
proposed questions.

For constrained systems, as well for singular Lagrangian functions, an
alternative approach for a better understanding of the geometry involving
mechanical systems was established by W.M. Tulczyjew: the so-called
Tulczyjew’s triple [54, 55], which makes strong use of Lagrangian submanifolds
of suitable symplectic manifolds. Lagrangian submanifolds are of great
interest in geometric mechanics, since they provide a way of describing both
Lagrangian and Hamiltonian dynamics from a purely geometric and intrinsic
point of view (see [54, 55]). In particular, let us recall that given a
mechanical system described by a Lagrangian function $L\colon
TQ\to\mathbb{R}$, the Lagrangian dynamics are “generated” by the Lagrangian
submanifold $dL(TQ)\subset T^{*}TQ$. On the other hand, if the system is
described by a Hamiltonian function $H\colon T^{*}Q\to\mathbb{R}$, then the
Hamiltonian dynamics are “generated” by the Lagrangian submanifold
$dH(T^{*}Q)\subset T^{*}T^{*}Q$. The relationship between these two
formulations is provided by the so-called Tulczyjew’s triple

$\textstyle{T^{*}TQ}$
$\textstyle{TT^{*}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{Q}}$$\scriptstyle{\beta_{Q}}$
$\textstyle{T^{*}T^{*}Q,}$

where $\alpha_{Q}$ and $\beta_{Q}$ are both vector bundle isomorphisms, and
$T^{*}TQ$, $TT^{*}Q$ and $T^{*}T^{*}Q$ are double vector bundles equipped with
suitable symplectic structures.

If we consider a $k$th-order Lagrangian system described by a $k$th-order
Lagrangian function $L\colon T^{(k)}Q\to\mathbb{R}$, a similar construction
can be carried out (with some additional technical issues arising from the
fact that $T^{(k)}Q$ is not a vector bundle in general, see [16, 17] for
details), thus obtaining a Lagrangian submanifold in $TT^{*}T^{(k-1)}Q$, which
completely determines the equations of motion for the dynamics. Moreover,
these equations of motion are of Hamiltonian type if the Lagrangian system is
regular. Our aim in this work is to study properties of fiber derivatives of
functions defined on Lagrangian submanifolds, thus pursuing the research lines
established in the works of M. de León and E. Lacomba [16, 17].

The paper is structured as follows. In Section 2 we introduce the necessary
geometric background in order to make the paper as much selfcontained as
possible. In particular, we review the definition and basic properties of
symplectic manifolds and Lagrangian submanifolds, fiber derivatives of fiber
preserving maps and their application to relate Lagrangian and Hamiltonian
dynamics, higher-order tangent bundles and some of its canonical structures
and, finally, a short review on the construction of Tulczyjew’s triple for
first-order dynamical systems. In Section 3 we study fiber derivatives of
higher-order Lagrangian systems, relating the classical Legendre-Ostrogradsky
map associated to a $k$th-order Lagrangian function $L$ and the $k$th-order
Legendre transformation defined on the Lagrangian submanifold
$\Sigma_{L}\subset T^{*}TT^{(k-1)}Q$ generated by the Lagrangian $L$. To close
this Section, we study the dynamics of the end of a thrown javelin as an
illustrative example. Finally, in Section 4 we study the problem of giving an
universal definition of higher-order Hamiltonian function, and the regularity
properties of the fiber derivative associated with such a Hamiltonian.

##  2 Geometric background

In this Section we introduce the geometric structures and definitions that we
use along this work. All the manifolds are real, second countable and
$C^{\infty}$. The maps and the structures are assumed to be $C^{\infty}$. Sum
over crossed repeated indices is understood. If $M$ denotes a finite-
dimensional smooth manifold, then $C^{\infty}(M)$, $\mathfrak{X}(M)$ and
$\Omega^{k}(M)$ denote the sets of smooth functions, smooth vector fields and
smooth $k$-forms on $M$, respectively.

###  2.1 Symplectic manifolds and Lagrangian submanifolds

Along this Subsection, $M$ denotes a finite-dimensional smooth manifold. We
refer to [8, 36, 57] for details and proofs.

######  Definition 1.

A symplectic form in $M$ is a closed $2$-form $\omega\in\Omega^{2}(M)$ which
is nondegenerate, that is, for every $p\in M$,
$\mathop{i}\nolimits_{X_{p}}\omega_{p}=0$ if, and only if, $X_{p}=0$ where
$X_{p}\in T_{p}M$. A symplectic manifold is a pair $(M,\omega)$, where $M$ is
a smooth manifold and $\omega$ is a symplectic form.

######  Remark.

If $(M,\omega)$ is a symplectic manifold, then the nondegeneracy of $\omega$
implies that $M$ has even dimension, that is, $\dim M=2n$. $\lozenge$

######  Definition 2.

Let $(M_{1},\omega_{1})$ and $(M_{2},\omega_{2})$ be two symplectic manifolds,
and $\Phi\colon M_{1}\to M_{2}$ a diffeomorphism. $\Phi$ is a
symplectomorphism if $\Phi^{*}\omega_{2}=\omega_{1}$, and it is an anti-
symplectomorphism if $\Phi^{*}\omega_{2}=-\omega_{1}$, where
$\Phi^{*}\omega_{2}$ denotes the pull-back of the $2$-form $\omega_{2}$ by the
diffeomorphism $\Phi$.

A distinguished symplectic manifold is the cotangent bundle $T^{*}Q$ of a
$n$-dimensional smooth manifold $Q$. Let $\pi_{Q}\colon T^{*}Q\to Q$ be the
canonical projection defined by $\pi_{Q}(\alpha_{q})=q\in Q$, where
$\alpha_{q}\in T^{*}_{q}Q$. The Liouville $1$-form, denoted by
$\theta_{Q}\in\Omega^{1}(T^{*}Q)$, is defined as

$\langle(\theta_{Q})_{\alpha_{q}},X_{\alpha_{q}}\rangle=\langle\alpha_{q},T_{\alpha_{q}}\pi_{T^{*}Q}(X_{\alpha_{q}})\rangle\,,\
\mbox{where }\alpha_{q}\in T^{*}Q\mbox{ and }X_{\alpha_{q}}\in
T_{\alpha_{q}}T^{*}Q\,.$

Observe that the Liouville $1$-form satisfies $\alpha^{*}\theta_{Q}=\alpha$
for every $\alpha\in\Omega^{1}(Q)$. Then, one can define the canonical
symplectic form of $T^{*}_{q}Q$, or Liouville $2$-form, as
$\omega_{Q}=-d\theta_{Q}\in\Omega^{2}(T^{*}Q)$. If $(U;(q^{i}))$, $1\leqslant
i\leqslant n$, are local coordinates in $Q$, the induced natural coordinates
in $\pi_{Q}^{-1}(U)\subseteq T^{*}Q$ are $(q^{i},p_{i})$, $1\leqslant
i\leqslant n$. In these coordinates, the local expression of the Liouville
$1$-form is $\theta_{Q}=p_{i}dq^{i}$, from where, the canonical symplectic
form has the following coordinate expression $\omega_{Q}=dq^{i}\wedge dp_{i}$.

The existence of a nondegenerate $2$-form on symplectic manifolds enables us
to define some special submanifolds. In particular, we are interested in the
study of Lagrangian submanifolds.

######  Definition 3.

Let $(M,\omega)$ be a symplectic manifold. An immersed submanifold
$i_{N}\colon N\hookrightarrow M$ is a Lagrangian submanifold if $\dim
N=\frac{1}{2}(\dim M)$ and $i_{N}^{*}\omega=0$.

Next, we introduce some particular Lagrangian submanifolds of the symplectic
manifold $(T^{*}Q,\omega_{Q})$. The first one is the image of a closed
$1$-form. Indeed, let $\lambda\in\Omega^{1}(Q)$ be a closed $1$-form, and let
us consider the subset $\Sigma_{\lambda}=\lambda(Q)\subset T^{*}Q$, which is a
submanifold of $T^{*}Q$ with canonical embedding $\lambda\colon
Q\hookrightarrow T^{*}Q$. Then we have
$\lambda^{*}\omega_{Q}=\lambda^{*}(-d\theta_{Q})=-d\lambda^{*}\theta_{Q}=-d\lambda=0$
since $\lambda$ is closed. Hence, $\Sigma_{\lambda}$ is a Lagrangian
submanifold. If, moreover, $\lambda$ is exact, that is, $\lambda=df$, with
$f\in C^{\infty}(Q)$, we say that $f$ is a generating function of the
Lagrangian submanifold ${\Sigma}_{\lambda}$, and we denote it by $\Sigma_{f}$
(see [57] for details).

There is a more general construction of Lagrangian submanifolds given by J.
Śniatycki and W.M. Tulczyjew in [53] (see also [54, 55]) which we use in
Subsection 3.1 to generate the dynamics of a higher-order dynamical system
through Lagrangian submanifolds.

######  Theorem 1 (Śniatycki & Tulczyjew).

Let $Q$ be a smooth manifold, $\tau_{Q}\colon TQ\to Q$ its tangent bundle,
$i_{N}\colon N\hookrightarrow Q$ a $k$-dimensional submanifold, and $f\colon
N\to\mathbb{R}$ a smooth function. Then

$\displaystyle\Sigma_{f,N}$ $\displaystyle=\left\\{\mu\in
T^{*}Q\mid\pi_{Q}(\mu)\in N\mbox{ and
}\left\langle\mu,v\right\rangle=\left\langle df,v\right\rangle\mbox{ for every
}v\in T_{\pi_{Q}(\mu)}N\right\\}$ $\displaystyle=\left\\{\mu\in T^{*}Q\mid
i_{N}^{*}\mu=df\right\\}\,,$

is a Lagrangian submanifold of $(T^{*}Q,\omega_{Q})$.

Let $(q^{i})$, $1\leqslant i\leqslant n$, be local coordinates in $Q$ adapted
to $N$, that is, such that $N$ is locally defined by the constraints
$q^{k+1}=\ldots=q^{n}=0$. Then, the smooth function $f\colon N\to\mathbb{R}$
depends only on the coordinates $q^{1},\ldots,q^{k}$, and the submanifold
$\Sigma_{f,N}\hookrightarrow T^{*}Q$ is locally defined by

$\Sigma_{f,N}=\left\\{(q^{i},p_{i})\in T^{*}Q\mid q^{i}=0\,,\
p_{j}=\displaystyle\frac{\partial{f}}{\partial{q^{j}}}\mbox{ for }k+1\leqslant
i\leqslant n-k\,,\,1\leqslant j\leqslant k\right\\}.$

Thus, it follows that $\dim\Sigma_{f,N}=n=\dim Q=\frac{1}{2}\dim T^{*}Q.$
Moreover, taking into account the local expression of the canonical symplectic
form $\omega_{Q}$, if we denote
$i_{\Sigma_{f,N}}\colon\Sigma_{f,N}\hookrightarrow T^{*}Q$ the canonical
embedding, it follows that $i_{\Sigma_{f,N}}^{*}\omega_{Q}=0$. Therefore,
$\Sigma_{f,N}$ is a Lagrangian submanifold of the symplectic manifold
$(T^{*}Q,\omega_{Q})$ (see [54] for an intrinsic proof).

The importance of this result lies in the fact that Lagrangian submanifolds
are associated to the dynamics of Lagrangian and Hamiltonian systems subject
or not to constraints as we show in Subsection 3.1.

###  2.2 Fiber derivative of a Hamiltonian function

Along this Subsection, we consider a first-order dynamical system with $n$
degrees of freedom whose configuration space is modeled by a $n$-dimensional
smooth manifold $Q$, and let $H\in C^{\infty}(T^{*}Q)$ be a Hamiltonian
function describing the dynamics of the system (see [1], §3.5 and §3.6, for
details).

######  Definition 4.

Let $\pi\colon E\to M$ and $\rho\colon F\to M$ be vector bundles over the
common base manifold $M$, and let $f\colon E\to F$ be a smooth fiber
preserving map (not necessarily a vector bundle morphism). Let $f_{x}$ denote
$\left.f\right|_{E_{x}}$, where $E_{x}=\pi^{-1}(x)$ is the fiber over $x\in
M$. The fiber derivative of $f$ is defined to be the map

$\begin{array}[]{rcl}\mathcal{F}f\colon
E&\longrightarrow&\displaystyle{\bigcup_{x\in M}}L(E_{x},F_{x})\\\
v_{x}&\longmapsto&Df_{x}(v_{x})\end{array}$

where $L(E_{x},F_{x})$ denotes the vector space of linear mappings from
$E_{x}$ to $F_{x}$.

Next, we apply the construction given in Definition 4 to the Hamiltonian
function $H$. Let $E$ be the cotangent bundle of $Q$, $\pi_{Q}\colon T^{*}Q\to
Q$ the canonical projection, $F=Q\times\mathbb{R}$ the trivial vector bundle
with projection $\operatorname{pr}_{1}\colon Q\times\mathbb{R}\to Q$, and $f$
the map $\widetilde{H}\colon T^{*}Q\to Q\times\mathbb{R}$ defined by

$\widetilde{H}(\alpha_{q})=(\pi_{Q}(\alpha_{q}),H(\alpha_{q}))\,.$

The map $\widetilde{H}$ is smooth and fiber preserving, since

$\operatorname{pr}_{1}(\widetilde{H}(\alpha_{q}))=\operatorname{pr}_{1}(\pi_{Q}(\alpha_{q}),H(\alpha_{q}))=\pi_{Q}(\alpha_{q})\,,$

although $\widetilde{H}$ is not a vector bundle morphism in general. Taking
into account that $L(T^{*}_{q}Q,\mathbb{R})=T_{q}^{**}Q\cong T_{q}Q$, the
fiber derivative of $H$, denoted by $\mathcal{F}H\colon T^{*}Q\to TQ$, is
defined as the fiber derivative of the map $\widetilde{H}$ in the sense of
Definition 4.

######  Remark.

This same procedure can be carried out with a Lagrangian function $L\in
C^{\infty}(TQ)$. As it is well-known, the fiber derivative of $L$,
$\mathcal{F}L\colon TQ\to T^{*}Q$, is the Legendre map
$\operatorname{leg}_{L}$ relating the Lagrangian and Hamiltonian formalisms of
dynamical systems (see [1], §3.5 for details). $\lozenge$

The map $\mathcal{F}H\colon T^{*}Q\to TQ$ is smooth and fiber preserving, that
is, $\tau_{Q}\circ\mathcal{F}H=\pi_{Q}$. Let $(U,(q^{i}))$ be a local chart in
$Q$, and $(q^{i},p_{i})$ the induced natural coordinates in
$\pi_{Q}^{-1}(U)\subseteq T^{*}Q$. Then, the coordinate expression of
$\mathcal{F}H$ is determined by

$\mathcal{F}H(q^{i},p_{i})=\left(q^{i},\frac{\partial H}{\partial
p_{i}}\right)\,,$

from where we can observe that $\mathcal{F}H$ is smooth and fiber preserving.

######  Definition 5.

A Hamiltonian function $H\in C^{\infty}(T^{*}Q)$ is regular if the map
$\mathcal{F}H\colon T^{*}Q\to TQ$ is a local diffeomorphism, and it is
hyperregular if $\mathcal{F}H$ is a global diffeomorphism. Otherwise, the
Hamiltonian function is said to be singular.

Locally, the regularity condition for $H$ is equivalent to

$\det\left(\displaystyle\frac{\partial^{2}{H}}{\partial{p_{i}}\partial{p_{j}}}\right)(\alpha_{q})\neq
0\,,\ \mbox{for every }\alpha_{q}\in T^{*}_{q}Q\,,$

that is, a Hamiltonian function is regular if, and only if, its Hessian matrix
with respect to the momenta is invertible at every point of $T^{*}Q$.

Next, we give a brief review of the relation between Hamiltonian and
Lagrangian formalisms in terms of the fiber derivative of $H$ (see [1]
details). First, let us recall how to define a Lagrangian $L\in
C^{\infty}(TQ)$ describing the dynamics of the system starting from a
Hamiltonian function.

