Higher-order Mechanics: Variational Principles and other topics

After reviewing the Lagrangian-Hamiltonian unified formalism (i.e, the Skinner-Rusk formalism) for higher-order (non-autonomous) dynamical systems, we state a unified geometrical version of the Variational Principles which allows us to derive the Lagrangian and Hamiltonian equations for these kinds of systems. Then, the standard Lagrangian and Hamiltonian formulations of these principles and the corresponding dynamical equations are recovered from this unified framework.


Introduction
Higher-order systems appear in many models in theoretical and mathematical physics, such as in the mathematical description of relativistic particles with spin, string theories, gravitation, Podolsky's generalization of electromagnetism and others. They also appear in some problems of fluid mechanics, electric networks and classical physics, and in numerical models arising from the geometric discretization of first-order dynamical systems that preserve their inherent geometric structures (see [20,21] for a long but non-exhaustive list of references).
In these kinds of systems and, in particular, in higher-order mechanics, the dynamics have explicit dependence on accelerations or higher-order derivatives of the generalized coordinates of position. So, for Lagrangian systems, if the Lagrangian function depends on derivatives of order k, the corresponding Euler-Lagrange equations are of order 2k. Thus, the geometric descriptions of these systems use higher-order tangent and jet bundles as the main tool (see, for instance, [5,9,10,17,18,19,23]).
Furthermore, a generalization of the Lagrangian and Hamiltonian formalisms of first-order autonomous mechanical systems exists that compresses them into a single formalism: the Skinner-Rusk or Lagrangian-Hamiltonian unified formalism, proposed by R. Skinner and R. Rusk for first-order autonomous mechanical systems [24]. It was generalized to non-autonomous dynamical systems, control systems, first-order classical field theories and higher-order classical field theories [2,3,7,12]. The generalization of the Skinner-Rusk unified formalism for higher-order mechanical systems has been developed in recent papers [6,20,21].
The aim of this lecture is twofold: first to review this unified formalism for higher-order mechanical systems and second to state the variational principles for higher-order systems and derive the higher-order Euler-Lagrange and Hamilton equations using this unified framework. These geometric variational techniques are based on those introduced for for first-order field theories in [1,13,14,15]. Our study is made for non-autonomous higher-order mechanical systems (the autonomous case can thereby be obtained by using trivial bundles and removing the time-dependence).
In particular, we start by introducing some basic geometrical background in Section 2, and then reviewing the construction of the framework for the unified Lagrangian-Hamiltonian formalism for non-autonomous higher-order systems (developed in [21]) in Section 3. The main contributions of the paper begin in Section 4, where we establish the variational principle and use it to derive the higher-order equations for the Lagrangian-Hamiltonian unified formalism, which are written in several equivalent geometric ways. Then, in Section 5, these equations are analyzed in detail, showing how they compress not only the dynamical evolution equations but also the equations of the Legendre-Ostrogradsky transformation connecting the Lagrangian and Hamiltonian formalism, which appear as compatibility and consistency conditions for the equations. Other relevant results are presented in Sections 6 and 7, where first we recover the generalization to higher-order systems of the Hamilton Variational Principle of the Lagrangian formalism and the Hamilton-Jacobi Variational Principle of the Hamiltonian formalism, and then the higher-order Euler-Lagrange and the Hamilton equations. All these results are obtained in a straightforward way from this unified formalism. Finally, some conclusions and further research on these topics are discussed in Section 8.
All the manifolds are real, second countable and C ∞ . The maps and the structures are assumed to be C ∞ . Sum over repeated indices is understood.
2 Higher-order jet bundles over R (See [23] for details on jet bundles and [21] for details and proofs on the unified formalism).
Let E π −→ R be a fiber bundle with dim E = n+1, and let η ∈ Ω 1 (R) be the canonical volume form in R. If k ∈ N, the kth order jet bundle of the projection π, J k π, is the ((k + 1)n + 1)dimensional manifold of the k-jets of sections φ ∈ Γ(π). A point in J k π is denoted by j k φ, where φ ∈ Γ(π) is any representative of the equivalence class. We have the following natural projections: if r k, Notice that π k 0 = π k , where J 0 π is canonically identified with E, and π k k = Id J k π . Furthermore, if φ ∈ Γ(π) is a section of π, we also denote the canonical lifting of φ to J k π by j k φ ∈ Γ(π k ).
Let t be the global coordinate in R such that η = dt, and (t, q A 0 ), 1 A n, local coordinates in E adapted to the bundle structure. Then, natural coordinates in J k π are (t, Using these coordinates, the local expressions of the natural projections are ; that is, the section ψ is the lifting of a section of π to J k−r+1 π. In particular, a section ψ is holonomic of type 1 if j k φ = ψ; that is, ψ is the canonical k-jet lifting of a section φ ∈ Γ(π), where φ = π k • ψ. A vector field X ∈ X(J k π) is a semispray of type r if every integral section of X is holonomic of type r. Throughout this paper, sections that are holonomic of type 1 are simply called holonomic.
In natural coordinates, the local expression of a holonomic section of type r is . Thus, the local expression of a semispray of type r is

