7. Hyperbolicity and Anosov systems. Vakonomic mechanics

2002 ◽  
pp. 127-143
Author(s):  
Waldyr Muniz Oliva
Keyword(s):  
Vakonomic Mechanics ◽  
Anosov Systems

Centralizers of derived-from-Anosov systems on : rigidity versus triviality

2021 ◽  
pp. 1-25
Author(s):  
SHAOBO GAN ◽  
YI SHI ◽  
DISHENG XU ◽  
JINHUA ZHANG
Keyword(s):  
Linear Part ◽  
Anosov Systems ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphism

Abstract In this paper, we study the centralizer of a partially hyperbolic diffeomorphism on ${\mathbb T}^3$ which is homotopic to an Anosov automorphism, and we show that either its centralizer is virtually trivial or such diffeomorphism is smoothly conjugate to its linear part.

Subsystems of Anosov Systems

1995 ◽  
Vol 117 (6) ◽  
pp. 1431 ◽  
Author(s):  
A. Zeghib
Keyword(s):  
Anosov Systems

Sur une notion d'autonomie de systèmes dynamiques, appliquée aux ensembles invariants des flots d' Anosov algébriques

1995 ◽  
Vol 15 (1) ◽  
pp. 175-207 ◽  
Author(s):  
A. Zeghib
Keyword(s):  
Dynamical Systems ◽  
Finite Volume ◽  
Algebraic Dynamical Systems ◽  
Anosov Systems ◽  
Anosov System ◽  
Autonomous Dynamical Systems

AbstractWe introduce a notion of autonomous dynamical systems which generalizes algebraic dynamical systems. We show by giving examples and by describing some properties that this generalization is not a trivial one. We apply the methods then developed to algebraic Anosov systems. We prove that a C1-submanifold of finite volume, which is invariant by an algebraic Anosov system is ‘essentially’ algebraic.

Discrete vakonomic mechanics

2005 ◽  
Vol 46 (8) ◽  
pp. 083521 ◽  
Author(s):  
Roberto Benito ◽  
David Martín de Diego
Keyword(s):  
Vakonomic Mechanics

scholarly journals Lagrangian and Hamiltonian formalism for constrained variational problems

2002 ◽  
Vol 132 (6) ◽  
pp. 1417-1437 ◽  
Author(s):  
Paolo Piccione ◽  
Daniel V. Tausk
Keyword(s):  
Critical Points ◽  
Vector Bundles ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Variational Problems ◽  
Linear Constraints ◽  
Legendre Transform ◽  
General Notion ◽  
Horizontal Curves ◽  
Vakonomic Mechanics

We consider solutions of Lagrangian variational problems with linear constraints on the derivative. More precisely, given a smooth distribution D ⊂ TM on M and a time-dependent Lagrangian L defined on D, we consider an action functional L defined on the set ΩPQ(M, D) of horizontal curves in M connecting two fixed submanifolds P, Q ⊂ M. Under suitable assumptions, the set ΩPQ(M, D) has the structure of a smooth Banach manifold and we can thus study the critical points of L. If the Lagrangian L satisfies an appropriate hyper-regularity condition, we associate to it a degenerate Hamiltonian H on TM* using a general notion of Legendre transform for maps on vector bundles. We prove that the solutions of the Hamilton equations of H are precisely the critical points of L. In the particular case where L is given by the quadratic form corresponding to a positive-definite metric on D, we obtain the well-known characterization of the normal geodesics in sub-Riemannian geometry (see [8]). By adding a potential energy term to L, we obtain again the equations of motion for the Vakonomic mechanics with non-holonomic constraints (see [6]).

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