Reduction of a Hamilton — Jacobi Equation for Nonholonomic Systems

Esen, Oğul; Jiménez, Víctor Manuel; León, Manuel de; Sardón, Cristina

doi:10.1134/s156035471905006x

Cited by 4 publications

(7 citation statements)

References 30 publications

(75 reference statements)

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“…Nonintegrable (also known as nonholonomic) constraints are important for engineering applications. One of the application areas of the geometric HJ theory is nonholonomic mechanics [99] and it has been studied by various authors in different perspectives [10,38,75,109].…”

Section: An Incomplete Literature On Geometric Hamilton-jacobi Theorymentioning

confidence: 99%

Reviewing the geometric Hamilton–Jacobi theory concerning Jacobi and Leibniz identities

Esen¹,

León²,

Valcázar³

et al. 2022

J. Phys. A: Math. Theor.

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In this survey, we review the classical Hamilton–Jacobi theory from a geometric point of view in different geometric backgrounds. We propose a Hamilton-Jacobi equation for different geometric structures attending to one particular characterization: whether they fulfill the Jacobi and Leibniz identities simultaneously, or if at least they satisfy one of them. In this regard, we review the case of time-dependent ($t$-dependent in the sequel) and dissipative physical systems as systems that fulfill the Jacobi identity but not the Leibnitz identity. Furthermore, we review the contact-evolution Hamilton-Jacobi theory as a split off the regular contact geometry, and that actually satisfies the Leibniz rule instead of Jacobi. Furthermore, we include a novel result, which is the Hamilton-Jacobi equation for conformal Hamiltonian vector fields as a generalization of the well-known Hamilton-Jacobi on a symplectic manifold, that is retrieved in the case of a zero conformal factor. The interest of a geometric Hamilton–Jacobi equation is the primordial observation that if a Hamiltonian vector field $X_H$ can be projected into a configuration manifold by means of a $1$-form $dW$, then the integral curves of the projected vector field $X_{H}^{dW}$ can be transformed into integral curves of $X_H$ provided that $W$ is a solution of the Hamilton-Jacobi equation. Geometrically, the solution of the Hamilton-Jacobi equation plays the role of a Lagrangian submanifold of a certain bundle. Exploiting these features in different geometric scenarios we propose a geometric theory for multiple physical systems depending on the fundamental identities that their dynamic satisfies. Different examples are pictured to reflect the results provided, being all of them new, except for one that is reassessment of a previously considered example.

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Section: An Incomplete Literature On Geometric Hamilton-jacobi Theorymentioning

confidence: 99%

Reviewing the geometric Hamilton–Jacobi theory concerning Jacobi and Leibniz identities

Esen¹,

León²,

Valcázar³

et al. 2022

J. Phys. A: Math. Theor.

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“…In the literature, the geometric Hamilton-Jacobi theorem 2.1 has been extended in various different dynamical formulations. For example, geometric Hamilton-Jacobi theorem has been studied for higher order systems in [12], for field theories in [15,19,16], for implicit dynamics in [22,23], for non-holonomic dynamics in [10,28]. We refer to a recent review [21] for a more complete picture.…”

Section: Hamilton-jacobi Theorymentioning

confidence: 99%

On the role of geometry in statistical mechanics and thermodynamics I: Geometric perspective

Esen¹,

Grmela²,

Pavelka³

2022

Preprint

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This paper contains a fully geometric formulation of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). Although GENERIC, which is the sum of Hamiltonian mechanics and gradient dynamics, is a framework unifying a vast range of models in nonequilibrium thermodynamics, it has unclear geometric structure, due to the diverse geometric origins of Hamiltonian mechanics and gradient dynamics. The difference can be overcome by cotangent lifts of the dynamics, which leads, for instance, to a Hamiltonian form of gradient dynamics. Moreover, the lifted vector fields can be split into their holonomic and vertical representatives, which provides a geometric method of dynamic reduction. The lifted dynamics can be also given physical meaning, here called the rate-GENERIC. Finally, the lifts can be formulated within contact geometry, where the second law of thermodynamics is explicitly contained within the evolution equations.

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“…The HJ theory is rooted in the idea of finding an appropriate canonical transformation [4,9,34] that leads the system to equilibrium and pairs of action-angle variables that render the dynamics trivial. This philosophy has brought many interesting results, deriving into integrability theories, reduction, KAM theory, among others [20,24,61,62]. Our interest resides in the geometric interpretation of this theory [1,46,49], its formulation, and applications.…”

Section: Hamilton-jacobi Theorymentioning

confidence: 99%

“…Geometric HJ theories in the literature. This realization of the Hamilton-Jacobi theory has been devised in various situations, as it is the case of nonholonomic systems [13,24,32,36,52,53], geometric mechanics on Lie algebroids [5], almost-Poisson manifolds [38], singular systems [40], Nambu-Poisson framework [44], control theory [7], classical field theories [37,39,45], partial differential equations in general [68], the geometric discretization of the Hamilton-Jacobi equation [43,54], and others [6,12].…”

Section: Geometric Hamilton-jacobi Theorymentioning

confidence: 99%

“…Therefore, a HJ theory finds solutions on the lower dimensional manifold Q and retrieves them on the higher dimensional manifold T * Q by the existence of a section γ of the cotangent bundle which is the solution γ of the Hamilton-Jacobi equation (1). This picture (3) can be devised in different situations, as it is the case of nonholonomic systems [14,23,32,37,39,50,51], geometric mechanics on Lie algebroids [5] and almost-Poisson manifolds, singular systems [41], Nambu-Poisson framework [44], control theory [7], classical field theories [38,40,45], partial differential equations in general [64], the geometric discretization of the Hamilton-Jacobi equation [43,52], and others [6,13].…”

Section: Introductionmentioning

confidence: 99%

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A Hamilton–Jacobi formalism for higher order implicit Lagrangians

Esen¹,

León²,

Sardón³

2020

J. Phys. A: Math. Theor.

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In this paper, we present a generalization of a Hamilton-Jacobi theory to higher order implicit differential equations. We propose two different backgrounds to deal with higher order implicit Lagrangian theories: the Ostrogradsky approach and the Schmidt transform, which convert a higher order Lagrangian into a first order one. The Ostrogradsky approach involves the addition of new independent variables to account for higher order derivatives, whilst the Schmidt transform adds gauge invariant terms to the Lagrangian function. In these two settings, the implicit character of the resulting equations will be treated in two different ways in order to provide a Hamilton-Jacobi equation. On one hand, the implicit differential equation will be a Lagrangian submanifold of a higher order tangent bundle and it is generated by a Morse family. On the other hand, we will rely on the existence of an auxiliary section of a certain bundle that allows the construction of local vector fields, even if the differential equations are implicit. We will illustrate some examples of our proposed schemes, and discuss the applicability of the proposal.

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Reduction of a Hamilton — Jacobi Equation for Nonholonomic Systems

Cited by 4 publications

References 30 publications

Reviewing the geometric Hamilton–Jacobi theory concerning Jacobi and Leibniz identities

Reviewing the geometric Hamilton–Jacobi theory concerning Jacobi and Leibniz identities

On the role of geometry in statistical mechanics and thermodynamics I: Geometric perspective

A Hamilton–Jacobi formalism for higher order implicit Lagrangians

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