Lagrangian-Hamiltonian unified formalism
Let π : E → R be the configuration bundle of a kth order dynamical system, with dim E = n + 1. The higher-order extended jet-momentum bundle and the higher-order restricted jet-momentum bundle are where J k−1 π * = T * (J k−1 π)/(π k−1 ) * T * R.
These bundles are endowed with the canonical projections In addition, the natural quotient map µ : T * (J k−1 π) → J k−1 π * induces a natural projection If (t, q A 0 ) are local coordinates in E adapted to the bundle structure, the induced coordinates in all these bundles are Remark: The last coordinates are the real dynamical variables and so W r is the real phase space in the unified formalism.
A vector field X ∈ X(W) is a semispray of type r in W, 1 r k, if every integral section ψ of X is holonomic of type r in W.
In natural coordinates, the above forms are given by and ker Ω is locally given by

Definition 2
The higher-order coupling 1-formĈ ∈ Ω 1 (W) is the ρ R -semibasic form defined as follows: for every w = (ȳ, α q ) ∈ W, whereȳ ∈ J 2k−1 π, q = π 2k−1 k−1 (ȳ), and α q ∈ T * q (J k−1 π); if φ ∈ Γ(π) is any representative ofȳ, and u ∈ T w W, then AsĈ is a ρ R -semibasic form, there is a coupling functionĈ ∈ C ∞ (W) such thatĈ =Ĉρ * R η = Cdt. In natural coordinates the coupling function iŝ The dynamical information is introduced by giving a kth-order Lagrangian density L ∈ Ω 1 (J k π), which is aπ k -semibasic form. So we can write L = L · (π k ) * η, where L ∈ C ∞ (J k π) is the Lagrangian function. Then we denoteL = (π 2k−1 k • ρ 1 ) * L. As the Lagrangian density is ā π k -semibasic form, thenL is a ρ R -semibasic 1-form, and we have thatL =Lρ * R η =Ldt, wherê In order to have a geometric structure in W r we define the so-called Hamiltonian submanifold SinceĈ andL are both ρ R -semibasic forms, the submanifold W o is defined by the constraint C −L = 0. In natural coordinates, bearing in mind the local expression (1) ofĈ, the constraint function is given by [21] we have that: As a consequence of this, if Υ : W r → W o denotes this diffeomorphism, we have an induced sectionĥ = j o • Υ ∈ Γ(µ W ), which is specified by giving the local The sectionĥ is called a Hamiltonian section of µ W , or a Hamiltonian µ W -section.

Variational Principle for the unified formalism
Next we establish the variational principle from which the dynamical equations for the unified formalism are derived. Our starting point is the precosymplectic Hamiltonian system where the convergence of the integral is assumed.

