The Hamilton–Jacobi Theory for Contact Hamiltonian Systems

León, Manuel de; Valcázar, Manuel Lainz; Muñiz-Brea, Álvaro

doi:10.3390/math9161993

Cited by 16 publications

(15 citation statements)

References 24 publications

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“…HJ Theory has been discussed in the context of contact geometry from various perspectives [8,14,18,30,39]. First, we consider the trivial line bundle M × R → M over a manifold M .…”

Section: Geometric Hamilton-jacobi Theories In Contact Geometrymentioning

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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“…HJ Theory has been discussed in the context of contact geometry from various perspectives [8,14,18,30,39]. First, we consider the trivial line bundle M × R → M over a manifold M .…”

Section: Geometric Hamilton-jacobi Theories In Contact Geometrymentioning

confidence: 99%

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

Esen¹,

Grmela²,

Pavelka³

2022

Preprint

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“…We shall call the non-horizontal Legendrian submanifolds of T T * Q as implicit contact Hamiltonian dynamics, since the dynamics that they determine is a system of implicit differential equations. Our first goal in this paper to introduce the notion of implicit contact Hamiltonian dynamics (see Subsection 3.3) and write a proper Hamilton-Jacobi theory (the geometric Hamilton-Jacobi theory for explicit Hamiltonian contact dynamics (1.14) has been recently examined in [13,17,22]). In Subsection 3.1, we shall provide a particular instance (as Theorem 3.1) of the contact HJ theorem.…”

Section: Theorymentioning

confidence: 99%

“…We refer to [13,17,22] for a more general picture of the Hamilton-Jacobi theorem for contact Hamiltonian dynamics. See that in the general case the momentum components of the sections γ in (3.2) depend on the fiber coordinate z as well, i.e, γ = γ(q, z).…”

Section: Remarkmentioning

confidence: 99%

Implicit Contact Dynamics and Hamilton-Jacobi Theory

Esen¹,

Valcázar²,

León³

et al. 2021

Preprint

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In this paper we propose a Hamilton-Jacobi theory for implicit contact Hamiltonian systems in two different ways. One is the understanding of implicit contact Hamiltonian dynamics as a Legendrian submanifold of the tangent contact space, and another is as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. In these two scenarios we propose a Hamilton-Jacobi theory specifically derived with the aid of Herglotz Lagrangian dynamics generated by non-regular Lagrangian functions.

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“…Generally, solutions of (1), the so-called viscosity solutions, are only Lipschitz continuous so that one have to seek for a variational principle to construct a semigroup expression of viscosity solutions. The Hamilton-Jacobi theory for contact Hamiltonian systems has been discussed in his recent paper [15]. In [31], K. Wang, L. Wang and J. Yan have established a novelty implicit variational principle for the contact Hamiltonian systems generated by the Hamiltonian H(x, u, p) with respect to the contact 1-form α = du − pdx under Tonelli and Lipschitz continuity conditions.…”

Section: Introduction Contact Hamiltonian System Appears Naturally In...mentioning

confidence: 99%

Orbital dynamics on invariant sets of contact Hamiltonian systems

Liu¹,

Torres

2022

DCDS-B

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<p style='text-indent:20px;'>In this paper, we shall give new insights on dynamics of contact Hamiltonian flows, which are gaining importance in several branches of physics as they model a dissipative behaviour. We divide the contact phase space into three parts, which are corresponding to three differential invariant sets <inline-formula><tex-math id="M1">\begin{document}$ \Omega_\pm, \Omega_0 $\end{document}</tex-math></inline-formula>. On the invariant sets <inline-formula><tex-math id="M2">\begin{document}$ \Omega_\pm $\end{document}</tex-math></inline-formula>, under some geometric conditions, the contact Hamiltonian system is equivalent to a Hamiltonian system via the Hölder transformation. The invariant set <inline-formula><tex-math id="M3">\begin{document}$ \Omega_0 $\end{document}</tex-math></inline-formula> may be composed of several equilibrium points and heteroclinic orbits connecting them, on which contact Hamiltonian system is conservative. Moreover, we have shown that, under general conditions, the zero energy level domain is a domain of attraction. In some cases, such a domain of attraction does not have nontrivial periodic orbits. Some interesting examples are presented.</p>

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The Hamilton–Jacobi Theory for Contact Hamiltonian Systems

Cited by 16 publications

References 24 publications

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

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

Implicit Contact Dynamics and Hamilton-Jacobi Theory

Orbital dynamics on invariant sets of contact Hamiltonian systems

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