Vakonomic Mechanics on Lie Affgebroids

Inverse Problem in Vakonomic Mechanics

Progress in Mathematics - Inverse Problems in Ordinary Differential Equations and Applications ◽

10.1007/978-3-319-26339-7_7 ◽

2016 ◽

pp. 201-251

Author(s):

Jaume Llibre ◽

Rafael Ramírez

Keyword(s):

Inverse Problem ◽

Vakonomic Mechanics

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Discrete vakonomic mechanics

Journal of Mathematical Physics ◽

10.1063/1.2008214 ◽

2005 ◽

Vol 46 (8) ◽

pp. 083521 ◽

Cited By ~ 17

Author(s):

Roberto Benito ◽

David Martín de Diego

Keyword(s):

Vakonomic Mechanics

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Lagrangian and Hamiltonian formalism for constrained variational problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500002183 ◽

2002 ◽

Vol 132 (6) ◽

pp. 1417-1437 ◽

Cited By ~ 7

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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