Definition 3 (Generalized Variational Principle)
The Lagrangian-Hamiltonian variational problem for the system (W r , Ω r , (ρ r R ) * η) is the search for the critical (or stationary) holonomic sections of the functional LH with respect to the variations of ψ given by ψ t = σ t • ψ, where {σ t } is a local one-parameter group of any compact-supported vector field The main result of the calculus of variations in this context is the following: The following assertions on a section ψ ∈ Γ(ρ r R ) are equivalent: 1. ψ is a solution to the Lagrangian-Hamiltonian variational problem.
2. ψ is a holonomic section solution to the equation 3. ψ is a holonomic section solution to the equation 4. ψ is an integral curve of a vector field contained in a class of ρ r R -transverse semisprays of type 1, {X} ⊂ X(W r ), satisfying the equation (Proof ) We prove the equivalence 1 ⇐⇒ 2 following the patterns taken from [11]. For the proof of the other equivalences, see [21] (Theorem 1).
as a consequence of Stoke's theorem and the assumptions made on the supports of the vertical vector fields. Thus, by the fundamental theorem of the variational calculus, we conclude However, since the compact-supported vector fields generate locally the C ∞ (W r )-module of vector fields in W r , it follows that the last equality holds for every ρ r R -vertical vector field Z in W r . Now, recall that for every point w ∈ Imψ, we have a canonical splitting of the tangent space of W r at w in a ρ r R -vertical subspace and a ρ r R -horizontal subspace, that is, since ψ * i(Y V )Ω r = 0, by the conclusion in the above paragraph. Now, as Y ψ w ∈ T w (Imψ) for every w ∈ Imψ, then the vector field Y ψ is tangent to Imψ, and hence there exists a vector field However, as dim Imψ = dim R = 1 and Ω r is a 2-form, we obtain that Taking into account the reasoning of the first paragraph, the converse is obvious since the condition ψ * i(Y )Ω r = 0, for every Y ∈ X(W r ), holds, in particular, for every Z ∈ X V (ρ r R ) (W r ).

Analysis of the unified dynamical equations
In order to complete the Lagrangian-Hamiltonian unified formalism, it is necessary to analyze the dynamical equations. We start by using the equations written for vector fields. Thus, equations (2), with the ρ r R -transverse condition, are It is usual to fix the ρ r R -transverse condition by demanding that This selects a representative in the class {X}. We will do this in the sequel.
The first important result is [21]: The above equations are compatible only on the points of the following submanifold of W r In natural coordinates, a generic vector field X ∈ X(W r ) is given by and the ρ r R -transverse condition implies f = 0, and in particular, using (3), we get f = 1. Therefore, the dynamical equation (2) first gives which are the compatibility relations (constraints) defining locally W 1 . Furthermore, for 0 l k − 1, k j 2k − 1; Observe that, in a natural way, X is a semispray of type k. Nevertheless, the variational principle requires that X must be a semispray of type 1, thus Next we must require X to be tangent to W 1 . Thus, it is necessary to impose that L(X)ξ| W 1 = 0, for every constraint function ξ defining W 1 : ). Repeating this procedure (k − 1 steps), we get Thus we obtain a sequence of submanifolds (which can also be obtained by applying any other constraint algorithm [8,16]), As a consequence of the last equalities we conclude that Proposition 3 The submanifold W L is the graph of a map FL : J 2k−1 π → J k−1 π * locally defined by Definition 4 The map FL : J 2k−1 π → J k−1 π * is the (restricted) Legendre-Ostrogradsky map.
A Lagrangian density L is regular if the map FL is a local diffeomorphism. Otherwise, L is said to be a singular Lagrangian. If FL is a global diffeomorphism, then L is said to be hyperregular.
In natural coordinates, the regularity condition for L is equivalent to Observe that X is not necessarily tangent to W L . Thus, imposing the tangency condition to the last generation of constraints defining W L , these conditions give the following equations (on W L ): And, as a consequence of (4), we have: Proposition 4 If L is regular, then there is a unique semispray of type 1, X ∈ X(W r ), tangent to W L , which is a solution to the dynamical equations (on W L ).
If L is not regular, new constraints could appear and the algorithm continues until arriving (in the best cases) at a final constraint submanifold W f ֒→ W L .
is an integral section of X, the above equations lead tȯ and after some calculations we reach (on the points of W L , or W f ), These equations compress both the higher-order Euler-Lagrange and Hamilton equations, as can be seen in the following sections.
Remark: It is interesting to point out that the variational principle for higher-order autonomous dynamical systems, and the corresponding dynamical equations, can be obtained as a particular case of these results when the Lagrangian function does not depend explicitly on the coordinate t.