######  Proposition 1 ([1], Prop. 3.6.7).

Let $H\in C^{\infty}(T^{*}Q)$ be a hyperregular Hamiltonian function,
$\theta_{Q}\in\Omega^{1}(T^{*}Q)$ the Liouville $1$-form,
$\omega_{Q}\in\Omega^{2}(T^{*}Q)$ the canonical symplectic form and
$X_{H}\in\mathfrak{X}(T^{*}Q)$ the unique vector field solution to the
dynamical equation
$\displaystyle{\mathop{i}\nolimits_{X_{H}}\omega_{Q}=dH\,}$. The function
$L\in C^{\infty}(TQ)$ defined by
$L=\theta_{Q}(X_{H})\circ\mathcal{F}H^{-1}-H\circ\mathcal{F}H^{-1}$ is
hyperregular, and
$\mathcal{F}L\equiv\operatorname{leg}_{L}=\mathcal{F}H^{-1}$.

Observe that, up to this point, the Hamiltonian function $H\in
C^{\infty}(T^{*}Q)$ could be any function defined on the cotangent bundle.
Nevertheless, if $L\in C^{\infty}(TQ)$ is a hyperregular Lagrangian function
describing the dynamics of the system, and $E_{L}=\Delta(L)-L\in
C^{\infty}(TQ)$ is the energy of the system, with $\Delta\in\mathfrak{X}(TQ)$
being the Liouville vector field, then we can define a Hamiltonian function
$H=E_{L}\circ\operatorname{leg}_{L}^{-1}\in C^{\infty}(T^{*}Q)$. Then, the
following result holds.

######  Proposition 2 ([1]. Prop. 3.6.8 and Thm. 3.6.9).

Let $L\in C^{\infty}(TQ)$ be a hyperregular Lagrangian and
$H=E_{L}\circ\operatorname{leg}_{L}^{-1}\in C^{\infty}(T^{*}Q)$ the associated
Hamiltonian function. Then $H$ is hyperregular and
$\mathcal{F}H=\operatorname{leg}_{L}^{-1}$. In addition, if
$\tilde{L}=\theta_{Q}(X_{H})\circ\mathcal{F}H^{-1}-H\circ\mathcal{F}H^{-1}\in
C^{\infty}(TQ)$ is the hyperregular Lagrangian function associated to $H$ by
Proposition 1, then $\tilde{L}=L$.

###  2.3 Tulczyjew’s triple

In [54, 55], Tulczyjew established two identifications, the first one between
$TT^{*}Q$ and $T^{*}TQ$ (useful to describe Lagrangian mechanics) and the
second one between $TT^{*}Q$ and $T^{*}T^{*}Q$ (useful to describe Hamiltonian
mechanics), giving rise to the so-called Tulczyjew’s triple. In this
Subsection we summarize these results. Along this Subsection, $Q$ denotes a
$n$-dimensional smooth manifold.

Let us recall that the double tangent bundle $TTQ$ of a manifold $Q$ is
endowed with two vector bundle structures over the base $TQ$, given by the
canonical projection $\tau_{TQ}\colon TTQ\to TQ$ arising from the tangent
bundle structure, and the tangent map $T\tau_{Q}\colon TTQ\to TQ$ of
$\tau_{Q}$ of the canonical projection $\tau_{Q}\colon TQ\to Q$ arising from
the starting tangent bundle structure. These two structures are related by the
canonical flip $\kappa_{Q}\colon TTQ\to TTQ$, which is an isomorphism of
double vector bundles. If $(U;(q^{i}))$, $1\leqslant i\leqslant n$, is a local
chart in $Q$ and $(q^{i},v^{i},\dot{q}^{i},\dot{v}^{i})$ the induced local
coordinates in a suitable open set of $TTQ$, then $\kappa_{Q}$ is given
locally by
$\kappa_{Q}(q^{i},v^{i},\dot{q}^{i},\dot{v}^{i})=(q^{i},\dot{q}^{i},v^{i},\dot{v}^{i})$.
It is clear from this coordinate expression that $\kappa_{Q}$ is an
involution. From this, we can give the following definition.

######  Definition 6.

The Tulczyjew’s isomorphisms are the diffeomorphisms $\alpha_{Q}\colon
TT^{*}Q\to T^{*}TQ$ and $\beta_{Q}\colon TT^{*}Q\to T^{*}T^{*}Q$ defined as
follows:

  1. 1.

$\alpha_{Q}$ is the dual map of $\kappa_{Q}$ (as a double vector bundle
morphism).

  2. 2.

If $\omega_{Q}\in\Omega^{2}(T^{*}Q)$ is the canonical symplectic form, then
$\beta_{Q}(X)=i_{X}\omega_{Q}\,,\ X\in TT^{*}Q$.

Let $(U;(q^{i}))$ be a local chart in $Q$, and $(q^{i},p_{i})$ the induced
natural coordinates in $\pi_{Q}^{-1}(U)\subseteq T^{*}Q$. The induced natural
coordinates in $\tau_{T^{*}Q}^{-1}(\pi_{Q}^{-1}(U))\subseteq TT^{*}Q$ are
$(q^{i},p_{i},\dot{q}^{i},\dot{p}_{i})$. In these coordinates, the maps
$\alpha_{Q}$ and $\beta_{Q}$ are given by
$\alpha_{Q}(q^{i},p_{i},\dot{q}^{i},\dot{p}_{i})=(q^{i},\dot{q}^{i},\dot{p}_{i},p_{i})$
and
$\beta_{Q}(q^{i},p_{i},\dot{q}^{i},\dot{p}_{i})=(q^{i},\dot{q}^{i},\dot{p}_{i},p_{i})$,
respectively.

The map $\alpha_{Q}$ is a symplectomorphism when we consider on $TT^{*}Q$ the
symplectic structure given by the complete lift $\omega_{Q}^{c}$ of the
canonical symplectic form $\omega_{Q}$ on $T^{*}Q$ and on $T^{*}TQ$ the
canonical symplectic form $\omega_{TQ}$. On the other hand, the map
$\beta_{Q}$ is an anti-symplectomorphism when we consider on $TT^{*}Q$ the
same symplectic structure $\omega_{Q}^{c}$ and we consider on $T^{*}T^{*}Q$
the canonical symplectic structure $\omega_{T^{*}Q}$.

The maps $\beta_{Q}$ and $\alpha_{Q}$ give rise to the Tulczyjew triple,
summarized in the following diagram

$\textstyle{T^{*}TQ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{TQ}}$
$\textstyle{TT^{*}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{T^{*}Q}}$$\scriptstyle{T\pi_{Q}}$$\scriptstyle{\alpha_{Q}}$$\scriptstyle{\beta_{Q}}$
$\textstyle{T^{*}T^{*}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{T^{*}Q}}$
$\textstyle{TQ\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{leg}_{{L}}}$$\scriptstyle{\tau_{Q}}$
$\textstyle{T^{*}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{Q}}$
$\textstyle{Q}$

where $\operatorname{leg}_{L}\colon TQ\to T^{*}Q$ denotes the Legendre
transformation associated to a given Lagrangian function ${L}\in
C^{\infty}(TQ)$.

###  2.4 Higher-order tangent bundles

(See [18, 51] for details).

Let $Q$ be a $n$-dimensional smooth manifold. We introduce an equivalence
relation in the set $C^{\infty}(\mathbb{R},Q)$ of smooth curves
$\gamma\colon\mathbb{R}\to Q$ as follows: given two curves
$\gamma_{1},\gamma_{2}\colon(-a,a)\to Q$, with $a>0$, we say that $\gamma_{1}$
and $\gamma_{2}$ have contact of order $k$ at
$q_{0}=\gamma_{1}(0)=\gamma_{2}(0)$ if there exists a local chart
$(U,\varphi)$ of $Q$ such that $q_{0}\in U$ and

$\left.\frac{d^{j}}{dt^{j}}\right|_{t=0}\left(\varphi\circ\gamma_{1}(t)\right)=\left.\frac{d^{j}}{dt^{j}}\right|_{t=0}\left(\varphi\circ\gamma_{2}(t)\right)\,,$

for $j=0,\ldots,k$. This is a well defined equivalence relation in
$C^{\infty}(\mathbb{R},Q)$ and the equivalence class of a curve $\gamma$ is
denoted $[\gamma]_{0}^{(k)}$. The set of equivalence classes is denoted
$T^{(k)}Q$, and it can be proved that it is a smooth manifold. Moreover, the
map $\tau^{k}_{Q}\colon T^{(k)}Q\to Q$ defined by
$\tau^{k}_{Q}\left([\gamma]_{0}^{(k)}\right)=\gamma(0)$ endows $T^{(k)}Q$ with
a fiber bundle structure over $Q$, and therefore $T^{(k)}Q$ is called the
tangent bundle of order $k$ of $Q$, or $k$th-order tangent bundle of $Q$.

The manifold $T^{(k)}Q$ is endowed with some additional structure. In
particular, for every $0\leqslant r\leqslant k$ we define a surjective
submersion $\tau_{Q}^{(r,k)}\colon T^{(k)}Q\to T^{(r)}Q$ as
$\tau_{Q}^{(r,k)}\left([\gamma]_{0}^{(k)}\right)=[\gamma]_{0}^{(r)}$. It is
easy to see that for every $0\leqslant r\leqslant k$, the map
$\tau_{Q}^{(r,k)}$ defines a fiber bundle structure. Moreover, we have that
$T^{(1)}Q\equiv TQ$ is just the usual tangent bundle of $Q$, $T^{(0)}Q\equiv
Q$ and $\tau_{Q}^{(0,k)}=\tau^{k}_{Q}$.

The $r$-lift of a smooth function $f\in C^{\infty}(Q)$, for $0\leqslant
r\leqslant k$, is the smooth function $f^{(r,k)}\in C^{\infty}(T^{(k)}Q)$
defined as

$f^{(r,k)}\left([\gamma]^{(k)}_{0}\right)=\left.\frac{d^{r}}{dt^{r}}\right|_{t=0}\left(f\circ\gamma(t)\right)\,.$

Of course, these definitions can be applied to functions defined on open sets
of $Q$. Observe that the $0$-lift of $f$ coincides with $f$.

Local coordinates in $T^{(k)}Q$ are introduced as follows. Let $(U,\varphi)$ a
local chart in $Q$ with coordinates $(q^{i})$, $1\leqslant i\leqslant n$. Then
the induced natural coordinates in the open set $(\tau^{k}_{Q})^{-1}(U)\equiv
T^{(k)}U\subseteq T^{(k)}Q$ are
$\left(q_{(0)}^{i},q_{(1)}^{i},\ldots,q_{(k)}^{i}\right)\equiv\left(q^{i}_{(j)}\right)$
with $1\leqslant i\leqslant n$, $0\leqslant j\leqslant k$, where
$q_{(r)}^{i}=(q^{i})^{(r,k)}$ for $0\leqslant r\leqslant k$. Sometimes, we use
the standard conventions $q_{(0)}^{i}\equiv q^{i}$,
$q_{(1)}^{i}\equiv\dot{q}^{i}$ and $q_{(2)}^{i}\equiv\ddot{q}^{i}$.

The canonical immersion $j_{k}\colon T^{(k)}Q\to T(T^{(k-1)}Q)$ is defined as

$j_{k}\left([\gamma]_{0}^{(k)}\right)=[{\gamma}^{(k-1)}]_{0}^{(1)}\,,$ (1)

where ${\gamma}^{(k-1)}$ is the lift of the curve $\gamma$ to $T^{(k-1)}Q$;
that is, the curve ${\gamma}^{(k-1)}\colon\mathbb{R}\to T^{(k-1)}Q$ given by
$\gamma^{(k-1)}(t)=[\gamma_{t}]_{0}^{(k-1)}$ where
$\gamma_{t}(s)=\gamma(t+s)$. In the induced local coordinates of $T^{(k)}Q$,
the map $j_{k}$ is locally given by

$j_{k}\left(q_{(0)}^{i},q_{(1)}^{i},q_{(2)}^{i},\ldots,q_{(k)}^{i}\right)=\left(q_{(0)}^{i},q_{(1)}^{i},\ldots,q_{(k-1)}^{i};q_{(1)}^{i},q_{(2)}^{i},\ldots,q_{(k)}^{i}\right)\,,$

from where we can deduce that in the induced natural coordinates
$\left(q^{i}_{(j)},v^{i}_{(j)}\right)$ of $TT^{(k-1)}Q$, the submanifold
$T^{(k)}Q$ is defined locally by the $(k-1)n$ constraints
$v_{(j)}^{i}=q_{(j+1)}^{i}$.

Denote by $\Omega^{q}(T^{(k)}Q)$ the real vector space of $q$-forms on
$T^{(k)}Q$. In the exterior algebra of differential forms on $T^{(k)}Q$,
denoted $\displaystyle{\bigoplus_{q\geqslant 0}\Omega^{q}(T^{(k)}Q)}$, we
define an equivalence relation as follows: for $\alpha\in\Omega^{q}(T^{(k)}M)$
and $\beta\in\Omega^{q}(T^{(k^{\prime})}Q)$,

$\alpha\sim\beta\Longleftrightarrow\begin{cases}\alpha=\left(\tau_{Q}^{(k^{\prime},k)}\right)^{*}(\beta)&\mbox{
if }k^{\prime}\leqslant k\\\
\beta=\left(\tau_{Q}^{(k,k^{\prime})}\right)^{*}(\alpha)&\mbox{ if
}k^{\prime}\geqslant k.\end{cases}\ $

Consider the quotient set $\displaystyle{{\Omega}=\bigoplus_{k\geqslant 0}\
df^{q}(T^{(k)}Q)\,/\sim}$, which is a commutative graded algebra. In this set
we define the Tulczyjew’s derivation, denoted by $d_{T}$, as follows: for
every $f\in C^{\infty}(T^{(k)}Q)$ the function $d_{T}f\in
C^{\infty}(T^{(k+1)}Q)$ is defined as
$\displaystyle{d_{T}f\left([\gamma]^{(k+1)}_{0}\right)=\left\langle
d_{[\gamma]^{(k)}_{0}}f\,,j_{k+1}\left([\gamma]^{(k+1)}_{0}\right)\right\rangle}$
where $j_{k+1}\colon T^{(k+1)}Q\to T(T^{(k)}Q)$ is the canonical immersion,
and the covector $d_{[\gamma]^{(k)}_{0}}f\in
T^{*}_{[\gamma]^{(k)}_{0}}T^{(k)}Q$ is the exterior derivative of $f$ at
$[\gamma]^{(k)}_{0}\in T^{(k)}Q$. Using the coordinate expression for
$j_{k+1}$, the function $d_{T}$ is given locally by

$d_{T}f\left(q_{(0)}^{i},\ldots,q_{(k+1)}^{i}\right)=\sum_{j=0}^{k}q_{(j+1)}^{i}\displaystyle\frac{\partial{f}}{\partial{q_{(j)}^{i}}}\left(q_{(0)}^{i},\ldots,q_{(k)}^{i}\right)\,.$

The map $d_{T}$ extends to a derivation of degree $0$ in $\Omega$ and, as
$d_{T}d=dd_{T}$, it is determined by its action on functions and by the
property $d_{T}(dq_{(j)}^{i})=dq_{(j+1)}^{i}$.

######  Definition 7.

A curve $\psi\colon\mathbb{R}\to T^{(k)}Q$ is holonomic of type $r$,
$1\leqslant r\leqslant k$, if
$\phi^{(k-r+1,k)}=\tau_{Q}^{(k-r+1,k)}\circ\psi$, where
$\phi=\tau_{Q}^{k}\circ\psi\colon\mathbb{R}\to Q$; that is, the curve $\psi$
is the lifting of a curve in $Q$ up to $T^{(k-r+1)}Q$.

From Definition 7, a vector field $X\in\mathfrak{X}(T^{(k)}Q)$ is a semispray
of type $r$, $1\leqslant r\leqslant k$, if every integral curve $\psi$ of $X$
is holonomic of type $r$. In the natural cordinates of $T^{(k)}Q$, the local
expression of a semispray of type $r$ is

$X=q_{(1)}^{i}\displaystyle\frac{\partial{}}{\partial{q_{(0)}^{i}}}+q_{(2)}^{i}\displaystyle\frac{\partial{}}{\partial{q_{(1)}^{i}}}+\ldots+q_{(k-r+1)}^{i}\displaystyle\frac{\partial{}}{\partial{q_{(k-r)}^{i}}}+F_{(k-r+1)}^{i}\displaystyle\frac{\partial{}}{\partial{q_{(k-r+1)}^{i}}}+\ldots+F_{(k)}^{i}\displaystyle\frac{\partial{}}{\partial{q_{(k)}^{i}}}\,.$
(2)

######  Remark.

It is clear that every holonomic curve of type $r$ is also holonomic of type
$s$, for $s\geqslant r$. The same remark is true for semisprays. $\lozenge$

##  3 Geometric description of higher-order dynamical systems

In this Section we aim at studying fiber derivatives of higher-order
Lagrangian systems, relating the classical Legendre-Ostrogradsky map and the
$k$th-order Legendre transformation.

