Higher order LagrangePoincaré and HamiltonPoincaré reductions
Abstract
Motivated by the problem of longitudinal data assimilation, e.g., in the registration of a sequence of images, we develop the higherorder framework for Lagrangian and Hamiltonian reduction by symmetry in geometric mechanics. In particular, we obtain the reduced variational principles and the associated Poisson brackets. The special case of higher order EulerPoincaré and LiePoisson reduction is also studied in detail.
J. Braz. Math. Soc. 42(4), (2011), 579–606
AMS Classification: 70H50; 37J15; 70H25; 70H30.
Keywords: variational principle,
symmetry, connection, Poisson brackets, higher order
tangent bundle, LiePoisson reduction, EulerLagrange
equations, EulerPoincaré equations,
LagrangePoincaré equations, HamiltonPoincaré
equations
Contents
1 Introduction
Background.
Many interesting mechanical systems, such as the incompressible fluid, the rigid body, the KdV equation, or the CamassaHolm equations can be written as the EulerPoincaré equations on a Lie algebra of a Lie group . The corresponding Hamiltonian formulations are given by LiePoisson equations obtained by Poisson reduction of the canonical Hamilton equations on .
A more general situation occurs if the original configuration space is not a Lie group, but a configuration manifold on which a Lie group acts freely and properly, so that becomes a principal bundle. Starting with a Lagrangian system on invariant under the tangent lifted action of , the reduced equations on , appropriately identified, are the LagrangePoincaré equations derived in Cendra, Marsden, and Ratiu [2001]. Similarly, if we start with a Hamiltonian system on , invariant under the cotangent lifted action of , the resulting reduced equations on are called the HamiltonPoincaré equations, Cendra, Marsden, Pekarsky, and Ratiu [2003], with an interesting Poisson bracket, the gauged LiePoisson structure, involving a canonical bracket, a LiePoisson bracket, and a curvature term.
Goals.
The goal of this paper is to present the extension of this picture to the higher order case, that is, the case when the Lagrangian function is defined on the order tangent bundle and thus depends on the first order time derivatives of the curve. We thus derive the order LagrangePoincaré equations on and obtain the order EulerPoincaré equations on in the particular case , together with the associated constrained variational formulations.
On the Hamiltonian side, using the Legendre transform associated to the Ostrogradsky momenta, we obtain what we call the OstrogradskyHamiltonPoincaré equations on and, in the particular case , the OstrogradskyLiePoisson equations.
Motivation and approach.
Our motivation for making these extensions to higher order of the fundamental representations of dynamics in geometric mechanics is to cast light on the options available in this framework for potential applications, for example, in longitudinal data assimilation. However, these applications will not be pursued here and we shall stay in the context of the initial value problem, rather than formulating the boundary value problems needed for the applications of the optimal control methods, say, in longitudinal data assimilation. For further discussion of the motivation for developing the higherorder framework for geometric mechanics in the context of optimal control problems, in particular for registration of a sequence of images, see GayBalmaz, Holm, Meier, Ratiu, and Vialard [2012].
2 Geometric setting
We shall begin by reviewing the definition of higher order tangent bundles , the connectionlike structures defined on them, and the description of the quotient of by a free and proper group action. For more details and explanation of the geometric setting for higher order variational principles in the context that we follow here, see Cendra, Marsden, and Ratiu [2001]. We also recall the formulation of the order EulerLagrange equations and the associated Hamiltonian formulation obtained through the Ostrogradsky momenta. We refer to de Leon and Rodrigues [1985] for the geometric formulation of higher order Lagrangian and Hamiltonian dynamics.
2.1 Higher order tangent bundles
The order tangent bundle of a manifold is defined as the set of equivalence classes of curves in under the equivalence relation that identifies two given curves , if and in any local chart we have , for , where denotes the derivative of order . The equivalence class of the curve at is denoted . The projection
It is clear that , , and that, for , there is a well defined fiber bundle structure
Apart from the cases and , the bundles are not vector bundles. We shall use the natural coordinates on induced by a coordinate system on .
A smooth map induces a map
(2.1) 
In particular, a group action naturally lifts to a group action
(2.2) 
When the action is free and proper we get a principal bundle . The quotient is a fiber bundle over the base . The class of the element in the quotient is denoted .