Lagrangian formalism: Generalized Hamilton Principle
In this section we show how to recover the Lagrangian formalism for higher-order mechanical systems. In particular, we state the classical Hamilton Variational Principle of the Lagrangian formalism for higher-order systems and study its relation with the unified variational Principle.
First, consider the diagram As W L is the graph of the restricted Legendre-Ostrogadski map, we have that the map ρ L 1 = ρ r 1 • j L : W L → J 2k−1 π is a diffeomorphism. Then we can define the Poincaré-Cartan 1 and 2 forms in J 2k−1 π as These forms can also be introduced in several equivalent ways (see, for instance, [1,14,22,23]).

Definition 5 (Generalized Hamilton Variational Principle)
The Lagrangian variational problem (also called Hamilton variational problem) for the higher-order Lagrangian system (J 2k−1 π, Ω L , (π 2k−1 ) * η) is the search for the critical (or stationary) sections of the functional L with respect to the variations of φ given by φ t = σ t • φ, where {σ t } is a local one-parameter group of any compact-supported Z ∈ X V (π) (E); that is, Then, as in the above section, we have: The following assertions on a section φ ∈ Γ(π) are equivalent: 1. φ is a solution to the Lagrangian variational problem.
(Proof ) The proof of the equivalence 1 ⇐⇒ 2 follows the same patterns as in Theorem 1. For the proof of the other equivalences, see [21] (Theorem 3).
Section solutions to the Hamilton variational problem are recovered from section solutions to the Lagrangian-Hamiltonian variational problem in the unified formalism. In fact: Theorem 3 Let ψ ∈ Γ(ρ r R ) be a holonomic section which is a solution to the Lagrangian-Hamiltonian variational problem given by the functional LH. Then, the section ψ L = ρ r 1 • ψ ∈ Γ(π 2k−1 ) is holonomic, and its projection φ = π 2k−1 • ψ L ∈ Γ(π) is a solution to the Lagrangian variational problem given by the functional L; Conversely, from a holonomic section ψ L = j 2k−1 φ ∈ Γ(π 2k−1 ) which is a solution to the Lagrangian variational problem, we recover a solution ψ = (ψ L , ψ L • FL) to the Lagrangian-Hamiltonian variational problem.
is a holonomic section, by definition. Now, ρ r 1 being a submersion, for every compact-supported vector field X ∈ X V (π 2k−1 ) (J 2k−1 π) there exist compact-supported vector fields Y ∈ X V (ρ r R ) (W r ) such that ρ r 1 * Y = X; that is, X and Y are ρ r 1 -related. In particular, this holds if X is the (2k − 1)-jet lifting of a compact-supported π-vertical vector field in E; that is, if we have X = j 2k−1 Z, with Z ∈ X V (π) (E). We denote by {σ t } a local one-parameter group for the compact-supported vector fields Y ∈ X V (ρ o R ) (W r ). Then, using this particular choice of ρ r 1 -related vector fields, we have since ψ is a critical section for the Lagrangian-Hamiltonian variational problem.
Conversely, if we have a holonomic section ψ L = j 2k−1 φ which is a solution to the Lagrangian variational problem, then we can construct ψ = (ψ L , ψ L • FL), which is a section ψ : R → W L ⊂ W r of the projection ρ r R (remember that, in the unified formalism, the dynamical equations have solutions only on the points of W L , or in a subset of it). Then, the above reasoning also shows also that if ψ L is a solution to the Lagrangian variational problem, then ψ is a solution to the Lagrangian-Hamiltonian variational problem.
In natural coordinates, if ψ is given by ψ(t) = (t, q A i (t), q A j (t), p i A (t)), then ψ L = (ρ r 1 • ψ)(t) = (t, q A i (t), q A j (t)), and φ(t) = (π 2k−1 • ψ L )(t) = (t, q A 0 (t)) satisfies the kth-order Euler-Lagrange Finally, as a consequence of all the above results, we have the corresponding relation between vector field solutions to the unified dynamical equations and those which are solutions to the Lagrangian equations: Proposition 5 Let X ∈ X(W r ) be a vector field tangent to W L which is a solution to the equations i(X)Ωr = 0 ; i(X)(ρ r R ) * η = 1 .
In addition, if L is a regular Lagrangian density, then X L is a semispray of type 1.
Conversely, if X L ∈ X(J 2k−1 π) is a semispray of type k (resp., of type 1), which is a solution to the equations (6), then there exists a unique X ∈ X(W r ) which is a solution to the equations (5) and it is a semispray of type k in W r (resp., of type 1).
(Proof ) See also [21] (Theorem 2) for a detailed proof of this statement.