###  3.1 Higher-order Tulczyjew triple and dynamics generated by Lagrangian
submanifolds

In this Subsection we explore some new results in the construction of the
Tulczyjew’s triple for higher-order dynamical systems of M. de León and E.
Lacomba [16, 17]. In particular, we study fiber derivatives of higher-order
Lagrangian systems, relating the classical Legendre-Ostrogradsky map
associated to a $k$th-order Lagrangian function $L$ and the $k$th-order
Legendre transformation defined on the Lagrangian submanifold
$\Sigma_{L}\subset T^{*}TT^{(k-1)}Q$ generated by the Lagrangian $L$. We show
the theory with a simple but interesting example, the dynamics of the end of a
thrown javelin.

######  Definition 8.

The $k^{th}$-order Tulczyjew’s isomorphism is the map
$\beta_{T^{(k-1)}Q}\colon TT^{*}T^{(k-1)}Q\to T^{*}T^{*}T^{(k-1)}Q$ defined by
$\beta_{T^{(k-1)}Q}(V):=i_{V}\omega_{T^{(k-1)}Q}$ with $V\in
TT^{*}T^{(k-1)}Q$, and $\omega_{T^{(k-1)}Q}$ being the canonical symplectic
form of $T^{*}T^{(k-1)}Q$.

Let $(q^{i})$, $1\leqslant i\leqslant n$, be local coordinates in an open set
$U\subset Q$, and $(q_{(j)}^{i})$, $0\leqslant j\leqslant k-1$, the induced
coordinates in $(\pi_{Q}^{k-1})^{-1}(U)\subset T^{(k-1)}Q$ introduced in
Section 2.4. Then, natural coordinates in
$\left(\pi_{Q}^{k-1}\circ\pi_{T^{(k-1)}Q}\right)^{-1}(U)\subset
T^{*}T^{(k-1)}Q$ are $\left(q_{(j)}^{i},p^{(j)}_{i}\right)$, from where we
deduce that the induced local natural coordinates in $TT^{*}T^{(k-1)}Q$ are
$\left(q_{(j)}^{i},p^{(j)}_{i},\dot{q}_{(j)}^{i},\dot{p}^{(j)}_{i}\right)$,
with $1\leqslant i\leqslant n$ and $0\leqslant j\leqslant k-1$. In these
coordinates, the map $\beta_{T^{(k-1)}Q}$ is locally given by
$\beta_{T^{(k-1)}Q}\left(q_{(j)}^{i},p^{(j)}_{i},\dot{q}_{(j)}^{i},\dot{p}^{(j)}_{i}\right)=\left(q_{(j)}^{i},\dot{q}_{(j)}^{i},\dot{p}^{(j)}_{i},p^{(j)}_{i}\right)$.
This map is an anti-symplectomorphism when we consider $T^{*}T^{*}T^{(k-1)}Q$
endowed with the canonical symplectic structure and $TT^{*}T^{(k-1)}Q$ endowed
with the symplectic structure given by the complete lift
$\omega_{T^{(k-1)}Q}^{c}$ of the canonical symplectic form on
$T^{*}T^{(k-1)}Q$.

The cotangent bundles $T^{*}T^{*}T^{(k-1)}Q$ and $T^{*}TT^{(k-1)}Q$ are
examples of double vector bundles (see [29] for details). In particular, the
double vector bundles $T^{*}T^{*}T^{(k-1)}Q$ and $T^{*}TT^{(k-1)}Q$ are
canonically isomorphic via a vector bundle isomorphism over $T^{*}T^{(k-1)}Q$

${\mathcal{R}}_{k}\colon T^{*}TT^{(k-1)}Q\to T^{*}T^{*}T^{(k-1)}Q\,.$

This map is an anti-symplectomorphism of symplectic manifolds (considering in
both cotangent bundles the canonical symplectic structures), and also an
isomorphism of double vector bundles. It is completely determined by the
condition

$\left\langle\mathcal{R}_{k}(\alpha_{u}),W_{T^{*}\tau_{T^{(k-1)}Q}(\alpha_{u})}\right\rangle=-\left\langle\alpha_{u},\widetilde{W}_{u}\right\rangle+\left\langle
W_{T^{*}\tau_{T^{(k-1)}Q}(\alpha_{u})},\widetilde{W}_{u}\right\rangle^{T}\,,$

for every $\alpha_{u}\in T^{*}_{u}TT^{(k-1)}Q$, $\widetilde{W}_{u}\in
T_{u}TT^{(k-1)}Q$ and $W_{T^{*}\tau_{T^{(k-1)}Q}(\alpha_{u})}\in
TT^{*}T^{(k-1)}Q$ satisfying the relation
$\displaystyle{T\tau_{T^{(k-1)}Q}(\widetilde{W}_{u})=T\pi_{T^{(k-1)}Q}(W_{T^{*}\tau_{Q}(\alpha_{u})})\,}$.

Here, $\langle\cdot,\cdot\rangle^{T}\colon
TT^{*}T^{(k-1)}Q\times_{TT^{(k-1)}Q}TTT^{(k-1)}Q\to\mathbb{R}$ is the pairing
defined by the tangent map of the usual pairing
$\langle\cdot,\cdot\rangle\colon
T^{*}T^{(k-1)}Q\times_{T^{(k-1)}Q}TT^{(k-1)}Q\to\mathbb{R}$, and the vector
bundle projection $T^{*}\tau_{T^{(k-1)}Q}\colon T^{*}TT^{(k-1)}Q\to
T^{*}T^{(k-1)}Q$ is characterized by $\langle
T^{*}\tau_{T^{(k-1)}Q}(\alpha_{u}),w\rangle=\langle\alpha_{u},w_{u}^{\vee}\rangle$
where $u,w\in T_{[q]_{(0)}^{(k-1)}}T^{(k-1)}Q$, $\alpha_{u}\in
T^{*}_{u}TT^{(k-1)}Q$, and $w_{u}^{\vee}\in T_{u}TT^{(k-1)}Q$ is the vertical
lift of the tangent vector $w$ (see [21] for first order systems. The
derivation for higher-order systems is derived straightforwardly from the
definition given for first order systems).

Let $(q^{i})$, $1\leqslant i\leqslant n$, be local coordinates in $Q$, and
$(q_{(j)}^{i})$, $1\leqslant i\leqslant n$, $0\leqslant j\leqslant k-1$, the
induced coordinates in $T^{(k-1)}Q$ introduced in Section 2.4. Then, natural
coordinates in $TT^{(k-1)}Q$ are $\left(q_{(j)}^{i},v_{(j)}^{i}\right)$, from
where we deduce that the induced natural coordinates in $T^{*}TT^{(k-1)}Q$ are
$\left(q_{(j)}^{i},v_{(j)}^{i},p^{(j)}_{i},\tilde{p}^{(j)}_{i}\right)$, with
$1\leqslant i\leqslant n$ and $0\leqslant j\leqslant k-1$. In these
coordinates, the map $\mathcal{R}_{k}$ is locally given by

${\mathcal{R}}_{k}\left(q_{(j)}^{i},v_{(j)}^{i},p^{(j)}_{i},\tilde{p}^{(j)}_{i}\right)=\left(q_{(j)}^{i},\tilde{p}^{(j)}_{i},-p^{(j)}_{i},v_{(j)}^{i}\right)\,.$

Then, composing $\beta_{T^{(k-1)}Q}$ with $\mathcal{R}_{k}^{-1}$ we obtain a
map $\alpha_{T^{(k-1)}Q}\colon TT^{*}T^{(k-1)}Q\to T^{*}TT^{(k-1)}Q$, which is
given in the natural coordinates
$\left(q_{(j)}^{i},p^{(j)}_{i},\dot{q}_{(j)}^{i},\dot{p}^{(j)}_{i}\right)$ in
$TT^{*}T^{(k-1)}Q$ by

$\alpha_{T^{(k-1)}Q}\left(q_{(j)}^{i},p^{(j)}_{i},\dot{q}_{(j)}^{i},\dot{p}^{(j)}_{i}\right)=\left(q_{(j)}^{i},\dot{q}_{(j)}^{i},\dot{p}^{(j)}_{i},p^{(j)}_{i}\right)\,.$
(3)

This map is a symplectomorphism when we consider in $TT^{*}T^{(k-1)}Q$ the
symplectic structure given by the complete lift $\omega_{T^{(k-1)}Q}^{c}$ of
the canonical symplectic form $\omega_{T^{(k-1)}Q}$ on $T^{*}T^{(k-1)}Q$ and
on $T^{*}TT^{(k-1)}Q$ the canonical symplectic form $\omega_{TT^{(k-1)}Q}$.
The maps $\beta_{T^{(k-1)}Q}$ and $\alpha_{T^{(k-1)}Q}$ give rise to the
$k$th-order Tulczyjew triple

$\textstyle{T^{*}TT^{(k-1)}Q}$
$\textstyle{TT^{*}T^{(k-1)}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{T^{(k-1)}Q}}$$\scriptstyle{\beta_{T^{(k-1)}Q}}$
$\textstyle{T^{*}T^{*}T^{(k-1)}Q,}$

######  Remark.

The map $\alpha_{T^{(k-1)}Q}\colon TT^{*}T^{(k-1)}Q\to T^{*}TT^{(k-1)}Q$ can
be obtained directly as the dual to the canonical flip
$\kappa_{T^{(k-1)}Q}\colon TT^{(k-1)}Q\to TT^{(k-1)}Q$, which is an
isomorphism of double vector bundle structures on $TT^{(k-1)}Q$ (see [17] for
more details). We prefer to avoid the use of the canonical flip by using
$\mathcal{R}_{k}^{-1}$, as in [28]. $\lozenge$

Now we introduce the dynamics using a suitable Lagrangian submanifold in
$T^{*}TT^{(k-1)}Q$ and the $k$th-order Tulczyjew’s triple. First, let
$j_{k}\colon T^{(k)}Q\hookrightarrow TT^{(k-1)}Q$ be the canonical immersion
defined in (1). Then, if $x\in T^{(k)}Q$, the map $j_{k}^{*}\colon
T_{j_{k}(x)}^{*}(TT^{(k-1)}Q)\to T_{x}^{*}(T^{(k)}Q)$ is given by

$j_{k}^{*}\mu=\mu\circ Tj_{k}\,,\ \mbox{for every }\mu\in
T_{j_{k}(x)}^{*}TT^{(k-1)}Q\,.$

Using this map, if $L\in C^{\infty}(T^{(k)}Q)$ is a $k$th-order Lagrangian
function, by Śniatycki and Tulczyjew’s construction given in Theorem 1 we
define a Lagrangian submanifold in the cotangent bundle $T^{*}TT^{(k-1)}Q$,
endowed with the canonical symplectic structure, as follows

$\Sigma_{L}=\left\\{\mu\in T^{*}TT^{(k-1)}Q\mid
j_{k}^{*}\mu=dL\right\\}\hookrightarrow T^{*}TT^{(k-1)}Q\,.$

The Lagrangian submanifold $\Sigma_{L}$ fibers onto $j_{k}(T^{(k)}Q)$, it is
locally parametrized by the $2kn$ coordinate functions
$\left(q_{(0)}^{i},\ldots,q_{(k)}^{i},\tilde{p}^{(0)}_{i},\ldots,\tilde{p}^{(k-2)}_{i}\right)$,
$1\leqslant i\leqslant n$, and it is immersed into $T^{*}TT^{(k-1)}Q$ as

$\left\\{\left(q_{(j)}^{i}\,;\,q_{(j+1)}^{i}\,;\,\displaystyle\frac{\partial{L}}{\partial{q_{(0)}^{i}}},\displaystyle\frac{\partial{L}}{\partial{q_{(1)}^{i}}}-\tilde{p}^{(0)}_{i},\ldots,\displaystyle\frac{\partial{L}}{\partial{q_{(k-1)}^{i}}}-\tilde{p}^{(k-2)}_{i}\,;\,\tilde{p}^{(0)}_{i},\ldots,\tilde{p}^{(k-2)}_{i},\displaystyle\frac{\partial{L}}{\partial{q_{(k)}^{i}}}\right)\right\\}\,.$

Therefore, taking this into account, the Lagrangian dynamics is given by the
Lagrangian submanifold
$N_{L}=\alpha_{T^{(k-1)}Q}^{-1}(\Sigma_{L})\hookrightarrow TT^{*}T^{(k-1)}Q$.
Locally, $N_{L}$ is the set of elements in $TT^{*}T^{(k-1)}Q$ of the form

$\left(q_{(j)}^{i}\,;\,\tilde{p}^{(0)}_{i},\ldots,\tilde{p}^{(k-2)}_{i},\displaystyle\frac{\partial{L}}{\partial{q_{(k)}^{i}}}\,;\,q_{(j+1)}^{i}\,;\,\displaystyle\frac{\partial{L}}{\partial{q_{(0)}^{i}}},\displaystyle\frac{\partial{L}}{\partial{q_{(1)}^{i}}}-\tilde{p}^{(0)},\ldots,\displaystyle\frac{\partial{L}}{\partial{q_{(k-1)}^{i}}}-\tilde{p}^{(k-2)}_{i}\right)\,.$

From this, the submanifold $N_{L}$ determines the following set of
differential equations

$\displaystyle\frac{d}{dt}\tilde{p}^{(0)}_{i}=\displaystyle\frac{\partial{L}}{\partial{q_{(0)}^{i}}}\,,$
(4)
$\displaystyle\frac{d}{dt}\tilde{p}^{(j)}_{i}+\tilde{p}^{(j-1)}_{i}=\displaystyle\frac{\partial{L}}{\partial{q_{(j)}^{i}}}\,,$
(5)
$\displaystyle\displaystyle\frac{\partial{L}}{\partial{q_{(k-1)}^{i}}}-\tilde{p}^{(k-2)}_{i}=\frac{d}{dt}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(k)}^{i}}}\right)\,,$
(6)

where $1\leqslant j\leqslant k-2$ in (5), and $1\leqslant i\leqslant n$ in
every set. Differentiating the $n$ equations (6) with respect to the time $t$,
and replacing into equation (5) for $j=k-2$, we obtain the following equations

$\frac{d^{2}}{dt^{2}}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(k)}^{i}}}\right)=\frac{d}{dt}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(k-1)}^{i}}}\right)-\displaystyle\frac{\partial{L}}{\partial{q_{(k-2)}^{i}}}-\tilde{p}^{(k-3)}_{i}\,.$

Differentiating the last set of equation with respect to the time $t$ and
replacing the result into (5) when $j=k-3$ we have

$\frac{d^{3}}{dt^{3}}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(k)}^{i}}}\right)=\frac{d^{2}}{dt^{2}}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(k-1)}^{i}}}\right)-\frac{d}{dt}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(k-2)}^{i}}}\right)+\displaystyle\frac{\partial{L}}{\partial{q_{(k-3)}^{i}}}-\tilde{p}^{(k-4)}_{i}\,.$

Iterating the process $k-4$ times, we obtain the following set of $n$
equations

$\frac{d^{k}}{dt^{k}}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(k)}^{i}}}\right)=\frac{d^{k-1}}{dt^{k-1}}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(k-1)}^{i}}}\right)-\frac{d^{k-2}}{dt^{k-2}}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(k-2)}^{i}}}\right)+\ldots-\frac{d}{dt}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(1)}^{i}}}\right)+\frac{d}{dt}\tilde{p}^{(0)}_{i}\,.$

Using equations (4) we obtain the following $n$ differential equations

$\sum_{j=0}^{k}(-1)^{j}\frac{d^{j}}{dt^{j}}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(j)}^{i}}}\right)=0\,,$

which are exactly the higher-order Euler-Lagrange equations for the higher-
order Lagrangian function $L$ (see [17]). From the computations and
considerations given above, we have the following result.

