# Variational calculus on Lie algebroids

@article{Martnez2006VariationalCO, title={Variational calculus on Lie algebroids}, author={Eduardo Mart{\'i}nez}, journal={ESAIM: Control, Optimisation and Calculus of Variations}, year={2006}, volume={14}, pages={356-380} }

It is shown that the Lagrange's equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of Lagrangian reduction and the relation with the method of Lagrange multipliers are also studied.

## 28 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…

### Lie algebroids in classical mechanics and optimal control.

- Mathematics
- 2007

We review some recent results on the theory of Lagrangian systems on Lie algebroids. In particular we consider the symplectic and variational formalism and we study reduction. Finally we also…

### Singular Lagrangian systems and variational constrained mechanics on Lie algebroids

- Mathematics
- 2007

The purpose of this article is to describe Lagrangian mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups,…

### Variational calculus with constraints on general algebroids

- Mathematics
- 2008

Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler–Lagrange equations. Constrained systems are…

### JACOBI FIELDS FOR SECOND-ORDER DIFFERENTIAL EQUATIONS ON LIE ALGEBROIDS

- Mathematics
- 2015

We generalize the concept of Jacobi field for general second-order differential equations on a manifold and on a Lie algebroid. The Jacobi equation is expressed in terms of the dynamical covariant…

### Local convexity for second order differential equations on a Lie algebroid

- MathematicsJournal of Geometric Mechanics
- 2021

<p style='text-indent:20px;'>A theory of local convexity for a second order differential equation (${\text{sode}}$) on a Lie algebroid is developed. The particular case when the ${\text{sode}}$ is…

### On the exact discrete Lagrangian function for variational integrators: theory and applications

- Mathematics
- 2016

In this paper, we will give a rigorous construction of the exact discrete Lagrangian formulation associated to a continuous Lagrangian problem. Moreover, we work in the setting of Lie groupoids and…

### Lagrangian and Hamiltonian formalism in Field Theory: a simple model

- Mathematics
- 2010

The static of smooth maps from the two-dimensional disc to a smooth manifold can be regarded as a simplified version of the Classical Field Theory. In this paper we construct the Tulczyjew triple for…

### Higher algebroids via differential relations

- Mathematics
- 2017

We introduce the concept of a higher algebroid, naturally generalizing the notions of an algebroid and a higher tangent bundle. Our ideas are based on a description of (Lie) algebroids as Zakrzewski…

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A geometric description of Lagrangian Mechanics on Lie algebroids is developed in a parallel way to the usual formalism of Lagrangian Mechanics on the tangent bundle of a manifold. The dynamical…

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- Mathematics
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The variational formalism for classical field theories is extended to the setting of Lie algebroids. Given a Lagrangian function, we study the problem of finding critical points of the action…

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In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to…

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