Contact Dynamics: Legendrian and Lagrangian Submanifolds

Esen, Oğul; Valcázar, Manuel Lainz; León, Manuel de; Marrero, Juan Carlos

doi:10.3390/math9212704

Cited by 12 publications

(11 citation statements)

References 64 publications

(111 reference statements)

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“…This is achieved in the realm of locally conformal cosymplectic manifolds. The symplectization and different geometric frameworks, as it is the case of time-dependent dynamics in [20], kcosymplectic framework in [66] and recently, on contact manifolds [36]. We wish to construct Tulczyjew triples for LCS and LCC manifolds.…”

Section: Discussionmentioning

confidence: 99%

On Locally Conformally Cosymplectic Hamiltonian Dynamics and Hamilton-Jacobi Theory

Ateşli¹,

Esen²,

León³

et al. 2022

Preprint

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Cosymplectic geometry has been proven to be a very useful geometric background to describe time-dependent Hamiltonian dynamics. In this work, we address the globalization problem of locally cosymplectic Hamiltonian dynamics that failed to be globally defined. We investigate both the geometry of locally conformally cosymplectic (abbreviated as LCC) manifolds and the Hamiltonian dynamics constructed on such LCC manifolds. Further, we provide a geometric Hamilton-Jacobi theory on this geometric framework.[12]. In the light of these observations, we shall establish Proposition 3.5 determining a Darboux theorem for LCC manifolds.(3) Jacobi Structure of LCC Manifolds. LCS manifolds are Jacobi manifolds, we shall explicitly show the Jacobi character of LCC manifolds in Proposition 4.1.(4) Algebra of One-Forms: Lie Algebroid Realizations. It is known that Jacobi manifolds admit Lie algebroid formalisms [22]. We shall review this for LCS manifolds in Section 2.3 and show that Lichnerowicz-deRham (abbreviated as LdR) exact one-forms constitute a subalgebra.In Section 4.3 we shall write an algebra on the space of one-form sections on LCC. Then we shall discuss the Lie algebroid realization of LCC manifolds.(5) Hamilton-Jacobi Formalism for LCC Hamiltonian Dynamics. Hamilton-Jacobi theory provides a way to solve Hamilton equations. In a recent paper [35], we examined the geometric Hamilton-Jacobi theory for Hamiltonian dynamics on LCS manifolds. Additionally, we took the locally conformal discussions to k-symplectic formalism [34] and jet bundle formalism [33] in order to define locally conformally Hamiltonian field theories. In that work, we provided the locally conformal Hamilton-De Donder-Weyl formalism, as well as the geometric Hamilton-Jacobi theories for these extensions. In this work, we shall present two versions (namely in Theorem 4.4 and Theorem 4.5) of the geometric Hamilton-Jacobi theorem in the realm of LCC Hamiltonian dynamics. These results are a generalization of the HJ theorems obtained for cosymplectic Hamiltonian dynamics in [28].The content. In the following section we summarize Hamiltonian dynamics on symplectic and LCS manifolds and their corresponding geometric HJ theorems. In Section 3, we shall recall cosymplectic manifolds and the Hamilton-Jacobi theorem for cosymplectic Hamiltonian dynamics.Then we shall explain the basics on LCC geometry, and it is in this same section where the symplectization problem of LCC manifolds and Darboux coordinates of LCC are established. In Section 4, dynamics on LCC manifolds is explained, Jacobi and Lie algebroid characters of LCC manifolds will be obtained, and finally the Hamilton-Jacobi theorems for LCC Hamiltonian dynamics will be written.Notation. From now on we consider X(M) to be the space of vector fields on M, whereas Γ k (M) is the space of k-form sections on M. L X is the Lie derivative with respect to the vector field X. An arbitrary almost symplectic manifold is represented by the pair (N, ω) while an arbitrary almost cosymplectic manifold is denoted by ...

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Section: Discussionmentioning

confidence: 99%

On Locally Conformally Cosymplectic Hamiltonian Dynamics and Hamilton-Jacobi Theory

Ateşli¹,

Esen²,

León³

et al. 2022

Preprint

Self Cite

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“…Unlike Hamiltonian mechanics, where the Hamiltonian vector field is the only geometrically distinguished vector field on the cotangent bundle, contact geometry admits two alternatives. In the preceding Section, we have recalled the contact Hamiltonian vector field X H , while in the current Section we discuss the evolution Hamiltonian vector field ε H [29,76,77], determined through the following equalities,…”

Section: Hamiltonian Dynamics On Contact Manifoldmentioning

confidence: 99%

“…Consider a (2n + 1)−dimensional contact manifold equipped with a contact one-form η. A maximally integrable submanifold of the contact manifold where the contact form vanishes is called Legendrian submanifold, see, for example, [2,5,29,42]. It is possible to see that a Legendrian submanifold is necessarily of dimension n. In Darboux' coordinates (x, x * , z), consider a partition A ∪ B of the set of indices (1, .…”

Section: The Legendrian Submanifolds and Rate-generic Flowmentioning

confidence: 99%

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

Esen¹,

Grmela²,

Pavelka³

2022

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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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“…In virtue of conditions ( 21), ( 22), (23), the vector fields Z solution to equations ( 17) have the local expression…”

Section: The Canonical 1-form Is the ρmentioning

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%

Lagrangian-Hamiltonian formalism for time-dependent dissipative mechanical systems

Rivas¹,

Torres²

2022

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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 Dynamics: Legendrian and Lagrangian Submanifolds

Cited by 12 publications

References 64 publications

On Locally Conformally Cosymplectic Hamiltonian Dynamics and Hamilton-Jacobi Theory

On Locally Conformally Cosymplectic Hamiltonian Dynamics and Hamilton-Jacobi Theory

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

Lagrangian-Hamiltonian formalism for time-dependent dissipative mechanical systems

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