######  Proposition 3.

The solutions of a $k$th-order Lagrangian system described by a $k$th-order
Lagrangian function $L\in C^{\infty}(T^{(k)}Q)$ are curves $\mu\colon
I\subset\mathbb{R}\to\Sigma_{L}$ satisfying

$\left.\pi_{TT^{(k-1)}Q}\right|_{\Sigma_{L}}\circ\mu=j_{k}\circ\gamma^{(k)}\,,$

where $\gamma^{(k)}\colon I\to T^{(k)}Q$ is the $k$-lift of a curve
$\gamma\colon I\to Q$, $j_{k}\colon T^{(k)}Q\to TT^{(k-1)}Q$ the canonical
immersion, and $\left.\pi_{TT^{(k-1)}Q}\right|_{\Sigma_{L}}\colon\Sigma_{L}\to
TT^{(k-1)}Q$ denotes the restriction of the canonical projection
$\pi_{TT^{k-1}Q}\colon T^{*}TT^{(k-1)}Q\to TT^{(k-1)}Q$ to $\Sigma_{L}$.

######  Remark.

Observe that the three spaces $T(T^{*}T^{(k-1)}Q),$ $T^{*}(T^{*}T^{(k-1)}Q)$
and $T^{*}(T^{(k-1)}Q)$ involved in the Tulczyjew triple are symplectic
manifolds; the two maps $\alpha_{T^{(k-1)}Q}$ and $\beta_{T^{(k-1)}Q}$
involved in the construction are a symplectomorphism and an anti-
symplectomorphism, respectively; and the dynamical equations (Euler-Lagrange
and Hamilton equations) are the local equations defining the Lagrangian
submanifolds

$N_{L}=\alpha_{T^{(k-1)}Q}^{-1}(\Sigma_{L})\ \mbox{ and }\
S_{H}=\beta_{T^{(k-1)}Q}^{-1}\left(dH(T^{*}(T^{(k-1)}Q))\right)\,,$

respectively. Moreover, the Lagrangian and Hamiltonian functions are not
involved in the definition of the triple. In this sense, the triple is
canonical. Finally, we would like to point out that the construction can be
applied to an arbitrary Lagrangian function, not necessarily to a regular one.
$\lozenge$

######  Remark (Higher-order variational constrained equations).

The natural extension for constrained (vakonomic) higher-order mechanical
systems can be studied by considering the embedded submanifold
$\mathcal{M}\subset T^{(k)}Q$ given by the vanishing of $m$ independent
constraint functions $\Phi^{\alpha}\colon T^{(k)}Q\to\mathbb{R}$,
$\alpha=1,\ldots,m$. We consider the extended Lagrangian
$\mathcal{L}=L+\lambda_{\alpha}\Phi^{\alpha}$ which includes the Lagrange
multipliers $\lambda_{\alpha}$ as a new extra variable. The equations of
motion for the higher-order constrined variational problem are the higher-
order Euler-Lagrange equations for $\mathcal{L}$, that is,

$\displaystyle\sum_{r=0}^{k}(-1)^{r}\frac{d^{r}}{dt^{r}}\left(\displaystyle\frac{\partial{L}}{\partial{q^{(r)i}}}+\lambda_{\alpha}\displaystyle\frac{\partial{\Phi^{\alpha}}}{\partial{q^{(r)i}}}\right)=$
$\displaystyle 0$ (7)
$\displaystyle\Phi^{\alpha}\left(q_{(0)}^{i},\ldots,q^{i}_{(k-1)},q_{(k)}^{i}\right)=$
$\displaystyle 0$ (8)

From the geometrical point of view, these kind of higher-order variationally
constrained problems are determined by a submanifold $\mathcal{M}\subset
T^{(k)}Q$ with inclusion $i:\mathcal{M}\hookrightarrow T^{(k)}Q$ and by a
Lagrangian $L_{\mathcal{M}}:\mathcal{M}\to\mathbb{R}$. Using Theorem (1) we
deduce that $\Sigma_{L_{\mathcal{M}}}\subset T^{*}TT^{(k-1)}Q$ is a Lagrangian
submanifold. Moreover, using the Tulczyjew’s symplectomorphism one induce a
new Lagrangian submanifold
$\alpha_{T^{(k-1)}Q}^{-1}(\Sigma_{L_{\mathcal{M}}})\subset TT^{*}T^{(k-1)}Q$
which completely determines the constrained variational dynamics. The case of
unconstrained mechanics is generated taking the whole space $T^{(k)}Q$ instead
of $\mathcal{M}$ and a Lagrangian function $L:T^{(k)}Q\to\mathbb{R}$. Indeed,
this procedure gives the correct dynamics for the higher-order constrained
variational problem using the same ideas as in the previous subsection only by
changing the Lagrangian $L:T^{(k)}Q\to\mathbb{R}$ by $\mathcal{L}$. This is
basically because the powerful of Tulczyjew’s triple does not depends on the
Lagrangian function.

We assume that the restriction of the projection
$(\tau_{Q}^{(k-1,k)})\Big{|}_{{\mathcal{M}}}:{\mathcal{M}}\to T^{(k-1)}Q$ is a
submersion. Locally, this conditions means that the $m\times n$-matrix

$\left(\frac{\partial(\Phi^{1},...,\Phi^{m})}{\partial(q^{1}_{(k)},...,q^{n}_{(k)})}\right)$

is of rank $m$ at all points of ${\mathcal{M}}$.

Consequently, by the implicit function theorem, we can locally express the
constraints (reordering coordinates if necessary) as

$\phi^{\alpha}\left(q_{(0)}^{i},\ldots,q^{i}_{(k-1)},q_{(k)}^{a}\right)=q_{(k)}^{\alpha}\,,\
1\leqslant\alpha\leqslant m\,,\ m+1\leqslant a\leqslant n\,,\ i=1,\ldots,n\,,$
(9)

and therefore, we can define a Lagrangian function $L\in
C^{\infty}(\mathcal{M})$ in these new adapted coordinates to $\mathcal{M}$.
Observe that $\mathcal{M}\hookrightarrow T^{(k)}Q\hookrightarrow TT^{(k-1)}Q$,
and let us denote by $i_{\mathcal{M}}\colon\mathcal{M}\hookrightarrow
TT^{(k-1)}Q$ the composition of both inclusions. Now construct

$\Sigma_{L,\mathcal{M}}=\left\\{\mu\in T^{*}TT^{(k-1)}Q\mid
i_{\mathcal{M}}^{*}\mu=dL\right\\}$

and proceeding as in the unconstrained case we obtain that the higher-order
constrained dynamics in governed by the following set of $n$ ordinary
differential equations of order $2k$

$\sum_{r=0}^{k}(-1)^{r}\frac{d^{r}}{dt^{r}}\left(\displaystyle\frac{\partial{L}}{\partial{q^{(r)i}}}-\mu_{\alpha}\displaystyle\frac{\partial{\Phi^{\alpha}}}{\partial{q^{(r)i}}}\right)=0\,,$
(10)

satisfying the constraints. Observe that $\mu\in T^{*}TT^{(k-1)}Q$ plays the
role of Lagrange multipliers for the equations, forcing the dynamics to
satisfy the constraints imposed by the submanifold $\mathcal{M}\subset
T^{(k)}Q$. $\lozenge$

###  3.2 On the Legendre maps for higher-order dynamical systems

In this Subsection we introduce a Legendre transformation (a fiber derivative)
$\mathbb{F}L\colon\Sigma_{L}\to T^{*}T^{(k-1)}Q$ in the Lagrangian submanifold
generated by a $k$th-order Lagrangian function $L\in C^{\infty}(T^{(k)}Q)$ and
we study its relationship with the Legendre-Ostrogradsky map
$\operatorname{leg}_{L}\colon T^{(2k-1)}Q\to T^{*}T^{(k-1)}Q$.

Let $Q$ be the configuration space of an autonomous dynamical system of order
$k$ with $n$ degrees of freedom, and let $L\in C^{\infty}(T^{(k)}Q)$ be the
Lagrangian function for this system. From the Lagrangian function $L$ we
construct the Poincaré-Cartan $1$-form $\theta_{L}\in\Omega^{1}(T^{(2k-1)}Q)$,
whose coordinate expression is

$\theta_{L}=\sum_{r=1}^{k}\sum_{j=0}^{k-r}(-1)^{j}d_{T}^{j}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(r+j)}^{i}}}\right)dq_{(r-1)}^{i}\,.$
(11)

The Poincaré-Cartan $1$-form $\theta_{L}\in\Omega^{1}(T^{(2k-1)}Q)$ allows to
define the Legendre-Ostrogradsky map as follows:

######  Definition 9.

The Legendre-Ostrogradsky map associated to the $k$th-order Lagrangian
function $L$ is the fiber bundle morphism $\operatorname{leg}_{L}\colon
T^{(2k-1)}Q\to T^{*}T^{(k-1)}Q$ over $T^{(k-1)}Q$ defined as follows: for
every $u\in TT^{(2k-1)}Q$,

$\theta_{L}(u)=\left\langle
T\tau_{Q}^{(k-1,2k-1)}(u)\,,\,\operatorname{leg}_{L}(\tau_{T^{(2k-1)}Q}(u)\right\rangle\,.$
(12)

Besides the condition
$\pi_{T^{(k-1)}Q}\circ\operatorname{leg}_{L}=\tau_{Q}^{(k-1,2k-1)}$ stated in
the definition, the Legendre-Ostrogradsky map relates the Liouville form in
$T^{*}T^{(k-1)}Q$ to the Poincaré-Cartan $1$-form. That is, if
$\theta_{T^{(k-1)}Q}\in\Omega^{1}(T^{*}T^{(k-1)}Q)$ is the canonical form of
the cotangent bundle $T^{*}T^{(k-1)}Q$, then
$\operatorname{leg}_{L}^{*}\theta_{T^{(k-1)}Q}=\theta_{L}$.

In the local coordinates $\left(q_{(j)}^{i}\right)$, $1\leqslant i\leqslant
n$, $0\leqslant j\leqslant 2k-1$, of $T^{(2k-1)}Q$ introduced in Section 2.4,
we define the following local functions

$\hat{p}^{(r-1)}_{i}=\sum_{j=0}^{k-r}(-1)^{j}d_{T}^{j}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(r+j)}^{i}}}\right)\,,$
(13)

which are called the Jacobi-Ostrogradsky momenta. Observe that we have the
following relation between $\hat{p}^{(r)}_{i}$ and $\hat{p}^{(r-1)}_{i}$

$\hat{p}^{(r-1)}_{i}=\displaystyle\frac{\partial{L}}{\partial{q_{(r)}^{i}}}-d_{T}\left(\hat{p}^{(r)}_{i}\right)\,,\
\mbox{for }1\leqslant r\leqslant k-1\,.$ (14)

######  Remark.

The relation (14) means that we can recover all the Jacobi-Ostrogradsky
momenta coordinates from the set of highest order momenta
$(\hat{p}^{(k-1)}_{i})$. $\lozenge$

Bearing in mind the local expression of the form $\theta_{L}$, we can write
$\theta_{L}=\hat{p}^{(j)}_{i}dq_{(j)}^{i}$. Let $(U;(q^{i}))$, $1\leqslant
i\leqslant n$, be a local chart of $Q$, and $\left(q_{(j)}^{i}\right)$,
$0\leqslant j\leqslant 2k-1$, the induced local coordinates in
$(\tau_{Q}^{2k-1})^{-1}(U)\subset T^{(2k-1)}Q$ introduced in Section 2.4. From
this, it is clear that the local expression of the Legendre-Ostrogradsky map
$\operatorname{leg}_{L}$ is

$\operatorname{leg}_{L}^{*}\left(q_{(r-1)}^{i}\right)=q_{(r-1)}^{i}\quad;\quad\operatorname{leg}_{L}^{*}\left(p_{i}^{(r-1)}\right)=\hat{p}_{i}^{(r-1)}=\sum_{j=0}^{k-r}(-1)^{j}d_{T}^{j}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(r+j)}^{i}}}\right)\,,$

where $1\leqslant r\leqslant k$, that is,

$\operatorname{leg}_{L}\left(q_{(0)}^{i},\ldots,q_{(2k-1)}^{i}\right)=\left(q_{(0)}^{i},\ldots,q_{(k-1)}^{i},\hat{p}_{i}^{(0)},\ldots,\hat{p}_{i}^{(k-1)}\right)\,.$

Consider the tangent map $T\operatorname{leg}_{L}\colon T(T^{(2k-1)}Q)\to
T(T^{*}T^{(k-1)}Q)$. Given an arbitrary point $[\gamma]_{0}^{(2k-1)}\in
T^{(2k-1)}Q$, the tangent map of $\operatorname{leg}_{L}$ at
$[\gamma]_{0}^{(2k-1)}$ is locally given by the following $2kn\times 2kn$
matrix

$T_{[\gamma]_{0}^{(2k-1)}}\operatorname{leg}_{L}=\left(\begin{array}[]{cccc|cccc}\textnormal{Id}_{n}&\mathbf{0}_{n}&\ldots&\mathbf{0}_{n}&\mathbf{0}_{n}&\mathbf{0}_{n}&\ldots&\mathbf{0}_{n}\\\
\mathbf{0}_{n}&\textnormal{Id}_{n}&\ldots&\mathbf{0}_{n}&\mathbf{0}_{n}&\mathbf{0}_{n}&\ldots&\mathbf{0}_{n}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
\mathbf{0}_{n}&\mathbf{0}_{n}&\ldots&\textnormal{Id}_{n}&\mathbf{0}_{n}&\mathbf{0}_{n}&\ldots&\mathbf{0}_{n}\\\
\hline\cr&&&&&&&\\\\[-10.0pt]
\displaystyle\frac{\partial{\hat{p}_{i}^{(0)}}}{\partial{q_{(0)}^{j}}}&\displaystyle\frac{\partial{\hat{p}_{i}^{(0)}}}{\partial{q_{(1)}^{j}}}&\ldots&\displaystyle\frac{\partial{\hat{p}_{i}^{(0)}}}{\partial{q_{(k-1)}^{j}}}&\displaystyle\frac{\partial{\hat{p}_{i}^{(0)}}}{\partial{q_{(k)}^{j}}}&\displaystyle\frac{\partial{\hat{p}_{i}^{(0)}}}{\partial{q_{(k+1)}^{j}}}&\ldots&\displaystyle\frac{\partial{\hat{p}_{i}^{(0)}}}{\partial{q_{(2k-1)}^{j}}}\\\\[15.0pt]
\displaystyle\frac{\partial{\hat{p}_{i}^{(1)}}}{\partial{q_{(0)}^{j}}}&\displaystyle\frac{\partial{\hat{p}_{i}^{(1)}}}{\partial{q_{(1)}^{j}}}&\ldots&\displaystyle\frac{\partial{\hat{p}_{i}^{(1)}}}{\partial{q_{(k-1)}^{j}}}&\displaystyle\frac{\partial{\hat{p}_{i}^{(1)}}}{\partial{q_{(k)}^{j}}}&\displaystyle\frac{\partial{\hat{p}_{i}^{(1)}}}{\partial{q_{(k+1)}^{j}}}&\ldots&\displaystyle\frac{\partial{\hat{p}_{i}^{(1)}}}{\partial{q_{(2k-1)}^{j}}}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
\displaystyle\frac{\partial{\hat{p}_{i}^{(k-1)}}}{\partial{q_{(0)}^{j}}}&\displaystyle\frac{\partial{\hat{p}_{i}^{(k-1)}}}{\partial{q_{(1)}^{j}}}&\ldots&\displaystyle\frac{\partial{\hat{p}_{i}^{(k-1)}}}{\partial{q_{(k-1)}^{j}}}&\displaystyle\frac{\partial{\hat{p}_{i}^{(k-1)}}}{\partial{q_{(k)}^{j}}}&\displaystyle\frac{\partial{\hat{p}_{i}^{(k-1)}}}{\partial{q_{(k+1)}^{j}}}&\ldots&\displaystyle\frac{\partial{\hat{p}_{i}^{(k-1)}}}{\partial{q_{(2k-1)}^{j}}}\end{array}\right)\,,$

where $\textnormal{Id}_{n}$ denotes the $n\times n$ identity matrix,
$\mathbf{0}_{n}$ the $n\times n$ null matrix and
$(\partial{\hat{p}_{i}^{(r)}}/\partial{q_{(s)}^{j}})$ the $n\times n$ Jacobian
matrix of the vector function $(\hat{p}_{1}^{(r)},\ldots,\hat{p}_{n}^{(r)})$
with respect to the $n$ variables $(q_{(s)}^{1},\ldots,q_{(s)}^{n})$.
Moreover, using the relation (14) among the momenta function in combination
with the coordinate expression of the Tulczyjew’s derivation, a long but
straightforward computation shows that

$\displaystyle\frac{\partial{\hat{p}^{(s)}_{i}}}{\partial{q_{(2k-1-s)}^{j}}}=\displaystyle\frac{\partial^{2}{L}}{\partial{q_{(k)}^{i}}\partial{q_{(k)}^{j}}}\,,$

from where we deduce that the antidiagonal $n\times n$ blocks in the lower
right submatrix of $T_{[\gamma]_{0}^{(2k-1)}}\operatorname{leg}_{L}$ coincide
with the Hessian matrix of the Lagrangian function. Therefore, it is clear
that the Lagrangian function $L\in C^{\infty}(T^{(k)}Q)$ is regular if, and
only if, the bundle morphism $\operatorname{leg}_{L}\colon T^{(2k-1)}Q\to
T^{*}T^{(k-1)}Q$ is a local diffeomorphism. As a consequence of this, we have
that if $L$ is a $k$th-order regular Lagrangian then the set
$(q^{A}_{(i)},\hat{p}_{A}^{(i)})$, $0\leqslant i\leqslant k-1$, is a set of
local coordinates in $T^{(2k-1)}Q$, and $(\hat{p}^{(i)}_{A})$ are called the
Jacobi-Ostrogradsky momenta coordinates.