Hamiltonian formalism: Generalized Hamilton-Jacobi Principle
In this section we state the Hamiltonian variational problem (Hamilton-Jacobi Principle) for higher-order systems, recovering it from the unified formalism. (See [21] for the proofs and details on the higher-order Hamiltonian formalism).
As stated in Section 5, if these conditions are fulfilled, the Lagrangan density L is said to be regular, and when the restricted Legendre-Ostrogradsky map is a global diffeomorphism, then L is hyperregular.
Now, let P = Im( FL) ֒→ T * (J k−1 π) and P = Im(FL) is the higher-order non-autonomous Hamiltonian system associated with (W r , Ω r , (ρ r R ) * η). In what follows, we consider that the Lagrangian density L ∈ Ω 1 (J k π) is, at least, almostregular. However, all the following results also hold for regular or hyperregular Lagrangian densities, replacing P by the corresponding open subset of J k−1 π * , or by J k−1 π * , respectively.
First, we establish the variational principle from which we can obtain the dynamical equations for the Hamiltonian formalism, and then we show how to obtain the Hamiltonian dynamical equations. Given the Hamiltonian system (P, Ω P ,τ * o η), let Γ(τ o ) be the set of sections ofτ o , that is, curves ϕ : R → P. Consider the functional where the convergence of the integral is assumed.
Definition 6 (Generalized Hamilton-Jacobi Variational Principle) The Hamiltonian or Hamilton-Jacobi variational problem for the higher-order Hamiltonian system (P, Ω P ,τ * o η) is the search for the critical (or stationary) sections of the functional H with respect to the variations of ϕ given by ϕ t = σ t • ϕ, where {σ t } is a local one-parameter group of any compact-supported Z ∈ X V (τo) (P); that is d dt t=0 R ϕ * t Θ P = 0 (7) Then, as in the above sections, we have: The following assertions on a section ϕ ∈ Γ(τ o ) are equivalent: 1. ϕ is a solution to the Hamiltonian variational problem.
2. ϕ is a solution to the equation

