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Variational calculus on Lie algebroids

Eduardo Martínez (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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.

Variational formulations I: Statics of mechanical systems

Włodzimierz M. Tulczyjew (2011)

Communications in Mathematics

Two improvements of variational formulations of mechanics are proposed. The first consists in a modification of the definition of equilibrium. The second consists in adding elements of control by external devices. In the present note the proposed improvements are applied to variational principles of statics. Numerous examples are given.

Variational principles and symmetries on fibered multisymplectic manifolds

Jordi Gaset, Pedro D. Prieto-Martínez, Narciso Román-Roy (2016)

Communications in Mathematics

The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multisymplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is equivalent, conservation laws), symmetries, Cartan (Noether) symmetries, gauge symmetries and different versions of Noether's theorem are studied in this ambient. In this way, this constitutes a general geometric framework for all these topics that includes, as special...

Viscosity solutions methods for converse KAM theory

Diogo A. Gomes, Adam Oberman (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

The main objective of this paper is to prove new necessary conditions to the existence of KAM tori. To do so, we develop a set of explicit a-priori estimates for smooth solutions of Hamilton-Jacobi equations, using a combination of methods from viscosity solutions, KAM and Aubry-Mather theories. These estimates are valid in any space dimension, and can be checked numerically to detect gaps between KAM tori and Aubry-Mather sets. We apply these results to detect non-integrable regions in several...

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