Now, we introduce a Legendre transformation $\mathbb{F}L\colon\Sigma_{L}\to
T^{*}(T^{(k-1)}Q)$ in the Lagrangian submanifold generated by a higher-order
Lagrangian function.

######  Definition 10.

The higher-order Legendre transformation on the Lagrangian submanifold
$\Sigma_{L}$, $\mathbb{F}L\colon\Sigma_{L}\to T^{*}T^{(k-1)}Q$, is the map
defined by
$\mathbb{F}L=\left.\tau_{T^{*}(T^{(k-1)}Q)}\circ(\alpha_{T^{k-1}Q})^{-1}\right|_{\Sigma_{L}}$.

In the natural coordinates
$\left(q_{(0)}^{i},\ldots,q_{(k)}^{i},\tilde{p}^{(0)}_{i},\ldots,\tilde{p}^{(k-2)}_{i}\right)$
of $\Sigma_{L}$ introduced in Subsection 3.1, the map $\mathbb{F}L$ is locally
determined by

$\mathbb{F}L(q_{(0)}^{i},\ldots,q_{(k)}^{i},\tilde{p}^{(0)}_{i},\ldots,\tilde{p}^{(k-2)}_{i})=\left(q_{(0)}^{i},\ldots,q_{(k-1)}^{i},\tilde{p}^{(0)}_{i},\ldots,\tilde{p}^{(k-2)}_{i},\displaystyle\frac{\partial{L}}{\partial{q_{(k)}^{i}}}\right).$

######  Definition 11.

A higher-order Lagrangian system determined by $L:T^{(k)}Q\to\mathbb{R}$ is
regular if, and only if, $\mathbb{F}L$ is a local diffeomorphism.

######  Remark.

Observe that a higher-order Lagrangian system is regular if and only if
$\left(\displaystyle\frac{\partial^{2}{L}}{\partial{q_{(k)}^{i}}\partial{q_{(k)}^{j}}}\right)$
is a nondegenerate matrix. In such a case, since
$\tilde{p}^{(k-1)}_{i}=\displaystyle\frac{\partial{L}}{\partial{q_{(k)}^{i}}}$,
by the implicit function theorem, we can define the $n$ coordinate functions
$q_{(k)}^{j}$ as functions depending on
$q_{(0)}^{i},\ldots,q_{(k-1)}^{i},\tilde{p}^{(k-1)}_{i}$; that is,

$\tilde{q}_{(k)}^{\,j}=f(q_{(0)}^{i},\ldots,q_{(k-1)}^{i},\tilde{p}^{(k-1)}_{i})\,.$
(15)

$\lozenge$

Using the higher-order Legendre transformation we can give in an alternative
way the solutions of the higher-order Lagrangian system as follows.

######  Proposition 4.

The solutions of a $k$th-order Lagrangian system described by a $k$th-order
Lagrangian function $L\in C^{\infty}(T^{(k)}Q)$ are the curves $\mu\colon
I\subset\mathbb{R}\to\Sigma_{L}$ satisfying

$\alpha_{T^{(k-1)}Q}^{-1}(\mu(t))=\frac{d}{dt}\mathbb{F}L(\mu(t)),$

where $\mu$ satisfies
$\left.\pi_{T^{*}TT^{k-1}Q}\right|_{\Sigma_{L}}(\mu(t))=\gamma^{(k)}(t)$, and
where $\gamma^{(k)}$ is the $k$-lift of a curve $\gamma\colon I\to Q$.

Next, we give an alternative characterization of the dynamics in the
Lagrangian submanifold $\Sigma_{L}$ in terms of the solution of the higher-
order Euler-Lagrange equations.

######  Proposition 5.

A curve $\gamma\colon I\to Q$ is a solution of the higher-order Euler-Lagrange
equations derived from a $k$th-order Lagrangian function $L\in
C^{\infty}(T^{(k)}Q)$ if, and only if,

$\alpha_{T^{(k-1)}Q}\left(\frac{d}{dt}\left(\operatorname{leg}_{L}\circ\gamma^{(2k-1)}\right)\right)\in\Sigma_{L}\,,$

where $\gamma^{(2k-1)}\colon I\to T^{(2k-1)}Q$ is the $(2k-1)$-lift of
$\gamma$ and $\operatorname{leg}_{L}\colon T^{(2k-1)}Q\to T^{*}(T^{(k-1)}Q)$
is the Legendre-Ostrogradsky map defined in (12).

###### Proof.

This proof is easy in coordinates. Let $(U;(q^{i}))$ be a local chart in $Q$,
and let us denote $\gamma(t)=(q^{i}(t))$ in $U$. Then, the $(2k-1)$-lift of
$\gamma$ is given by
$\gamma^{(2k-1)}(t)=(q_{(0)}^{i}(t),\ldots,q_{(2k-1)}^{i}(t))$, and we have

$\left(\operatorname{leg}_{L}\circ\gamma^{(2k-1)}\right)(t)=\left(q_{(0)}^{i}(t),\ldots,q_{(k-1)}^{i}(t),\hat{p}^{(0)}_{i}(t),\ldots,\hat{p}^{(k-1)}_{i}(t)\right)\,,$

where $\hat{p}_{(r)}^{i}$, $1\leqslant i\leqslant n$ and $0\leqslant
r\leqslant k-1$, are the Jacobi-Ostrogradsky momenta coordinates defined in
(13). Then, bearing in mind the coordinate expression (3) of
$\alpha_{T^{k-1}Q}$ we have that

$\alpha_{T^{(k-1)}Q}\left(\frac{d}{dt}\left(\operatorname{leg}_{L}\circ\gamma^{(2k-1)}\right)\right)=\left(q_{(j)}^{i}(t);q_{(j+1)}^{i}(t);\frac{d}{dt}\hat{p}_{i}^{(j)}(t);\hat{p}_{i}^{(j)}(t)\right)\,,$

with $1\leqslant i\leqslant n$ and $0\leqslant j\leqslant k-1$. Requiring
$\alpha_{T^{(k-1)}Q}\left(\frac{d}{dt}\left(\operatorname{leg}_{L}\circ\gamma^{(2k-1)}\right)\right)\in\Sigma_{L}$,
we obtain the following system of $(k+1)n$ differential equations on the
component functions of $\gamma^{(2k-1)}$

$\frac{d}{dt}\hat{p}^{(0)}_{i}=\displaystyle\frac{\partial{L}}{\partial{q_{(0)}^{i}}},\quad\frac{d}{dt}\hat{p}^{(j)}_{i}+\hat{p}^{(j-1)}_{i}=\displaystyle\frac{\partial{L}}{\partial{q_{(j)}^{i}}},\quad\hat{p}^{(k-1)}_{i}=\displaystyle\frac{\partial{L}}{\partial{q_{(k)}^{i}}}$

with $1\leqslant j\leqslant k-1$ in the second set of equations (which,
observe, is exactly the relation (14) among the momenta). Combining these
equations following the same patterns as in the end of Subsection 3.1, we
obtain the higher-order Euler-Lagrange equations

$\sum_{j=0}^{k}(-1)^{j}\frac{d^{j}}{dt^{j}}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(j)}^{i}}}\right)=0\,.\qed$

From Propositions 4 and 5 it is clear that we can characterize the solutions
of the Euler-Lagrange equations in purely geometric way by means of the either
the higher-order Legendre map $\mathbb{F}L$ or the Legendre-Ostrogradsky map
$\operatorname{leg}_{L}$. Now, assume that the Lagrangian function $L\in
C^{\infty}(T^{(k)}Q)$ is regular, so both maps $\mathbb{F}L\colon\Sigma_{L}\to
T^{*}T^{(k-1)}Q$ and $\operatorname{leg}_{L}\colon T^{(2k-1)}Q\to
T^{*}T^{(k-1)}Q$ are local diffeomorphisms. Then consider the map
$\phi_{L}\colon T^{(2k-1)}Q\to\Sigma_{L}$ defined by the composition
$\phi_{L}=\mathbb{F}L^{-1}\circ\operatorname{leg}_{L}$, which is locally given
by

$\phi_{L}\left(q_{(0)}^{i},\ldots,q_{(2k-1)}^{i}\right)=\left(q_{(0)}^{i},\ldots,q_{(k-1)}^{i};\hat{p}^{(0)}_{i},\ldots,\hat{p}^{(k-2)}_{i},\displaystyle\frac{\partial{L}}{\partial{q_{(k)}^{i}}}\right)\,.$

This map is a local diffeomorphism with local inverse
$\phi_{L}^{-1}=\operatorname{leg}_{L}^{-1}\circ\mathbb{F}L$ in the
corresponding open sets. If, moreover, $L$ is hyperegular, then $\phi_{L}$ is
a global diffeomorphism, from which we can recover the dynamics using the
implicit function theorem and the inverse of the Tulczyjew’s isomorphism. In
addition, in this case we can establish the Hamiltonian formalism in
$T^{*}T^{(k-1)}Q$ by defining a canonical Hamiltonian function $H\in
C^{\infty}(T^{*}T^{(k-1)}Q)$ with coordinate expression

$H(q_{(0)}^{i},\ldots,q_{(k-1)}^{i},p^{(0)}_{i},\ldots,p^{(k-1)}_{i})=\sum_{j=0}^{k-2}p^{(j)}_{i}q_{(j+1)}^{i}+p^{(k-1)}_{i}\tilde{q}^{\,i}_{(k)}-L(q_{(0)}^{i},\ldots,q_{(k-1)}^{i},\tilde{q}^{\,i}_{(k)})\,,$

where $\tilde{q}^{\,i}_{(k)}$ is given implicitly by (15). The canonical
Hamiltonian function obtained in this way does not depend on the choice of the
Legendre transformation used to derive it, and it depends only on the starting
Lagrangian function. The corresponding Hamiltonian vector field $X_{H}$ is
determined by $\mathop{i}\nolimits_{X_{H}}\omega_{T^{(k-1)}Q}=dH$. In this
case we have that

$\hbox{Im}(X_{H})=X_{H}(T^{*}T^{(k-1)}Q)=\beta_{T^{k-1}Q}^{-1}(dH(T^{*}T^{(k-1)}Q))=\alpha_{T^{(k-1)}Q}^{-1}(\Sigma_{L})\,.$

In the singular case, the submanifold $\hbox{Im}(dH)$ is not transversal with
respect to $\pi_{T^{*}T^{(k-1)}Q}$, and therefore it is necessary to apply an
integrability algorithm to find, if it exists, a subset where there are
consistent solutions of the dynamics (see [26] and [27], for example).

###  3.3 An example: the dynamics of the end of a javelin

Let us consider the dynamical system that describes the motion of the end of a
thrown javelin. This gives rise to a $3$-dimensional second-order dynamical
system, which is a particular case of the problem of determining the
trajectory of a particle rotating about a translating center [14]. Let
$Q=\mathbb{R}^{3}$ be the manifold modeling the configuration space for this
system with coordinates $(q_{(0)}^{1},q_{(0)}^{2},q_{(0)}^{3})=(q_{(0)}^{i})$.
Using the induced coordinates in $T^{(2)}\mathbb{R}^{3}$, the Lagrangian
function for this system is

$L(q_{(0)}^{i},q_{(1)}^{i},q_{(2)}^{i})=\frac{1}{2}\sum_{i=1}^{3}\left(\left(q_{(1)}^{i}\right)^{2}-\left(q_{(2)}^{i}\right)^{2}\right)\,,$

which is a regular Lagrangian function since the Hessian matrix of $L$ with
respect to the second-order velocities is

$\left(\displaystyle\frac{\partial^{2}{L}}{\partial{q_{(2)}^{j}}\partial{q_{(2)}^{i}}}\right)=\begin{pmatrix}-1&0&0\\\
0&-1&0\\\ 0&0&-1\end{pmatrix}\,.$

The second-order Euler-Lagrange equations are

$\frac{d^{4}}{dt^{4}}q_{(0)}^{i}+\frac{d^{2}}{dt^{2}}q_{(0)}^{i}=0\,,$

for $1\leqslant i\leqslant 3$. The general solution for the second-order
Euler-Lagrange equations is a curve $\gamma\colon
I\subset\mathbb{R}\to\mathbb{R}^{3}$ with component functions
$\gamma(t)=(\gamma^{1}(t),\gamma^{2}(t),\gamma^{3}(t))$ given by

$\gamma^{i}(t)=c_{1}^{i}+c_{2}^{i}t+c_{3}^{i}\sin(t)+c_{4}^{i}\cos(t)\,,$

with $c_{j}^{i}$ constants, $1\leqslant i\leqslant 3$ and $1\leqslant
j\leqslant 4$.

The Jacobi-Ostrogradsky momenta are given by

$\hat{p}^{(0)}_{i}=\displaystyle\frac{\partial{L}}{\partial{q_{(1)}^{i}}}-\frac{d}{dt}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(2)}^{i}}}\right)=q_{(1)}^{i}+q_{(3)}^{i}\,,\quad\hat{p}^{(1)}_{i}=\displaystyle\frac{\partial{L}}{\partial{q_{(2)}^{i}}}=-q_{(2)}^{i}\,.$

Therefore, the Legendre-Ostrogradsky map is given by

$\operatorname{leg}_{L}\left(q_{(0)}^{i},q_{(1)}^{i},q_{(2)}^{i},q_{(3)}^{i}\right)=\left(q_{(0)}^{i},q_{(1)}^{i},\hat{p}^{(0)}_{i},\hat{p}^{(1)}_{i}\right)=\left(q_{(0)}^{i},q_{(1)}^{i},q_{(1)}^{i}+q_{(3)}^{i},-q_{(2)}^{i}\right)\,.$

Consider a Lagrangian submanifold $\Sigma_{L}\hookrightarrow T^{*}TTQ$
associated with the Lagrangian function $L$. Local coordinates in the
Lagrangian submanifold are
$(q_{(0)}^{i},q_{(1)}^{i},q_{(2)}^{i},p^{(0)}_{i})$. The Legendre
transformation on $\Sigma_{L}$ is given locally by

$\mathbb{F}L\left(q_{(0)}^{i},q_{(1)}^{i},q_{(2)}^{i},p^{(0)}_{i}\right)=\left(q_{(0)}^{i},q_{(1)}^{i},p^{(0)}_{i},\displaystyle\frac{\partial{L}}{\partial{q_{(2)}^{i}}}\right)=\left(q_{(0)}^{i},q_{(1)}^{i},p^{(0)}_{i},-q_{(2)}^{i}\right)\,.$

$\mathbb{F}L$ is a diffeomorphism, and thus the second-order system is
regular. Therefore, we can define the second-order derivative in terms of the
momenta as $q_{(2)}^{i}=-p^{(1)}_{i}$ and define a Hamiltonian function on
$T^{*}TQ$, which is given in coordinates by

$H\left(q_{(0)}^{i},q_{(1)}^{i};p^{(0)}_{i},p^{(1)}_{i}\right)=p^{(0)}_{i}q_{(1)}^{i}-\frac{1}{2}\left(\left(q_{(1)}^{i}\right)^{2}+\left(p^{(1)}_{i}\right)^{2}\right)\,.$

Hamilton’s equations for this second-order dynamical system are

$\frac{d}{dt}p^{(0)}_{i}=0\,,\quad\frac{d}{dt}p^{(1)}_{i}=-p^{(0)}_{i}+q_{(1)}^{i}\,,\quad\frac{d}{dt}q_{(0)}^{i}=q_{(1)}^{i}\,,\quad\frac{d}{dt}q_{(1)}^{i}=-p^{(1)}_{i}\,.$

From the equations given above, in the regular case, one can obtain in a
straightforward way the second-order Euler-Lagrange equations.