ϕ is a solution to the equation
where ϕ ′ : R → TP is the canonical lifting of ϕ to TP. Conversely, from a section ψ h solution to the Hamiltonian variational problem, we recover a solution ψ to the Lagrangian-Hamiltonian variational problem.
(Proof ) Observe that FL o • ρ r 1 is a submersion, since it is a composition of submersions, and (FL o • ρ r 1 ) * Θ P = (ρ r 1 ) * (FL * o Θ h ) = (ρ r 1 ) * Θ L = Θ r . Then, for every compact-supported vector field Z ∈ X V (τo) (P), there exist compact-supported vector fields Y ∈ X V (ρ r R ) (W r ) such that We denote by {σ r t } a local oneparameter group for the compact-supported vector fields Y ∈ X V (ρ r R ) (W r ). Then, using this particular choice of (FL o • ρ r 1 )-related vector fields, we have since ψ is a critical section for the Lagrangian-Hamiltonian variational problem.
For the converse, following the same patterns as in the theory of singular non-autonomous first-order mechanical systems [8], it can be proved that there are holonomic sections ψ L : R → J 2k−1 π of the projectionπ 2k−1 such that ψ h = FL o • ψ L , and they are solutions to the Lagrangian dynamical equations. Then, the sections ψ = (ψ L , ψ h ) are solutions to the Lagrangian-Hamiltonian variational problem (see the proof of Theorem 3).
In natural coordinates, if ψ h is given by ψ h (t) = (t, q A i (t), p i A (t)), 0 i k − 1, then the above equations give the classical higher-order Hamilton equations: Finally, as a consequence of all the above results, we have the corresponding relation between vector field solutions to the unified dynamical equations and those which are solutions to the Hamiltonian equations: Proposition 6 Let X ∈ X(W r ) be a vector field tangent to W L and solution to the equations i(X)Ωr = 0 ; i(X)(ρ r R ) * η = 1 , Then there exist vector fields X h ∈ X(P), which are solutions to the equations Conversely, if X h ∈ X(P) is a vector field which is a solution to the equations (9), then there exist vector fields X ∈ X(W r ) which are solutions to the equations (8).
(Proof ) : See also [21] (Theorem 4) for a detailed proof of this statement.
Remark: It is interesting to point out that, for almost-regular systems, if the unified dynamical equations have consistent solutions on a final constraint submanifold W f ֒→ W r , then the Lagrangian and Hamiltonian equations have consistent solutions on final constraint submanifolds S f = ρ r 1 (W f ) ֒→ J 2k−1 π and P f =ρ r 1 (W f ) ֒→ P, respectively. Then all the above results hold on the points of these submanifolds instead of W r , J 2k−1 π, and P, respectively.
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Conclusions and further research
We have made an accurate revision of the generalization of the Lagrangian-Hamiltonian unified formalism of R. Skinner and R. Rusk to higher-order dynamical systems. We have analyzed the non-autonomous case, since the autonomous case can be obtained as a particular situation of this. This particular situation consists in using trivial bundles and removing the time-dependence (see [20]). This unified formalism constitutes a nice framework which allows us to study different kinds of problems in a simpler way. In particular, singular (constrained) systems can be analyzed more easily.
In particular, as a new contribution, the classical variational principles of first-order mechanics are generalized to this framework, in order to state the dynamical equations for higher-order mechanics in several equivalent ways. Therefore, the Lagrangian and Hamiltonian structures, equations and solutions of higherorder mechanics are recovered from those obtained in the unified formalism, which also includes the corresponding Lagrangian and Hamiltonian variational principles: the generalized Hamilton and Hamilton-Jacobi Principles respectively.
Several interesting physical examples have been studied using this formalism; for instance the Pais-Uhlenbeck oscillator and the shape of a deformed elastic cylindrical beam with fixed ends, as regular systems; the second-order relativistic particle, first as a free particle and later subjected to a potential, as singular systems [20,21], and also underactuated control systems [6].
This generalization of the Lagrangian-Hamiltonian unified formalism to higher-order dynamical systems using a general fibre bundle E over R (instead of the classical approach using trivial bundles) is a first step towards the study of higher-order classical field theory. However, replacing the base manifold R with an orientable m-dimensional manifold M gives rise to new difficulties, such as defining a suitable fiber bundle to act as the phase space; obtaining the Legendre map without ambiguities, or obtaining the relation between the momenta (which is crucial in our formulation). Nevertheless+, our future aim is to obtain an unambiguous Lagrangian-Hamiltonian unified formalism for higher-order classical field theory, thus completing previous works [4,25].