Basic Principles of Classical Mechanics

Arnold, Vladimir I.; Kozlov, Valery Vasil'evich; Neishtadt, Anatoly

doi:10.1007/978-3-540-48926-9_1

Cited by 24 publications

(61 citation statements)

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“…Integration by parts in (2.8) and the Fundamental Lemma of Calculus of Variations imply then that equation (2.7) is satisfied. Observe also that the coordinate representation of the map t → FL (2) t,γ(t)…”

Section: Time Dependent Lagrangians and Hamiltonians On Manifoldsmentioning

confidence: 99%

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Lagrangian and Hamiltonian formalism for constrained variational problems

Piccione

Tausk

2002

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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ABSTRACT. We consider solutions of Lagrangian variational problems with linear constraints on the derivative. More precisely, given a smooth distribution D ⊂ T M on M and a time-dependent Lagrangian L defined on D, we consider an action functional L defined on the set Ω P Q (M, D) of horizontal curves in M connecting two fixed submanifolds P, Q ⊂ M . Under suitable assumptions, the set Ω P Q (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 T M * 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 [10]); by adding a potential energy term to L, we reobtain the equations of motion for the Vakonomic mechanics with non holonomic constraints (see [8]).

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Section: Time Dependent Lagrangians and Hamiltonians On Manifoldsmentioning

confidence: 99%

“…Particular cases of this theory are the sub-Riemannian geodesic problem (see for instance [10,11,13,17]), and the so called Vakonomic approach to the non holonomic mechanics (see for instance [2,5,8,19]). …”

Section: Introductionmentioning

confidence: 99%

Lagrangian and Hamiltonian formalism for constrained variational problems

Piccione

Tausk

2002

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…The transformation of the map to a normal form is inspired by singularity theory of Lagrangian mappings [AGZV85, AKN88,Arn95]. The class of transformations is, however, more restricted.…”

Section: Introductionmentioning

confidence: 99%

Normal forms for 4D symplectic maps with twist singularities

Dullin

Ivanov

Meiss

2006

Physica D: Nonlinear Phenomena

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We derive a normal form for a near-integrable, four-dimensional symplectic map with a fold or cusp singularity in its frequency mapping. The normal form is obtained for when the frequency is near a resonance and the mapping is approximately given by the time-T mapping of a two-degree-of freedom Hamiltonian flow. Consequently there is an energy-like invariant. The fold Hamiltonian is similar to the well-studied, one-degree-of freedom case but is essentially nonintegrable when the direction of the singular curve in action does not coincide with curves of the resonance module. We show that many familiar features, such as multiple island chains and reconnecting invariant manifolds, are retained even in this case. The cusp Hamiltonian has an essential coupling between its two degrees of freedom even when the singular set is aligned with the resonance module. Using averaging, we approximately reduced this case to one degree of freedom as well. The resulting Hamiltonian and its perturbation with small cusp-angle is analyzed in detail.

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“…the subspace of virtual velocities (following the nomenclature of [3]) at v q ; C vq is the subspace of T q M which is the image of the tangent map at v q of the inclusion C q → T q M.…”

Section: Definition 2 (Marle)mentioning

confidence: 99%

“…This is Marle's [28] definition of a "regular constraint". Other formulations of systems with non-linear constraints may be found in [38], [39], [37], [24], [25], [26], [3], [40], [31], [8], [28], [29], [30], [4], [9], [18], among others.…”

Section: Introductionmentioning

confidence: 99%