The Herglotz Principle and Vakonomic Dynamics

León, Manuel de; Valcázar, Manuel Lainz; Muñoz-Lecanda, Miguel C.

doi:10.1007/978-3-030-80209-7_21

Cited by 7 publications

(8 citation statements)

References 16 publications

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“…In this case, we must take into account all the constraints (defining a submanifold C of N, where N is considered now as a submanifold in T(Q × R) in the obvious way) and we obtain different equations which we can call the dissipative nonholonomic and vakonomic equations, respectively. We also note that, if the constraints defining N do so from a submanifold of TQ × R, in the nonholonomic case we obtain the same equations as when there is no dependence on s, while in the vakonomic case we recover the equations obtained in [21].…”

Section: Introductionsupporting

confidence: 57%

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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: Introductionsupporting

confidence: 57%

“…Observe that we need to consider the constraints ψ β as additional equations in order to determine the Lagrange multipliers µ α and, consequently, to obtain the coefficients B i form equation (21). Then the dynamical vector field is completely determined except for the Lagrange multipliers.…”

Section: For Dsmentioning

confidence: 99%

Constrained Lagrangian dissipative contact dynamics

de León,

Laínz,

Muñoz-Lecanda

et al. 2021

Preprint

Self Cite

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“…We have that A = ṫ, B i = qi , C i = vi , D i = ṗi and E = ṡ. Then, equations ( 19), ( 20), ( 21), ( 22) and ( 23) lead to the local expression of (18). In particular,…”

Section: The Canonical 1-form Is the ρmentioning

confidence: 99%

“…• Conversely, let the path σ 0 : I ⊂ R → R × Q × R be a solution of (6) on S f . Then, the section σ = (σ L , σ H ) which is a solution of (18), where σ L := σ 0 and σ H := FL • σ 0 . Also path σ H is a solution to (1) on P f .…”

Section: Recovering the Lagrangian And Hamiltonian Formalismsmentioning

confidence: 99%

“…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%

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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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The Herglotz variational principle for dissipative field theories

Gaset,

Lainz,

Mas

et al. 2024

Geom. Mech.

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In the recent years, with the incorporation of contact geometry, there has been a renewed interest in the study of dissipative or non-conservative systems in physics and other areas of applied mathematics. The equations arising when studying contact Hamiltonian systems can also be obtained via the Herglotz variational principle. The contact Lagrangian and Hamiltonian formalisms for mechanical systems has also been generalized to field theories. The main goal of this paper is to develop a generalization of the Herglotz variational principle for first-order and higher-order field theories. In order to illustrate this, we study three examples: the damped vibrating string, the Korteweg–de Vries equation, and an academic example showing that the non-holonomic and the vakonomic variational principles are not fully equivalent.

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The Herglotz Principle and Vakonomic Dynamics

Cited by 7 publications

References 16 publications

Constrained Lagrangian dissipative contact dynamics

Constrained Lagrangian dissipative contact dynamics

Lagrangian-Hamiltonian formalism for time-dependent dissipative mechanical systems

The Herglotz variational principle for dissipative field theories

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