##  4 On the definition and the regularity of a higher-order Hamiltonian
function

In this Section we aim at studying the Hamiltonian functions that describe the
dynamics of higher-order systems. More particularly, we want to define the
notions of “higher-order Hamiltonian function” and “regularity” of these
functions in a precise way.

Henceforth, we consider a $k$th-order dynamical system with $n$ degrees of
freedom, and let $Q$ be a $n$-dimensional smooth manifold modeling the
configuration space of this system. From the results in [18] and in Section 3
we know that the Hamiltonian phase space for this system is the cotangent
bundle $T^{*}T^{(k-1)}Q$. Hence, let $H\in C^{\infty}(T^{*}T^{(k-1)}Q)$ be a
Hamiltonian function describing the dynamics of the system.

###  4.1 Statement of the problems

Following the patterns in Section 2.2, let $\mathcal{F}H\colon
T^{*}T^{(k-1)}Q\to TT^{(k-1)}Q$ be the fiber derivative of $H$. If
$(U;(q^{i}))$, $1\leqslant i\leqslant n$, is a local chart in $Q$ and
$(q^{i}_{(j)},p_{i}^{(j)})$, $0\leqslant j\leqslant k-1$, are the induced
local coordinates in a suitable open subset of $T^{*}T^{(k-1)}Q$, then the map
$\mathcal{F}H$ is given locally by

$\mathcal{F}H\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)};p_{i}^{(0)},\ldots,p_{i}^{(k-1)}\right)=\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)};\displaystyle\frac{\partial{H}}{\partial{p_{i}^{(0)}}},\ldots,\displaystyle\frac{\partial{H}}{\partial{p_{i}^{(k-1)}}}\right)\,.$

Moreover, the regularity condition for the Hamiltonian function $H$ is,
locally, equivalent to

$\det\left(\displaystyle\frac{\partial^{2}{H}}{\partial{p^{(j)}_{i}}\partial{p^{(j^{\prime})}_{i^{\prime}}}}\right)(\alpha_{q})\neq
0\,,\ \mbox{for every }\alpha_{q}\in T^{*}T^{(k-1)}Q\,.$

######  Proposition 6.

Let $H\in C^{\infty}(T^{*}T^{(k-1)}Q)$ be a hyperregular Hamiltonian function,
$\theta_{T^{(k-1)}Q}\in\Omega^{1}(T^{*}T^{(k-1)}Q)$ is the Liouville $1$-form
and $X_{H}\in\mathfrak{X}(T^{*}T^{(k-1)}Q)$ is the unique vector field
solution to the dynamical equation

$\mathop{i}\nolimits_{X_{H}}\omega_{T^{(k-1)}Q}=dH\,,$

where
$\omega_{T^{(k-1)}Q}=-d\theta_{T^{(k-1)}Q}\in\Omega^{2}(T^{*}T^{(k-1)}Q)$ is
the canonical symplectic form. Then there exists a global diffeomorphism $L\in
C^{\infty}(TT^{(k-1)}Q)$, locally determined by

$L\left(q^{i}_{(j)};v^{i}_{(j)}\right)=\tilde{p}_{i}^{(j)}\dot{q}^{i}_{(j)}-H\left(q^{i}_{(j)};\tilde{p}_{i}^{(j)}\right)\,,$

where $\tilde{p}_{i}^{(j)}=(\mathcal{F}H^{-1})^{*}p_{i}^{(j)}\in TT^{(k-1)}Q$.

###### Proof.

Using that the Hamiltonian function $H$ is hyperregular and using Proposition
1 with $T^{(k-1)}Q$ as the base manifold, we define a hyperregular Lagrangian
function (a global diffeomorphism) ${L}\in C^{\infty}(TT^{(k-1)}Q)$ as follows

${L}=\theta_{T^{(k-1)}Q}(X_{H})\circ\mathcal{F}H^{-1}-H\circ\mathcal{F}H^{-1}\,,$

where $\theta_{T^{(k-1)}Q}\in\Omega^{1}(T^{*}T^{(k-1)}Q)$ is the Liouville
$1$-form and $X_{H}\in\mathfrak{X}(T^{*}T^{(k-1)}Q)$ is the unique vector
field solution to the dynamical equation

$\mathop{i}\nolimits_{X_{H}}\omega_{T^{(k-1)}Q}=dH\,,$

where
$\omega_{T^{(k-1)}Q}=-d\theta_{T^{(k-1)}Q}\in\Omega^{2}(T^{*}T^{(k-1)}Q)$ is
the canonical symplectic form. In the induced local coordinates of
$T^{*}T^{(k-1)}Q$, the Liouville $1$-form is locally given by
$\theta_{T^{(k-1)}Q}=p_{i}^{(j)}dq^{i}_{(j)}$. From where we deduce that the
vector field $X_{H}\in\mathfrak{X}(T^{*}T^{(k-1)}Q)$ solution to the previous
equation is locally given by

$X_{H}=\displaystyle\frac{\partial{H}}{\partial{p_{i}^{(j)}}}\displaystyle\frac{\partial{}}{\partial{q^{i}_{(j)}}}-\displaystyle\frac{\partial{H}}{\partial{q^{i}_{(j)}}}\displaystyle\frac{\partial{}}{\partial{p_{i}^{(j)}}}\,.$
(16)

Then, the function $\mathcal{F}H^{*}{L}=\theta_{T^{(k-1)}Q}(X_{H})-H\in
C^{\infty}(T^{*}T^{(k-1)}Q)$ is given in coordinates by

$\mathcal{F}H^{*}{L}\left(q^{i}_{(j)};p_{i}^{(j)}\right)=p_{i}^{(j)}\displaystyle\frac{\partial{H}}{\partial{p_{i}^{(j)}}}-H\left(q^{i}_{(j)};p_{i}^{(j)}\right)\,.$

From this, the coordinate expression of the Lagrangian function ${L}$ in the
induced natural coordinates $\left(q^{i}_{(j)},v^{i}_{(j)}\right)$,
$1\leqslant i\leqslant n$, $0\leqslant j\leqslant k-1$, of $TT^{(k-1)}Q$ is

$L\left(q^{i}_{(j)};v^{i}_{(j)}\right)=\tilde{p}_{i}^{(j)}\dot{q}^{i}_{(j)}-H\left(q^{i}_{(j)};\tilde{p}_{i}^{(j)}\right)\,,$

where $\tilde{p}_{i}^{(j)}=(\mathcal{F}H^{-1})^{*}p_{i}^{(j)}$ are local
functions in $TT^{(k-1)}Q$. ∎

It is important to point out that the Lagrangian function obtained may not be
a $k$th-order Lagrangian function, in the physical sense: the coordinate
functions in the basis $T^{(k-1)}Q$ may not be well-related to the coordinate
functions in the fibers, in the sense that, in general, we have
$q^{i}_{(j+1)}\neq\dot{q}^{i}_{(j)}$, that is, they are independent variables.
From the geometric point of view, the problem is that ${L}$ is not defined in
the “holonomic” submanifold $j_{k}\colon T^{(k)}Q\hookrightarrow TT^{(k-1)}Q$.
In fact, observe that, up to this point, the order of the system has not been
taken into account at any step, since this condition is usually inherited from
the Lagrangian formulation. That is, the function $H\in
C^{\infty}(T^{*}T^{(k-1)}Q)$ is not considered as a Hamiltonian function for a
$k$th-order dynamical system, but just as a Hamiltonian function for a first-
order system defined on the cotangent bundle of a larger manifold, since we do
not require any relation among the momenta. This issue gives rise to the first
problem that we want to solve for Hamiltonian functions defined on
$T^{*}T^{(k-1)}Q$.

######  Problem 1.

Given a function $H\in C^{\infty}(T^{*}T^{(k-1)}Q)$, find conditions on $H$
such that it describes the dynamics of a $k$th-order dynamical system, that
is, find a suitable definition of $k$th-order Hamiltonian functions.

######  Remark.

Observe that the equivalent problem in the Lagrangian formalism (find
conditions on a function ${L}\in C^{\infty}(TT^{(k-1)}Q)$ such that ${L}$ is a
$k$th-order Lagrangian function) is solved straightforwardly, since there
exists a distinguished “holonomic” submanifold $j_{k}\colon
T^{(k)}Q\hookrightarrow TT^{(k-1)}Q$. Nevertheless, there is not such a
submanifold in $T^{*}T^{(k-1)}Q$, and hence the problem is not trivial.
$\lozenge$

Notice that a sufficient condition to ensure that ${L}$ is indeed a $k$th-
order Lagrangian function in the usual sense, that is, ${L}\in
C^{\infty}(T^{(k)}Q)$ (as a submanifold of $TT^{(k-1)}Q$), is to require
$\operatorname{Im}(\mathcal{F}H)\subseteq j_{k}(T^{(k)}Q)$. However, since
$\dim T^{(k)}Q=(k+1)n<2kn=\dim T^{*}T^{(k-1)}Q$ for $k>1$, this requirement on
the fiber derivative of $H$ prevents the Hamiltonian function to be regular in
the sense of Definition 5. Nevertheless, recall that the regularity condition
for a $k$th-order Lagrangian function ${L}\in C^{\infty}(T^{(k)}Q)$ is locally
equivalent to

$\det\left(\displaystyle\frac{\partial^{2}{{L}}}{\partial{q^{i}_{(k)}}\partial{q^{j}_{(k)}}}\right)\left([\gamma]_{0}^{(k)}\right)\neq
0\,,\ \mbox{for every }[\gamma]_{0}^{(k)}\in T^{(k)}Q\,,$

with $1\leqslant i,j\leqslant n$, instead of

$\det\left(\displaystyle\frac{\partial^{2}{{L}}}{\partial{q^{i}_{(r)}}\partial{q^{j}_{(s)}}}\right)\left([\gamma]_{0}^{(k)}\right)\neq
0\,,\ \mbox{for every }[\gamma]_{0}^{(k)}\in T^{(k)}Q\,,$

with $1\leqslant i,j\leqslant n$ and $1\leqslant r,s\leqslant k$. That is, the
Hessian of ${L}$ is taken only with respect to the highest-order “velocities”
$q^{i}_{(k)}$, and not with respect to all the “velocities”. Therefore, we
deduce that the regularity condition given in Definition 5 is not suitable for
higher-order Hamiltonian functions, since too many “orders” of the momenta
coordinates are taken into account. This gives rise to the second problem that
we want to solve.

######  Problem 2.

To find a suitable definition of regularity for $k$th-order Hamiltonian
functions $H\in C^{\infty}(T^{*}T^{(k-1)}Q)$ in terms of the fiber derivative
of $H$ such that Propositions 1 and 2 hold for $k$th-order dynamical systems.

###  4.2 A particular case: the Hamiltonian function associated to a
(hyper)regular Lagrangian system

In order to solve Problems 1 and 2 stated in the previous Subsection, we first
consider the particular case of a well-known Hamiltonian that describes
properly the dynamics of a higher-order system: the Hamiltonian function
associated to a higher-order Lagrangian system.

######  Proposition 7.

Given a hyperregular $k$th-order Lagrangian function, ${L}\in
C^{\infty}(T^{(k)}Q)$, there exists a unique $k$th-order Hamiltonian function
associated to this Lagrangian given locally by

$H\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)};p_{i}^{(0)},\ldots,p_{i}^{(k-1)}\right)=\sum_{j=0}^{k-2}q^{i}_{(j+1)}p_{i}^{(j)}+\tilde{q}^{\,i}_{(k)}p_{i}^{(k-1)}-{L}\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)},\tilde{q}^{\,i}_{(k)}\right)\,,$

where $\tilde{q}^{\,i}_{(k)}=(\operatorname{leg}_{L}^{-1})^{*}q^{i}_{(k)}$.
Moreover, $\operatorname{Im}(\mathcal{F}H)\subset j_{k}(T^{(k)}Q)$.

###### Proof.

Let $\operatorname{leg}_{L}\colon T^{(2k-1)}Q\to T^{*}T^{(k-1)}Q$ be the
Legendre-Ostrogradsky map defined in (12). From the Lagrangian function ${L}$
we construct the Lagrangian energy $E_{L}\in C^{\infty}(T^{(2k-1)}Q)$, with
coordinate expression

$E_{L}\left(q^{i}_{(0)},\ldots,q^{i}_{(2k-1)}\right)=\sum_{r=1}^{k}q_{(r)}^{i}\sum_{j=0}^{k-r}(-1)^{j}d_{T}^{j}\left(\displaystyle\frac{\partial{{L}}}{\partial{q_{(r+j)}^{i}}}\right)-{L}\left(q^{i}_{(0)},\ldots,q^{i}_{(k)}\right)\,.$
(17)

Then, since ${L}$ is hyperregular, the Legendre-Ostrogradsky map is a
diffeomorphism, and thus there exists a unique $k$th-order Hamiltonian
function associated to this Lagrangian system defined by
$H=E_{L}\circ\operatorname{leg}_{L}^{-1}\in C^{\infty}(T^{*}T^{(k-1)}Q)$. This
Hamiltonian function is given locally by

$H\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)};p_{i}^{(0)},\ldots,p_{i}^{(k-1)}\right)=\sum_{j=0}^{k-2}q^{i}_{(j+1)}p_{i}^{(j)}+\tilde{q}^{\,i}_{(k)}p_{i}^{(k-1)}-{L}\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)},\tilde{q}^{\,i}_{(k)}\right)\,,$

where $\tilde{q}^{\,i}_{(k)}=(\operatorname{leg}_{L}^{-1})^{*}q^{i}_{(k)}$ are
local functions in $T^{*}T^{(k-1)}Q$. Observe that the derivative of $H$ with
respect to a momenta coordinate $p_{r}^{(j)}$ gives

$\displaystyle\frac{\partial{H}}{\partial{p_{r}^{(j)}}}=\begin{cases}q^{r}_{(j+1)}&\mbox{if
}0\leqslant j\leqslant k-2\,,\\\\[15.0pt]
\tilde{q}^{\,r}_{(k)}+p_{i}^{(k-1)}\displaystyle\frac{\partial{\tilde{q}^{\,i}_{(k)}}}{\partial{p_{r}^{(k-1)}}}-\displaystyle\frac{\partial{{L}}}{\partial{q^{i}_{(k)}}}\displaystyle\frac{\partial{\tilde{q}^{\,i}_{(k)}}}{\partial{p_{r}^{(k-1)}}}=\tilde{q}^{\,r}_{(k)}&\mbox{if
}j=k-1\,,\end{cases}$

where, since $T^{*}T^{(k-1)}Q=\operatorname{Im}(\operatorname{leg}_{L})$, we
have $p^{(k-1)}_{i}=\displaystyle\frac{\partial{{L}}}{\partial{q^{i}_{(k)}}}$,
and therefore the last two terms in the above sums cancel each other. From
this we deduce that the coordinate expression of the fiber derivative of $H$
in this particular case is

$\mathcal{F}H\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)};p_{i}^{(0)},\ldots,p_{i}^{(k-1)}\right)=\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)};q^{i}_{(1)},\ldots,q^{i}_{(k-1)},\tilde{q}^{\,i}_{(k)}\right)\,.$
(18)

