Implicit Contact Dynamics and Hamilton-Jacobi Theory

Esen, Oğul; Valcázar, Manuel Lainz; León, Manuel de; Sardón, Cristina

doi:10.48550/arxiv.2109.14921

Cited by 2 publications

(5 citation statements)

References 33 publications

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“…Remark 6.5. The above contact Hamilton-Jacobi equations for extended cotangent bundle reduce to the ones obtained in [22,25]. Below we extend them to the case of first jet bundles J 1 (L) for arbitrary line bundles.…”

Section: Data Availability Statementmentioning

confidence: 73%

“…A lot of papers in this subject deal with contact Hamiltonian mechanics, e.g. [3,5,16,17], including contact Hamilton-Jacobi theory [9,12,22,25,51,67].…”

Section: Introductionmentioning

confidence: 99%

“…first jets J 1 (L) of non-trivial line bundles or projectivized cotangent bundles P(T * M) which are non-orientable for odd-dimensional M. On the other hand, even authors working exclusively with globally defined contact forms (especially with extended cotangent bundles T * Q × R) accept contact transformations as preserving the contact form up to a factor, that is a slight inconsequence. Also the proposed Lagrangian formalisms in this context are usually defined only for regular 'contact Lagrangians' or Lagrangians on extended tangent bundles TQ × R, except for some recent attempts [20,25,26,60] to extend it for singular Lagrangians in the spirit of Tulczyjew triples, invented by Tulczyjew [70][71][72] as an effective geometrical tool to deal with singular Lagrangians. However, the corresponding contact Euler-Lagrange equations have connections to Hamilton-Jacobi theory and variational approach of Herglotz [54], who used a generalization of the well-known Hamilton principle, that miraculously provides the same equations that we can obtain using contact geometry.…”

Section: Introductionmentioning

confidence: 99%

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A geometric approach to contact Hamiltonians and contact Hamilton–Jacobi theory

Grabowska¹,

Grabowski²

2022

J. Phys. A: Math. Theor.

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We propose a novel approach to contact Hamiltonian mechanics which, in contrast to the one dominating in the literature, serves also for non-trivial contact structures. In this approach Hamiltonians are no longer functions on the contact manifold M itself but sections of a line bundle over M or, equivalently, 1-homogeneous functions on a certain GL(1, ℝ)-principal bundle τ : P → M, which is equipped with a homogeneous symplectic form ω. In other words, our understanding of contact geometry is that it is not an ‘odd-dimensional cousin’ of symplectic geometry but rather a part of the latter, namely ‘homogeneous symplec-Jtic geometry’. This understanding of contact structures is much simpler than the traditionalone and very eﬀective in applications, reducing the contact Hamiltonian formalism to thestandard symplectic picture. We develop in this language contact Hamiltonian mechanics inautonomous, as well as time-dependent case, and the corresponding Hamilton-Jacobi theory.Fundamental examples are based on canonical contact structures on the ﬁrst jet bundles1(L) of sections of line bundles L, which play in contact geometry a fundamental rˆole, similar to that played by cotangent bundles in symplectic geometry.

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Section: Data Availability Statementmentioning

confidence: 73%

“…A lot of papers in this subject deal with contact Hamiltonian mechanics, e.g. [3,5,16,17], including contact Hamilton-Jacobi theory [9,12,22,25,51,67].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A geometric approach to contact Hamiltonians and contact Hamilton–Jacobi theory

Grabowska¹,

Grabowski²

2022

J. Phys. A: Math. Theor.

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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%

“…Finally, we can formulate a version of the Hamilton-Jacobi theorem for the evolutionary vector field [30].…”

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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Implicit Contact Dynamics and Hamilton-Jacobi Theory

Cited by 2 publications

References 33 publications

A geometric approach to contact Hamiltonians and contact Hamilton–Jacobi theory

A geometric approach to contact Hamiltonians and contact Hamilton–Jacobi theory

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

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