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2 results

18 - Integrability of the Sub-Riemannian Geodesic Flowon 3D Lie Groups
Andrei Agrachev, Scuola Internazionale Superiore di Studi Avanzati, Trieste, Davide Barilari, Université de Paris VII (Denis Diderot), Ugo Boscain, Centre National de la Recherche Scientifique (CNRS), Paris
Book:

A Comprehensive Introduction to Sub-Riemannian Geometry

Published online:

28 October 2019

Print publication:

31 October 2019, pp 607-632
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Summary

In this chapter we show how to find certain firstintegrals, for Hamiltonian systems on Lie groups,that are automatically in involution with each otherand with the Hamiltonian. This theory will be usedto prove that the Hamiltonian system for normalPontryagin extremals for rank-2 left-invariantsub-Riemannian structures on three-dimensional Liegroups is completely integrable.

The $C^{0}$ integrability of symplectic twist maps without conjugate points
Part of
MARC ARCOSTANZO
Journal:

Ergodic Theory and Dynamical Systems / Volume 41 / Issue 1 / January 2021

Published online by Cambridge University Press:

15 August 2019, pp. 48-65

Print publication:

January 2021
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It is proved that a symplectic twist map of the cotangent bundle $T^{\ast }\mathbb{T}^{d}$ of the $d$-dimensional torus that is without conjugate points is $C^{0}$-integrable, that is $T^{\ast }\mathbb{T}^{d}$ is foliated by a family of invariant $C^{0}$ Lagrangian graphs.