2.2 Higher order EulerLagrange equations
Consider a Lagrangian defined on the order tangent bundle. We will often use the local notation instead of the intrinsic one . A curve is a critical curve of the action
(2.3) 
among all curves whose first derivatives , , , are fixed at the endpoints if and only if is a solution of the order EulerLagrange equations
(2.4) 
These equations follow from Hamilton’s variational principle,
In the notation, an infinitesimal variation of the curve is denoted by and defined by the variational derivative,
(2.5) 
where for all for which the curve is defined and , for all , , and . Thus for all .
Examples: Riemannian cubic polynomials and generalizations.
As originally introduced in Noakes, Heinzinger, and Paden [1989], Riemannian cubic polynomials (or splines) generalize Euclidean splines to Riemannian manifolds. Let be a Riemannian manifold and be the covariant derivative along curves associated with the LeviCivita connection for the metric . The Riemannian cubic polynomials are defined as minimizers of the functional in (2.3) for the Lagrangian defined by
(2.6) 
This Lagrangian is welldefined on the secondorder tangent bundle since, in coordinates,
(2.7) 
where are the Christoffel symbols at the point of the metric in the given basis. These Riemannian cubic polynomials have been generalized to the socalled elastic splines through the following class of Lagrangians
(2.8) 
where is a real constant, see Hussein and Bloch [2004]. Another extension are the order Riemannian splines, or geometric splines, where
(2.9) 
for . As for the Riemannian cubic splines, is welldefined on . Denoting by the curvature tensor defined as , the EulerLagrange equation for elastic splines () reads
(2.10) 
as proven in Noakes, Heinzinger, and Paden [1989], Hussein and Bloch [2004]. For the higherorder Lagrangians (2.9), the EulerLagrange equations read, Camarinha, Silva Leite, and Crouch [1995],
(2.11) 
2.3 Quotient space and reduced Lagrangian
We now review the geometry of the quotient space relative to the lifted action , in preparation for the reduction processes we shall present in the next sections.
The quotient space .
Consider a free and proper right (resp. left) Lie group action of on . Let us fix a principal connection on the principal bundle , that is, a oneform such that
where is the infinitesimal generator associated to the Lie algebra element . Recall that by choosing a principal connection on the principal bundle , we can construct a vector bundle isomorphism
(2.12) 
where denotes the Whitney sum and the adjoint bundle is the vector bundle defined by the quotient space relative to the diagonal action of .
We now recall from Cendra, Marsden, and Ratiu [2001] how this construction generalizes to the case of order tangent bundles. The covariant derivative of a curve relative to a given principal connection is given by
(2.13) 
where the upper (resp. lower) sign in corresponds to a right (resp. left) action. In the particular case when , we have
(2.14)  
(2.15) 
and, more generally,
(2.16) 
where and . The bundle isomorphism that generalizes (2.12) to the order case is defined by
(2.17) 
where , is any curve representing such that , and denotes the Witney sum of copies of the adjoint bundle. We refer to Cendra, Marsden, and Ratiu [2001] for further information and proofs. We will use the suggestive notation
(2.18) 
for the reduced variables.
Remark 2.2
It is important to observe that the notation in the quotient map (2.17) stands for the intrinsic expression obtained via (2.14)–(2.16) from the element .
In (2.18) the dot notations on and have not the same meaning: are natural coordinates on , whereas are elements in , all seen as independent variables. When dealing with curves, really means the ordinary time derivative in the local chart, whereas means the covariant derivative .
The reduced Lagrangian.
If is a invariant order Lagrangian, then it induces a Lagrangian defined on the quotient space . If a connection is chosen, then we can write the reduced Lagrangian as
The case of a Lie group.
Let us particularize the map to the case where is the Lie group . The adjoint bundle can be identified with the Lie algebra via the isomorphism
(2.19) 
The principal connection is the MaurerCartan connection
and one observes that the associated covariant derivative of a curve in corresponds, via the isomorphism (2.19), to the ordinary time derivative in :
Therefore, in the case , the bundle isomorphism becomes
(2.20) 
respectively,
(2.21) 
where denotes the sum of copies of . Note that in this particular case, one may choose since we have . The reduced Lagrangian is thus a map .
