Contact Hamiltonian and Lagrangian systems with nonholonomic constraints

León, Manuel de; Jiménez, Víctor Manuel; Valcázar, Manuel Lainz

doi:10.3934/jgm.2021001

Cited by 13 publications

(22 citation statements)

References 34 publications

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“…One can easily verify that, if the partial Hessian matrix of L with respect to the velocities is positive or negative definite, then there exists a unique solution to equation (11) (see [24] and [18] for the presymplectic and the contact cases, respectively).…”

Section: Nonholonomic Lagrangian Mechanicsmentioning

confidence: 99%

Constrained Lagrangian dissipative contact dynamics

de León,

Laínz,

Muñoz-Lecanda

et al. 2021

Preprint

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We show that the contact dynamics obtained from the Herglotz variational principle can be described as a constrained nonholonomic or vakonomic ordinary Lagrangian system depending on a dissipative variable with an adequate choice of one constraint. As a consequence we obtain the dynamics of contact nonholonomic and vakonomic systems as ordinary variational calculus with constraints on a Lagrangian with a dissipative variable. The variation of the energy and the other dissipative quantities are also obtained giving the usual results.

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Section: Nonholonomic Lagrangian Mechanicsmentioning

confidence: 99%

Constrained Lagrangian dissipative contact dynamics

de León,

Laínz,

Muñoz-Lecanda

et al. 2021

Preprint

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“…In recent years, there has been an increasing interest in the study of contact mechanical systems (see [6,8,13,22] and references therein). Contact geometry has been used in the last years to describe dissipative mechanical systems, as well as systems in thermodynamics [10,15,21,24], quantum mechanics [1], control theory [7], dissipative field theories [12,14,22], etc.…”

Section: Introductionmentioning

confidence: 99%

“…The main difference between both variational principles is that in the Herglotz variational principle the action is defined by a nonautonomous ODE instead of an integral. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints were introduced by de León, Jiménez and Lainz in [6].…”

Section: Introductionmentioning

confidence: 99%

Contact Lagrangian systems subject to impulsive constraints

Colombo¹,

León²,

Asier³

2022

Preprint

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We describe geometrically contact Lagrangian systems under impulsive forces and constraints, as well as instantaneous nonholonomic constraints which are not uniform along the configuration space. In both situations, the vector field describing the dynamics of a contact Lagrangian system is determined by defining projectors to evaluate the constraints by using a Riemannian metric. In particular, we introduce the Herglotz equations for contact Lagrangian systems subject to instantaneous nonholonomic constraints. Moreover, we provide a Carnot-type theorem for contact Lagrangian systems subject to impulsive forces and constraints, which characterizes the changes of energy due to contact-type dissipation and impulsive forces. We illustrate the applicability of the method with practical examples, in particular, a rolling cylinder on a springily surface and a rolling sphere on a non-uniform surface, both with dissipation.

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“…This is due to the fact that one can use contact structures to describe many different types of dynamical systems which can not be described by means of symplectic geometry and standard Hamiltonian dynamics in a natural way. The dynamical systems which can be modelled using contact structures include mechanical systems with certain types of damping [25,37,46], some systems in quantum mechanics [11], circuit theory [28], control theory [40] and thermodynamics [6,43], among many others [7,16,18,19,21,23,35,45].…”

Section: Introductionmentioning

confidence: 99%

Lagrangian-Hamiltonian formalism for time-dependent dissipative mechanical systems

Rivas¹,

Torres²

2022

Preprint

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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.

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Contact Hamiltonian and Lagrangian systems with nonholonomic constraints

Cited by 13 publications

References 34 publications

Constrained Lagrangian dissipative contact dynamics

Constrained Lagrangian dissipative contact dynamics

Contact Lagrangian systems subject to impulsive constraints

Lagrangian-Hamiltonian formalism for time-dependent dissipative mechanical systems

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