It is clear from this coordinate expression that
$\operatorname{Im}(\mathcal{F}H)\subseteq T^{(k)}Q\stackrel{{\scriptstyle
j_{k}}}{{\hookrightarrow}}TT^{(k-1)}Q$. ∎

Let $\mathcal{F}H_{o}\colon T^{*}T^{(k-1)}Q\to T^{(k)}Q$ be the map defined by
$\mathcal{F}H=\mathcal{F}H_{o}\circ j_{k}$, that is, the unique map such that
the following diagram commutes

$\textstyle{T^{*}T^{(k-1)}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}H}$$\scriptstyle{\mathcal{F}H_{o}}$
$\textstyle{TT^{(k-1)}Q}$
$\textstyle{T^{(k)}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{k}}$

with coordinate expression

$\mathcal{F}H_{o}\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)};p_{i}^{(0)},\ldots,p_{i}^{(k-1)}\right)=\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)},\tilde{q}^{\,i}_{(k)}\right)\,.$

Now, let us compute the coordinate expression of the tangent map of
$\mathcal{F}H$ in an arbitrary point $\alpha_{q}\in T^{*}T^{(k-1)}Q$. Bearing
in mind the coordinate expression (18) of $\mathcal{F}H$, the map
$T_{\alpha_{q}}\mathcal{F}H$ is given in coordinates by the following
$2kn\times 2kn$ real matrix

$T_{\alpha_{q}}\mathcal{F}H=\left(\begin{array}[]{cccc|ccccc}\textnormal{Id}_{n}&\mathbf{0}_{n}&\ldots&\mathbf{0}_{n}&\mathbf{0}_{n}&\mathbf{0}_{n}&\ldots&\mathbf{0}_{n}&\mathbf{0}_{n}\\\
\mathbf{0}_{n}&\textnormal{Id}_{n}&\ldots&\mathbf{0}_{n}&\mathbf{0}_{n}&\mathbf{0}_{n}&\ldots&\mathbf{0}_{n}&\mathbf{0}_{n}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
\mathbf{0}_{n}&\mathbf{0}_{n}&\ldots&\textnormal{Id}_{n}&\mathbf{0}_{n}&\mathbf{0}_{n}&\ldots&\mathbf{0}_{n}&\mathbf{0}_{n}\\\
\hline\cr\mathbf{0}_{n}&\textnormal{Id}_{n}&\ldots&\mathbf{0}_{n}&\mathbf{0}_{n}&\mathbf{0}_{n}&\ldots&\mathbf{0}_{n}&\mathbf{0}_{n}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
\mathbf{0}_{n}&\mathbf{0}_{n}&\ldots&\textnormal{Id}_{n}&\mathbf{0}_{n}&\mathbf{0}_{n}&\ldots&\mathbf{0}_{n}&\mathbf{0}_{n}\\\
\displaystyle\frac{\partial{\tilde{q}^{\,i}_{(k)}}}{\partial{q^{j}_{(0)}}}&\displaystyle\frac{\partial{\tilde{q}^{\,i}_{(k)}}}{\partial{q^{j}_{(1)}}}&\ldots&\displaystyle\frac{\partial{\tilde{q}^{\,i}_{(k)}}}{\partial{q^{j}_{(k-1)}}}&\displaystyle\frac{\partial{\tilde{q}^{\,i}_{(k)}}}{\partial{p_{j}^{(0)}}}&\displaystyle\frac{\partial{\tilde{q}^{\,i}_{(k)}}}{\partial{p_{j}^{(1)}}}&\ldots&\displaystyle\frac{\partial{\tilde{q}^{\,i}_{(k)}}}{\partial{p_{j}^{(k-2)}}}&\displaystyle\frac{\partial{\tilde{q}^{\,i}_{(k)}}}{\partial{p_{j}^{(k-1)}}}\end{array}\right)\,,$

where every entry is a $n\times n$ real matrix, and every block of the matrix
has size $kn\times kn$. In particular, $\textnormal{Id}_{n}$ denotes the
$n\times n$ identity matrix, $\mathbf{0}_{n}$ the $n\times n$ null matrix and,
in the last row, we have $1\leqslant i,j\leqslant n$. Observe that, in the
most favorable case, the map $\mathcal{F}H\colon T^{*}T^{(k-1)}Q\to
TT^{(k-1)}Q$ has rank $(k+1)n=\dim T^{(k)}Q$, that is, at the best the map
$\mathcal{F}H_{o}\colon T^{*}T^{(k-1)}Q\to T^{(k)}Q$ is a submersion onto
$T^{(k)}Q$. A long but straightforward calculation shows that

$\left(\displaystyle\frac{\partial^{2}{H}}{\partial{p_{i}^{(k-1)}}\partial{p_{j}^{(k-1)}}}\right)=\left(\displaystyle\frac{\partial{\tilde{q}^{\,i}_{(k)}}}{\partial{p_{j}^{(k-1)}}}\right)=\left(\displaystyle\frac{\partial{\left(q^{i}_{(k)}\circ\operatorname{leg}_{L}^{-1}\right)}}{\partial{p_{j}^{(k-1)}}}\right)=\left(\displaystyle\frac{\partial^{2}{{L}}}{\partial{q^{i}_{(k)}}\partial{q^{j}_{(k)}}}\right)^{-1}\,,$

which is a well-defined $n\times n$ matrix because ${L}$ is (hyper)regular,
and therefore the map $\mathcal{F}H_{o}$ is a submersion onto $T^{(k)}Q$.
Observe that, as a consequence, $\mathcal{F}H_{o}$ admits local sections, that
is, maps $\sigma\colon T^{(k)}Q\to T^{*}T^{(k-1)}Q$ satisfying
$\mathcal{F}H_{o}\circ\sigma=\textnormal{Id}_{T^{(k)}Q}$.

The following result gives the illuminating key idea for the “true” definition
of higher-order Hamiltonian function and regularity of a higher-order
Hamiltonian function that we will exploit in the next Subsection.

######  Theorem 2.

Let $L:T^{(k)}Q\to\mathbb{R}$ be a $kth$-order Lagrangian function

  * (i)

If $L$ is regular, there exists an open subset $U\subset T^{(2k-1)}Q$ such
that

$\left.\mathcal{F}H_{o}\right|_{U}=\left.\left(\tau_{Q}^{(k,2k-1)}\circ\operatorname{leg}_{L}^{-1}\right)\right|_{U}\,.$

  * (ii)

If $L$ is hyperregular then $\mathcal{F}H_{o}:T^{*}T^{(k-1)}Q\to T^{(k)}Q$ is
a submersion onto $T^{(k)}Q$.

  * (iii)

If $L$ is hyperregular then $\mathcal{F}H_{o}$ admits a global section
$\Upsilon:T^{(k)}Q\to T^{*}T^{(k-1)}Q$.

###### Proof.

Let us consider the map $\mathcal{F}H_{o}\circ\operatorname{leg}_{L}\colon
T^{(2k-1)}Q\to T^{(k)}Q$. In the natural coordinates
$\left(q^{i}_{(j)}\right)$ of $T^{(2k-1)}Q$ ($1\leqslant i\leqslant n$,
$0\leqslant j\leqslant k$), the local expression of this map is given by

$\displaystyle\left(\mathcal{F}H_{o}\circ\operatorname{leg}_{L}\right)\left(q^{i}_{(0)},\ldots,q^{i}_{(2k-1)}\right)$
$\displaystyle=\mathcal{F}H_{o}\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)};\hat{p}_{i}^{(0)},\ldots,\hat{p}_{i}^{(k-1)}\right)$
$\displaystyle=\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)},q^{i}_{(k)}\right)\,,$

since $\tilde{q}^{\,i}_{(k)}=(\operatorname{leg}_{L}^{-1})^{*}q_{(k)}^{i}$.
From this coordinate expression we deduce that for every point
$[\gamma]^{(2k-1)}_{0}\in T^{(2k-1)}Q$ there exists an open subset $U\subseteq
T^{(2k-1)}Q$ such that $[\gamma]^{(2k-1)}_{0}\in U$ and

$\left.\left(\mathcal{F}H_{o}\circ\operatorname{leg}_{L}\right)\right|_{U}=\left.\tau_{Q}^{(k,2k-1)}\right|_{U}\,,$

and, since ${L}\in C^{\infty}(T^{(k)}Q)$ is a regular Lagrangian function, the
Legendre-Ostrogradsky map is a local diffeomorphism, we have

$\left.\mathcal{F}H_{o}\right|_{U}=\left.\left(\tau_{Q}^{(k,2k-1)}\circ\operatorname{leg}_{L}^{-1}\right)\right|_{U}\,.$

Observe that, moreover, we assume that ${L}$ is a hyperregular Lagrangian
function. Therefore, the map $\operatorname{leg}_{L}^{-1}\colon
T^{*}T^{(k-1)}Q\to T^{(2k-1)}Q$ is bijective and defined in the entire
manifold $T^{*}T^{(k-1)}Q$. On the other hand, the map $\tau_{Q}^{(k,2k-1)}$
is surjective, from where we deduce that $\mathcal{F}H_{o}\colon
T^{*}T^{(k-1)}Q\to T^{(k)}Q$ is surjective in this case, since we have

$\mathcal{F}H_{o}=\tau_{Q}^{(k,2k-1)}\circ\operatorname{leg}_{L}^{-1}\,,$

that is, $\operatorname{Im}(\mathcal{F}H_{o})=T^{(k)}Q$.

Observe that, in addition, there exists a map $\Upsilon\colon T^{(k)}Q\to
T^{*}T^{(k-1)}Q$ defined by $\Upsilon=\operatorname{leg}_{L}\circ\Psi$, with
$\Psi\in\Gamma\left(\tau_{Q}^{(k,2k-1)}\right)$ being a global section of
$\tau_{Q}^{(k,2k-1)}$, which satisfies

$\mathcal{F}H_{o}\circ\Upsilon=\tau_{Q}^{(k,2k-1)}\circ\operatorname{leg}_{L}^{-1}\circ\operatorname{leg}_{L}\circ\Psi=\tau_{Q}^{(k,2k-1)}\circ\Psi=\textnormal{Id}_{T^{(k)}Q}\,,$

that is, $\Upsilon$ is a global section of $\mathcal{F}H_{o}$. ∎

###  4.3 Higher-order Hamiltonian functions: definition and regularity

Bearing in mind the results obtained in the previous Section, we are now able
to give a solution to Problems 1 and 2. From the last theorem the statement of
Problem 1 and the calculations for the particular case of a Hamiltonian system
associated to a hyperregular Lagrangian system, we can give the following
definition.

######  Definition 12.

A function $H\in C^{\infty}(T^{*}T^{(k-1)}Q)$ is a $k$th-order Hamiltonian
function if its fiber derivative, $\mathcal{F}H\colon T^{*}T^{(k-1)}Q\to
TT^{(k-1)}Q$, takes values in the “holonomic” submanifold $j_{k}\colon
T^{(k)}Q\hookrightarrow TT^{(k-1)}Q$, that is,
$\operatorname{Im}(\mathcal{F}H)\subseteq j_{k}(T^{(k)}Q)$.

In the induced natural coordinates $\left(q^{i}_{(j)},p_{i}^{(j)}\right)$,
$1\leqslant i\leqslant n$ and $0\leqslant j\leqslant k-1$, of
$T^{*}T^{(k-1)}Q$, taking into account that the submanifold $j_{k}\colon
T^{(k)}Q\to TT^{(k-1)}Q$ is defined locally by the $(k-1)n$ constraints
$v_{(j)}^{i}=q_{(j+1)}^{i}$ ($0\leqslant j\leqslant k-2$) and that the fiber
derivative of $H$ is a fiber bundle morphism, the condition for $H$ to be a
$k$th-order Hamiltonian function gives in coordinates

$\displaystyle\frac{\partial{H}}{\partial{p^{(j)}_{i}}}=\mathcal{F}H^{*}v_{(j)}^{i}=\mathcal{F}H^{*}q_{(j+1)}^{i}=q_{(j+1)}^{i}\,,\
\mbox{for every }1\leqslant i\leqslant n\,,\ 0\leqslant j\leqslant k-2\,.$
(19)

Observe that if $H\in C^{\infty}(T^{*}T^{(k-1)}Q)$ is a $k$th-order
Hamiltonian function, then the fiber derivative $\mathcal{F}H$ of $H$ induces
a map $\mathcal{F}H_{o}\colon T^{*}T^{(k-1)}Q\to T^{(k)}Q$ defined as
$\mathcal{F}H=j_{k}\circ\mathcal{F}H_{o}$. This map is given in coordinates by

$\mathcal{F}H_{o}\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)};p_{i}^{(0)},\ldots,p_{i}^{(k-1)}\right)=\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)},\displaystyle\frac{\partial{H}}{\partial{p_{i}^{(k-1)}}}\right)\,.$
(20)

The map $\mathcal{F}H_{o}\colon T^{*}T^{(k-1)}Q\to T^{(k)}Q$ enables us to
give a solution to Problem 2 in terms of its (local) “inverse map”.
Nevertheless, due to the restriction imposed by the dimensions of the
manifolds involved, there is no inverse map to $\mathcal{F}H_{o}$ (even
locally). Hence, instead of an inverse, we use the most similar approach,
which consists in considering inverses in just one way, that is, sections.
Therefore, following the patterns in [52], we give the following definition.

######  Definition 13.

A $k$th-order Hamiltonian function $H\in C^{\infty}(T^{*}T^{(k-1)}Q)$ is said
to be regular if the map $\mathcal{F}H_{o}\colon T^{*}T^{(k-1)}Q\to T^{(k)}Q$
is a submersion onto $T^{(k)}Q$. If, moreover, $\mathcal{F}H_{o}$ admits a
global section $\Upsilon\colon T^{(k)}Q\to T^{*}T^{(k-1)}Q$, the Hamiltonian
function is said to be hyperregular. Otherwise, the Hamiltonian function is
said to be singular.

In the induced natural coordinates $\left(q^{i}_{(j)},p_{i}^{(j)}\right)$,
$1\leqslant i\leqslant n$ and $0\leqslant j\leqslant k-1$, of
$T^{*}T^{(k-1)}Q$ from the relation (19) we have

$\displaystyle\frac{\partial^{2}{H}}{\partial{p^{(s)}_{j}}\partial{p_{i}^{(k-1)}}}=\displaystyle\frac{\partial{}}{\partial{p_{i}^{(k-1)}}}\left(\displaystyle\frac{\partial{H}}{\partial{p^{(s)}_{j}}}\right)=\displaystyle\frac{\partial{q_{(s+1)}^{j}}}{\partial{p_{i}^{(k-1)}}}=0\,,$
(21)

where $1\leqslant i,j\leqslant n$ and $0\leqslant s\leqslant k-2$. From those
and the local expression (20) of $\mathcal{F}H_{o}$ we deduce that its tangent
map in an arbitrary point $\alpha_{q}\in T^{*}T^{(k-1)}Q$ is given locally by
the following $(k+1)n\times 2kn$ real matrix

$T_{\alpha_{q}}\mathcal{F}H_{o}=\left(\begin{array}[]{ccc|cccc}\textnormal{Id}_{n}&\ldots&\mathbf{0}_{n}&\mathbf{0}_{n}&\ldots&\mathbf{0}_{n}&\mathbf{0}_{n}\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
\mathbf{0}_{n}&\ldots&\textnormal{Id}_{n}&\mathbf{0}_{n}&\ldots&\mathbf{0}_{n}&\mathbf{0}_{n}\\\
\hline\cr&&&&&\\\\[-10.0pt]
\displaystyle\frac{\partial^{2}{H}}{\partial{q^{j}_{(0)}}\partial{p_{i}^{(k-1)}}}&\ldots&\displaystyle\frac{\partial^{2}{H}}{\partial{q^{j}_{(k-1)}}\partial{p_{i}^{(k-1)}}}&\mathbf{0}_{n}&\ldots&\mathbf{0}_{n}&\displaystyle\frac{\partial^{2}{H}}{\partial{p^{(k-1)}_{j}}\partial{p_{i}^{(k-1)}}}\end{array}\right)\,,$
(22)

where every entry is a $n\times n$ real matrix. In particular,
$\textnormal{Id}_{n}$ denotes the $n\times n$ identity matrix,
$\mathbf{0}_{n}$ the $n\times n$ null matrix and, in the last row, we have
$1\leqslant i,j\leqslant n$. Therefore, the local condition for a $k$th-order
Hamiltonian function $H\in C^{\infty}(T^{*}T^{(k-1)}Q)$ to be a regular is

$\det\left(\displaystyle\frac{\partial^{2}{H}}{\partial{p^{(k-1)}_{j}}\partial{p_{i}^{(k-1)}}}\right)(\alpha_{q})\neq
0\,,\ \mbox{for every }\alpha_{q}\in T^{*}T^{(k-1)}Q\,.$

Using both Definitions 12 and 13 we can now state and prove the analogous
results to Propositions 1 and 2 in the higher-order setting. As for
Proposition 1, we have the following result.

