Mathematical Methods of Classical Mechanics

Arnold, Vladimir I.

doi:10.1007/978-1-4757-1693-1

Cited by 4,032 publications

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“…This curly bracket notation is due to Arnol'd, [1,Appendix 2]. This representation is defined by the relation,…”

Section: Conjugate Variablesmentioning

confidence: 99%

A Class of Explicitly Solvable Vehicle Motion Problems

Selig

2015

IEEE Trans. Robot.

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Abstract-A small but interesting result of Brockett is extended to the Euclidean group SE(3) and is illustrated by several examples. The result concerns the explicit solution of an optimal control problem on Lie groups, where the control belongs to a Lie triple system in the Lie algebra. The extension allows for an objective function based on an indefinite quadratic form.Applying the result requires explicit knowledge of the Lie triple systems of the Lie algebra se(3). Hence, a complete classification of the Lie triple systems of this Lie algebra is derived.Examples are considered for optimal trajectories in 3 cases. The first case concerns cars moving in the plane. The second looks at motions that rigidly follow the Bishop frame to a space curve. The final example does not have a particular name as it does not seem to have been studied before.The appendix gives a brief introduction to Screw theory. This is essentially the study of the Lie algebra se(3).

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“…This curly bracket notation is due to Arnol'd, [1,Appendix 2]. This representation is defined by the relation,…”

Section: Conjugate Variablesmentioning

confidence: 99%

A Class of Explicitly Solvable Vehicle Motion Problems

Selig

2015

IEEE Trans. Robot.

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“…To study the flow (3.1) on the manifold M (N ) , we shall make use of the Bogoyavlensky-Novikov [4,16] reduction scheme [4,6,8,12]. N ) ) be the standard finitely generated Grassmann algebra [2,6,13] of differential forms on the manifold M (N ) . Then the differential complex…”

Section: The Bogoyavlensky-novikov Finite-dimensional Reductionmentioning

confidence: 99%

“…satisfies the Nötherian conditions (1.6). We are studying the integrability [2,4,8,13] of the discrete dynamical system (1.1). Accordingly we need to construct invariants with respect to its functions, called conservation laws, which are mutually commuting with respect to the Poisson bracket (1.4).…”

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confidence: 99%

Isospectral integrability analysis of dynamical systems on discrete manifolds

Blackmore¹,

Prykarpatsky²,

Prykarpatsky³

2012

OpMath

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“…Thermodynamics In this section we shall briefly recall the basic concepts of contact geometry (following [8] and [2]) in the context of Reversible Thermodynamics (see [14] and the references herein). First we recall the canonical state space, called Thermodynamic Phase Space, in which the thermodynamic properties of a system are defined.…”

Section: Contact Structures For Reversiblementioning

confidence: 99%

“…The sketch of the paper is the following. In section 2 we briefly recall the basic concepts of contact geometry [8] and [2] in the context of Reversible Thermodynamics [7], [10], [14]. In the section 3, we define control contact systems on contact manifolds, which may be seen as the analogue of input-output Hamiltonian systems defined on symplectic manifolds [15], [16].…”

Section: Introductionmentioning

confidence: 99%

Port contact systems for irreversible thermodynamical systems

Ebérard

Maschke

Schaft

Proceedings of the 44th IEEE Conference on Decision and Control

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Abstract-In this paper we propose a definition of control contact systems, generalizing input-output Hamiltonian systems, to cope with models arising from irreversible Thermodynamics. We exhibit a particular subclass of these systems, called conservative, that leaves invariant some Legendre submanifold (the geometric structures associated with thermodynamic properties). These systems, both energy-preserving and irreversible, are then used to analyze the losslessness of these systems with respect to different generating functions.

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Mathematical Methods of Classical Mechanics

Cited by 4,032 publications

References 0 publications

A Class of Explicitly Solvable Vehicle Motion Problems

A Class of Explicitly Solvable Vehicle Motion Problems

Isospectral integrability analysis of dynamical systems on discrete manifolds

Port contact systems for irreversible thermodynamical systems

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