Lagrangian–Hamiltonian formalism for cocontact systems

Xavier Rivas; Daniel Torres; Xavier Rivas; Daniel Torres

doi:10.3934/jgm.2023001

Journal of Geometric Mechanics

2023, Volume 15, Issue 1: 1-26. doi: 10.3934/jgm.2023001

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Research article

Lagrangian–Hamiltonian formalism for cocontact systems

Xavier Rivas ^{1
,
,},
Daniel Torres ²

1.
Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, Logroño, Spain
2.
Universitat Politècnica de Catalunya, Barcelona, Spain

Received: 29 May 2022 Revised: 25 July 2022 Accepted: 29 July 2022 Published: 19 October 2022
Primary: 37J55; Secondary: 70H03, 70H05, 53D10, 70G45, 53Z05, 70H45

In this paper we present a unified Lagrangian–Hamiltonian geometric formalism to describe time-dependent contact mechanical systems, based on the one first introduced by K. Kamimura and later formalized by R. Skinner and R. Rusk. This formalism is especially interesting when dealing with systems described by singular Lagrangians, since the second-order condition is recovered from the constraint algorithm. In order to illustrate this formulation, some relevant examples are described in full detail: the Duffing equation, an ascending particle with time-dependent mass and quadratic drag, and a charged particle in a stationary electric field with a time-dependent constraint.
- contact structure,
- Lagrangian and Hamiltonian formalisms,
- timedependent system,
- dissipation
Citation: Xavier Rivas, Daniel Torres. Lagrangian–Hamiltonian formalism for cocontact systems[J]. Journal of Geometric Mechanics, 2023, 15(1): 1-26. doi: 10.3934/jgm.2023001

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Abstract

In this paper we present a unified Lagrangian–Hamiltonian geometric formalism to describe time-dependent contact mechanical systems, based on the one first introduced by K. Kamimura and later formalized by R. Skinner and R. Rusk. This formalism is especially interesting when dealing with systems described by singular Lagrangians, since the second-order condition is recovered from the constraint algorithm. In order to illustrate this formulation, some relevant examples are described in full detail: the Duffing equation, an ascending particle with time-dependent mass and quadratic drag, and a charged particle in a stationary electric field with a time-dependent constraint.

References

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