# Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions

@article{GayBalmaz2011HigherOL, title={Higher order Lagrange-Poincar{\'e} and Hamilton-Poincar{\'e} reductions}, author={François Gay‐Balmaz and Darryl D. Holm and Tudor S. Ratiu}, journal={Bulletin of the Brazilian Mathematical Society, New Series}, year={2011}, volume={42}, pages={579-606} }

Motivated by the problem of longitudinal data assimilation, e.g., in the registration of a sequence of images, we develop the higher-order 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 Euler-Poincaré and Lie-Poisson reduction is also studied in detail.

## 35 Citations

### Higher-order Variational Calculus on Lie algebroids

- Mathematics
- 2015

The equations for the critical points of the action functional defined by a Lagrangian depending on higher-order derivatives of admissible curves on a Lie algebroid are found. The relation with…

### Higher order mechanics on graded bundles

- Mathematics
- 2014

In this paper we develop a geometric approach to higher order mechanics on graded bundles in both, the Lagrangian and Hamiltonian formalism, via the recently discovered weighted algebroids. We…

### Invariant Higher-Order Variational Problems

- MathematicsCommunications in Mathematical Physics
- 2011

We investigate higher-order geometric k-splines for template matching on Lie groups. This is motivated by the need to apply diffeomorphic template matching to a series of images, e.g., in…

### Invariant Higher-Order Variational Problems

- Mathematics
- 2012

We investigate higher-order geometric k-splines for template matching on Lie groups. This is motivated by the need to apply diffeomorphic template matching to a series of images, e.g., in…

### Geometric integrators for higher-order mechanics on Lie groups

- Mathematics
- 2011

This paper develops a structure-preserving numerical integration scheme for a class of higher-order mechanical systems. The dynamics of these systems are governed by invariant variational principles…

### Matched Higher Order Lagrangian Dynamics

- Mathematics
- 2019

We write both the second order and the iterated tangent bundles of a Lie group as cocycle Lie group extensions. For two Lie groups under mutual interactions, matched pair group structures of these…

### Regularity properties of fiber derivatives associated with higher-order mechanical systems

- Mathematics
- 2016

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…

### Higher-order discrete variational problems with constraints

- Mathematics, Computer Science
- 2013

New variational integrators for higher-order Lagrangian mechanical system subjected to higher- order constraints are derived from the discretization of the variational principles, showing that their methods are automatically symplectic and, in consequence, with a very good energy behavior.

### Invariant higher-order variational problems: Reduction, geometry and applications

- Mathematics
- 2013

This thesis is centred around higher-order invariant variational problems defined on Lie groups. We are mainly motivated by applications in computational anatomy and quantum control, but the general…

### Second-order constrained variational problems on Lie algebroids: applications to optimal control

- Mathematics
- 2017

The aim of this work is to study, from an intrinsic and geometric point of view, second-order constrained variational problems on Lie algebroids, that is, optimization problems defined by a cost…

## References

SHOWING 1-10 OF 21 REFERENCES

### On the geometry of higher-order variational problems on Lie groups

- MathematicsArXiv
- 2011

Using left-trivialization of the higher- order tangent bundle of a Lie group and an adaptation of the classical Skinner-Rusk formalism, an intrinsic framework for higher-order lagrangian problems on Lie groups is deduced.

### Invariant Higher-Order Variational Problems

- Mathematics
- 2012

We investigate higher-order geometric k-splines for template matching on Lie groups. This is motivated by the need to apply diffeomorphic template matching to a series of images, e.g., in…

### Symplectic reduction of higher order Lagrangian systems with symmetry

- Mathematics
- 1994

One can develop a symplectic reduction procedure for higher order Lagrangian systems with symmetry. The reconstruction procedure of the dynamics is also studied and an application (spinning particle)…

### Gauged Lie-Poisson structures

- Mathematics
- 1984

A global formula for Poisson brackets on reduced cotangent
bundles of principal bundles is derived. The result bears on the basic constructions for interacting systems due to Sternberg and Weinstein…

### Variational principles for Lie-Poisson and Hamilton-Poincaré equations

- Mathematics
- 2003

As is well-known, there is a variational principle for the
Euler–Poincare equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton’s principle on G by the action of G by, say,…

### The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories

- Mathematics
- 1998

We study Euler–Poincare systems (i.e., the Lagrangian analogue of Lie–Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincare…

### Lagrangian Reduction by Stages

- Mathematics
- 2001

This booklet studies the geometry of the reduction of Lagrangian systems with symmetry in a way that allows the reduction process to be repeated; that is, it develops a context for Lagrangian…

### Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1977

This note is to show how to use symplectic geometry to write equations of motion of a "classical particle" in the presence of a Yang-Mills field, for any gauge group, G, and any differentiable…

### Higher order constrained Hamiltonian systems

- Mathematics
- 2009

In this paper we study a Hamiltonian formulation of the higher order constrained systems (HOCSs), defined previously in Lagrangian terms. We shall focus on second order constraints in positions,…

### A universal phase space for particles in Yang-Mills fields

- Mathematics
- 1978

Given a principal G-bundle P over M and a Hamiltonian G-space Q, one may construct the reduced symplectic manifold (T*P x Q)0. When a connection on P is chosen, this manifold becomes a bundle over…