######  Proposition 8.

Let $H\in C^{\infty}(T^{*}T^{(k-1)}Q)$ be a regular $k$th-order Hamiltonian
function, $\theta_{T^{(k-1)}Q}\in\Omega^{1}(T^{*}T^{(k-1)}Q)$ the Liouville
1-form, $\omega_{T^{(k-1)}Q}\in\Omega^{2}(T^{*}T^{(k-1)}Q)$ the canonical
symplectic form, and $X_{H}\in\mathfrak{X}(T^{*}T^{(k-1)}Q)$ the unique vector
field solution to the equation

$\mathop{i}\nolimits_{X_{H}}\omega_{T^{(k-1)}Q}=dH\,.$ (23)

Then the function $\theta_{T^{(k-1)}Q}(X_{H})-H\in
C^{\infty}(T^{*}T^{(k-1)}Q)$ is $\mathcal{F}H_{o}$-projectable, and the
function ${L}\in C^{\infty}(T^{(k)}Q)$ defined by
$(\mathcal{F}H_{o})^{*}{L}=\theta_{T^{(k-1)}Q}(X_{H})-H$ is a regular $k$th-
order Lagrangian function. Moreover, for every $[\gamma]^{(2k-1)}_{0}\in
T^{(2k-1)}Q$ there exists an open set $U\subseteq T^{(2k-1)}Q$ such that
$[\gamma]^{(2k-1)}_{0}\in U$ and
$\left.\left(\mathcal{F}H_{o}\circ\operatorname{leg}_{L}\right)\right|_{U}=\left.\tau_{Q}^{(k,2k-1)}\right|_{U}$.

###### Proof.

Following the patterns in [1], the easiest proof of this result is done in
coordinates. Hence, let $\left(q^{i}_{(j)},p_{i}^{(j)}\right)$, $1\leqslant
i\leqslant n$ and $0\leqslant j\leqslant k-1$, be the induced natural
coordinates in a suitable open set of $T^{*}T^{(k-1)}Q$. Along this proof we
only consider these coordinates.

First, we must prove that the function $\theta_{T^{(k-1)}Q}(X_{H})-H\in
C^{\infty}(T^{*}T^{(k-1)}Q)$ is $\mathcal{F}H_{o}$-projectable, that is,

$\mathop{\rm L}\nolimits(Y)\left(\theta_{T^{(k-1)}Q}(X_{H})-H\right)=0\,,\
\mbox{for every }Y\in\ker T\mathcal{F}H_{o}\,.$

From the coordinate expression (22) of the tangent map $T\mathcal{F}H_{o}$ at
a point $\alpha_{q}\in T^{*}T^{(k-1)}Q$, it is clear that a local basis for
$\ker T\mathcal{F}H_{o}$ is given by

$\ker
T\mathcal{F}H_{o}=\left\langle\displaystyle\frac{\partial{}}{\partial{p_{i}^{(0)}}},\ldots,\displaystyle\frac{\partial{}}{\partial{p_{i}^{(k-2)}}}\right\rangle\,.$

On the other hand, from our calculations in Section 4.1 we know that the
Hamiltonian vector fiels solution to equation (23) is given locally by (16),
which in combination with the identities (19) gives

$X_{H}=\sum_{l=0}^{k-2}q_{(l+1)}^{i}\displaystyle\frac{\partial{}}{\partial{q_{(l)}^{i}}}+\displaystyle\frac{\partial{H}}{\partial{p_{i}^{(k-1)}}}\displaystyle\frac{\partial{}}{\partial{q^{i}_{(k-1)}}}-\displaystyle\frac{\partial{H}}{\partial{q^{i}_{(j)}}}\displaystyle\frac{\partial{}}{\partial{p_{i}^{(j)}}}\,.$

Thus, since the Liouville $1$-form is given in coordinates by
$\theta_{T^{(k-1)}Q}=p_{i}^{(j)}dq_{(j)}^{i}$, the function
$\theta_{T^{(k-1)}Q}(X_{H})-H\in C^{\infty}(T^{*}T^{(k-1)}Q)$ has the
following coordinate expression

$\theta_{T^{(k-1)}Q}(X_{H})-H=\sum_{l=0}^{k-2}p_{i}^{(l)}q_{(l+1)}^{i}+p_{i}^{(k-1)}\displaystyle\frac{\partial{H}}{\partial{p_{i}^{(k-1)}}}-H\,.$

Hence for every $Y=\partial/\partial p_{r}^{(s)}$, $1\leqslant r\leqslant n$
and $0\leqslant s\leqslant k-2$, we have

$\displaystyle\mathop{\rm
L}\nolimits(Y)\left(\theta_{T^{(k-1)}Q}(X_{H})-H\right)$
$\displaystyle=\mathop{\rm
L}\nolimits\left(\displaystyle\frac{\partial{}}{\partial{p_{r}^{(s)}}}\right)\left(\sum_{l=0}^{k-2}p_{i}^{(l)}q_{(l+1)}^{i}+p_{i}^{(k-1)}\displaystyle\frac{\partial{H}}{\partial{p_{i}^{(k-1)}}}-H\right)$
$\displaystyle=q_{(s+1)}^{r}+p_{i}^{(k-1)}\displaystyle\frac{\partial^{2}{H}}{\partial{p_{r}^{(s)}}\partial{p_{i}^{(k-1)}}}-\displaystyle\frac{\partial{H}}{\partial{p_{r}^{(s)}}}=0\,,$

where, in the last equality, we have used the identities (19) and (21).
Therefore, the function $\theta_{T^{(k-1)}Q}(X_{H})-H\in
C^{\infty}(T^{*}T^{(k-1)}Q)$ is $\mathcal{F}H_{o}$-projectable.

It is now a long but straightforward calculation to prove that the Lagrangian
function $L\in C^{\infty}(T^{(k)}Q)$ defined by
$(\mathcal{F}H_{o})^{*}{L}=\theta_{T^{(k-1)}Q}(X_{H})-H$ is regular. Indeed,
using implicit differentiation and the chain rule, we have

$\left(\displaystyle\frac{\partial^{2}{L}}{\partial{q_{(k)}^{i}}\partial{q_{(k)}^{j}}}\right)=\left(\displaystyle\frac{\partial{\hat{p}^{(k-1)}_{j}}}{\partial{q^{i}_{(k)}}}\right)=\left(\displaystyle\frac{\partial{p^{j}_{(k-1)}\circ\operatorname{leg}_{L}}}{\partial{q^{i}_{(k)}}}\right)=\left(\displaystyle\frac{\partial^{2}{H}}{\partial{p_{j}^{(k-1)}}\partial{p_{i}^{(k-1)}}}\right)^{-1}\,,$

where $\hat{p}^{(k-1)}_{j}$ are $n$ last Jacobi-Ostrogradsky momenta functions
defined in (13). Therefore, since $H$ is a $k$th-order regular Hamiltonian
function, the Hessian of $H$ with respect to the highest order momenta is
invertible at every point of $T^{*}T^{(k-1)}Q$, and hence so is the Hessian of
$L$ with respect to the highest order velocities. Thus, $L$ is regular.

Finally, let us consider the map
$\mathcal{F}H_{o}\circ\operatorname{leg}_{L}\colon T^{(2k-1)}Q\to T^{(k)}Q$.
In the natural coordinates of $T^{(2k-1)}Q$ the local expression of this map
is given by

$\left(\mathcal{F}H_{o}\circ\operatorname{leg}_{L}\right)\left(q^{i}_{(0)},\ldots,q^{i}_{(2k-1)}\right)=\mathcal{F}H_{o}\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)},\hat{p}_{i}^{(0)},\ldots,\hat{p}_{i}^{(k-1)}\right)=\left(q^{i}_{(0)},\ldots,q^{i}_{(k-1)},q^{i}_{(k)}\right)$

since $\tilde{q}^{\,i}_{(k)}=\mathcal{F}H_{o}^{*}q_{(k)}^{i}$. From this
coordinate expression we deduce that for every point $[\gamma]^{(2k-1)}_{0}\in
T^{(2k-1)}Q$ there exists an open set $U\subseteq T^{(2k-1)}Q$ such that
$[\gamma]^{(2k-1)}_{0}\in U$ and
$\left.\left(\mathcal{F}H_{o}\circ\operatorname{leg}_{L}\right)\right|_{U}=\left.\tau_{Q}^{(k,2k-1)}\right|_{U}$.
∎

######  Remark.

Since $\mathcal{F}H_{o}$ is at the best a submersion we can not directly pull-
back a function in $T^{*}T^{(k-1)}Q$ to $T^{(k)}Q$ by the inverse of
$\mathcal{F}H_{o}$ (since there isn’t), and hence we must first prove the
$\mathcal{F}H_{o}$-projectability of the function. $\lozenge$

######  Remark.

It is important to point out that in the conclusion of Proposition 8 we only
prove the regularity of $L$, instead of its hyperregularity (as in Proposition
1). Indeed, even if the Hamiltonian function is hyperregular, we can only
prove that $\operatorname{leg}_{L}$ is a surjective local diffeomorphism, but
not the injectivity. $\lozenge$

Before stating and proving the analogous to Proposition 2, we need the
following technical result, which we state and prove in a more general case
than we need.

######  Lemma 1.

Let $L\in C^{\infty}(T^{(k)}Q)$ be a $k$th-order Lagrangian function,
$\theta_{L}\in\Omega^{1}(T^{(2k-1)}Q)$ the associated Poincaré-Cartan $1$-form
and $E_{L}\in C^{\infty}(T^{(2k-1)}Q)$ the $k$th-order Lagrangian energy. If
$X\in\mathfrak{X}(T^{(2k-1)}Q)$ is a semispray of type $k$, then
$\left(\tau_{Q}^{(k,2k-1)}\right)^{*}L=\theta_{L}(X)-E_{L}$.

###### Proof.

This proof is easy in coordinates. Recall that the coordinate expression of a
semispray of type $r$ in $T^{(k)}Q$ is given by (2), from where we deduce that
a semispray of type $k$ $X$ in $T^{(2k-1)}Q$ is given locally by

$X=q_{(1)}^{i}\displaystyle\frac{\partial{}}{\partial{q_{(0)}^{i}}}+\ldots+q_{(k)}^{i}\displaystyle\frac{\partial{}}{\partial{q_{(k-1)}^{i}}}+F_{(k)}^{i}\displaystyle\frac{\partial{}}{\partial{q_{(k)}^{i}}}+\ldots+F_{(2k-1)}^{i}\displaystyle\frac{\partial{}}{\partial{q_{(2k-1)}^{i}}}\,.$

Hence, bearing in mind the coordinate expression (11) of the Poincaré-Cartan
$1$-form, we deduce that the smooth function $\theta_{L}(X)$ is given locally
by

$\theta_{L}(X)=\sum_{r=1}^{k}q_{(r)}^{i}\sum_{j=0}^{k-r}(-1)^{j}d_{T}^{j}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(r+j)}^{i}}}\right)\,.$

Hence, bearing in minda the coordinate expression (17) of the Lagrangian
energy $E_{L}$, we obtain

$\displaystyle\theta_{L}(X)-E_{L}$
$\displaystyle=\sum_{r=1}^{k}q_{(r)}^{i}\sum_{j=0}^{k-r}(-1)^{j}d_{T}^{j}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(r+j)}^{i}}}\right)-\sum_{r=1}^{k}q_{(r)}^{i}\sum_{j=0}^{k-r}(-1)^{j}d_{T}^{j}\left(\displaystyle\frac{\partial{L}}{\partial{q_{(r+j)}^{i}}}\right)+\left(\tau_{Q}^{(k,2k-1)}\right)^{*}L$
$\displaystyle=\left(\tau_{Q}^{(k,2k-1)}\right)^{*}L\,.$

as claimed. ∎

Finally, as for Proposition 2, we have the following result.

######  Proposition 9.

Let ${L}\in C^{\infty}(T^{(k)}Q)$ be a hyperregular $k$th-order Lagrangian
function and $H=E_{L}\circ\operatorname{leg}_{L}^{-1}\in
C^{\infty}(T^{*}T^{(k-1)}Q)$ the associated Hamiltonian function. Then $H$ is
a hyperregular $k$th-order Hamiltonian function and
$\mathcal{F}H_{o}=\tau_{Q}^{(k,2k-1)}\circ\operatorname{leg}_{L}^{-1}$. In
addition, if $\tilde{L}\in C^{\infty}(T^{(k)}Q)$ is the $k$th-order Lagrangian
function associated to $H$ by Proposition 8, then $\tilde{{L}}={L}$.

###### Proof.

The proof of $H$ being a hyperregular $k$th-order Hamiltonian functions
follows exactly the same patterns as the computations carried out in Section
4.2, and hence we omit them. Therefore, the only need to prove that if
$\tilde{L}\in C^{\infty}(T^{(k)}Q)$ is the $k$th-order Lagrangian function
associated to $H$ by Proposition 8, then $\tilde{{L}}={L}$. Observe that from
the properties of the map $\mathcal{F}H_{o}^{*}$ we have

$\mathcal{F}H_{o}^{*}\tilde{L}=(\tau_{Q}^{(k,2k-1)}\circ\operatorname{leg}_{L}^{-1})^{*}\tilde{L}=\left(\operatorname{leg}_{L}^{-1}\right)^{*}\left(\left(\tau_{Q}^{(k,2k-1)}\right)^{*}\tilde{L}\right)\,.$

On the other hand, from the definition of the Hamiltonian function and the
properties of the Legendre-Ostrogradsky map we have

$\displaystyle\theta_{T^{(k-1)}Q}(X_{H})-H$
$\displaystyle=\theta_{T^{(k-1)}Q}(X_{H})-(E_{L}\circ\operatorname{leg}_{L}^{-1})=\left(\operatorname{leg}_{L}^{-1}\right)^{*}\left(\theta_{T^{(k-1)}Q}(X_{H})\circ\operatorname{leg}_{L}-E_{L}\right)$
$\displaystyle=\left(\operatorname{leg}_{L}^{-1}\right)^{*}\left(\theta_{L}(X_{L})-E_{L}\right)=\left(\operatorname{leg}_{L}^{-1}\right)^{*}\left(\left(\tau_{Q}^{(k,2k-1)}\right)^{*}L\right)\,,$

where in the last step we have used Lemma 1, as the vector field $X_{L}$
solution to the Lagrangian dynamical equation
$\mathop{i}\nolimits_{X_{L}}\omega_{L}=dE_{L}$ is a semispray of type $1$ when
$L$ is a hyperregular higher-order Lagrangian function (see [18] for details).
Equating these two expressions, we have

$(\operatorname{leg}_{L}^{-1})^{*}\left(\left(\tau_{Q}^{(k,2k-1)}\right)^{*}\tilde{L}\right)=\left(\operatorname{leg}_{L}^{-1}\right)^{*}\left(\left(\tau_{Q}^{(k,2k-1)}\right)^{*}L\right)\,.$

Now, since $\operatorname{leg}_{L}^{-1}$ is a global diffeomorphism and
$\tau_{Q}^{(k,2k-1)}$ is the canonical projection, this last equality holds
if, and only if, $\tilde{L}=L$. ∎

## Acknowledgments

This work has been partially supported by NSF grant INSPIRE-1363720,
Ministerio de Economía y Competividad (Spain) grants MTM2013-42870-P and
MTM2014-54855-P, and Generalitat de Catalunya (Catalonia) project
2014-SGR-634. We would like to thank D. Martín de Diego and N. Román Roy for
providing the initial stimulus for this project. Also we wish to thanks M. de
León, J.C. Marrero and M.C. Muñoz Lecanda for fruitful comments and
discussions.

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