2.4 Ostrogradsky momenta and higher order Hamilton equations
It is wellknown that the Hamiltonian formulation of the order EulerLagrange equations is obtained via the Ostrogradsky momenta defined locally by
So, for example, we have
These momenta are encoded in the OstrogradskyLegendre transform that reads locally . We refer to de Leon and Rodrigues [1985] for the intrinsic definition of the Legendre transform as well as for the geometric formulation of higher order Lagrangian dynamics. In the same way as in the first order case, the PoincaréCartan forms associated to are defined by
where and are the canonical forms on . The energy function associated to is defined by
where the bracket denotes the duality pairing between and . Locally we have
The Lagrangian is said to be regular if is a symplectic form or, equivalently, if is a local diffeomorphism. In this case the solution of the order EulerLagrange are the integral curves of the Lagrangian vector field defined by
When is a global diffeomorphism, then is hyperregular and the associated Hamiltonian is defined by
In this case, the order EulerLagrange equations are equivalent to the canonical Hamilton equations associated to and are locally given by
The solution is the integral curve of the Hamiltonian vector field defined by
3 Higher order EulerPoincaré reduction
In this section by following GayBalmaz, Holm, Meier, Ratiu, and Vialard [2012], we derive the order EulerPoincaré equations by reducing the variational principle associated to a right (resp. left) invariant Lagrangian in the special case when the configuration manifold is the Lie group and the action is by right (resp. left) multiplication.
Constrained variations.
As we have seen, the reduced Lagrangian is induced by the quotient map given by
(3.1) 
and obtained by particularizing the quotient map of (2.17). The variations of the quantities in the quotient map (3.1) are thus given by
(3.2)  
where , resp. for right, resp. left invariance. Therefore, the variations are such that vanish at the endpoints, for all .
Hamilton’s principle.
The EulerPoincaré equations for the reduced Lagrangian follow from Hamilton’s principle with by using these variations, as
and applying the vanishing endpoint conditions when integrating by parts. Therefore, stationarity implies the korder EulerPoincaré equations,
(3.3) 
Formula (3.3) takes the following forms for various choices of :
If :
If :
(3.4) 
If :
The first of these is the usual EulerPoincaré equation. The others adopt a factorized form in which the EulerPoincaré operator is applied to the EulerLagrange operation on the reduced Lagrangian at the given order.
The results obtained above are summarized in the following theorem.
Theorem 3.1 (order EulerPoincaré reduction)
Let be a invariant Lagrangian and the associated reduced Lagrangian. Let be a curve in and , resp. be the reduced curve in the Lie algebra . Then the following assertions are equivalent.

The curve is a solution of the order EulerLagrange equations for .

Hamilton’s variational principle
holds using variations such that vanish at the endpoints for .

The order EulerPoincaré equations for hold:
(3.5) 
The constrained variational principle
holds for constrained variations , where is an arbitrary curve in such that vanish at the endpoints, for all .
We now quickly recall from GayBalmaz, Holm, Meier, Ratiu, and Vialard [2012] how the higher order EulerPoincaré theory applies to geodesic splines on Lie groups.
Example: Riemannian 2splines on Lie groups.
Fix a right, resp. left invariant Riemannian metric on the Lie group and let be its corresponding squared metric. Consider the Lagrangian for Riemannian splines, given by
(3.6) 
in which denotes covariant derivative in time. It is shown in GayBalmaz, Holm, Meier, Ratiu, and Vialard [2012] that the reduced Lagrangian associated to is given by
(3.7) 
where is the flat operator associated to . From formula (3.4) with one then finds the order EulerPoincaré equation
(3.8) 
If the metric is both left and right invariant (biinvariant) further simplifications arise. Indeed, in this case we have so that and the equations in (3.8) become
(3.9) 
as in Crouch and Silva Leite [1995]. Note that in this case, the reduced Lagrangian (3.7) is given simply by .
4 Higher order LagrangePoincaré reduction
Here we generalize the method of LagrangePoincaré reduction in Cendra, Marsden, and Ratiu [2001] to higher order invariant Lagrangians defined on . Recall from §2.1 that we consider a free and proper right (resp. left) action of on and its lift on the order tangent bundle . By fixing a principal connection on the principal bundle , the quotient space can be identified with the bundle .
Constrained variations.
The main departure point is to compute the constrained variations of
(4.1) 
induced by a variation of the curve . Since
the variations of are arbitrary except for the endpoint conditions , for all , . The variations of may be computed with the help of a fixed connection . For , we have
This computation implies
(4.2) 
for and where is the curvature form and is the reduced curvature. Our conventions are for all .
For