Symmetries, Conservation and Dissipation in Time‐Dependent Contact Systems

Gaset, Jordi; Asier, López-Gordón,; Rivas, Xavier

doi:10.1002/prop.202300048

Cited by 5 publications

(2 citation statements)

References 56 publications

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“…Starting from the results of the previous section, one can prove a contact version of Noether's theorem [13,[27][28][29], which we now recall (see also [15,16] for the Lagrangian counterpart and further discussion).…”

Section: A Noether Identity From Contact Geometrymentioning

confidence: 90%

Thermodynamic Entropy as a Noether Invariant from Contact Geometry

Bravetti

García-Ariza

Tapias

2023

Entropy

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We use a formulation of Noether’s theorem for contact Hamiltonian systems to derive a relation between the thermodynamic entropy and the Noether invariant associated with time-translational symmetry. In the particular case of thermostatted systems at equilibrium, we show that the total entropy of the system plus the reservoir are conserved as a consequence thereof. Our results contribute to understanding thermodynamic entropy from a geometric point of view.

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Section: A Noether Identity From Contact Geometrymentioning

confidence: 90%

Thermodynamic Entropy as a Noether Invariant from Contact Geometry

Bravetti

García-Ariza

Tapias

2023

Entropy

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“…Moreover, contact geometry has drawn, by itself, much attention in recent times [47][48][49]. Recently, the notion of cocontact manifold has also been developed to introduce explicit dependence on time [25,43,70].…”

Section: Introductionmentioning

confidence: 99%

Contact Lie systems: theory and applications

Lucas

Rivas

2023

J. Phys. A: Math. Theor.

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A Lie system is a time-dependent system of differential equations describing the integral curves of a time-dependent vector field that can be considered as a curve in a finite-dimensional Lie algebra of vector fields $V$. We call $V$ a Vessiot--Guldberg Lie algebra.We define and analyse contact Lie systems, namely Lie systems admitting a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields relative to a contact manifold. We also study contact Lie systems of Liouville type, which are invariant relative to the flow of a Reeb vector field. Liouville theorems, contact Marsden--Weinstein reductions, and Gromov non-squeezing theorems are developed and applied to contact Lie systems. Contact Lie systems on three-dimensional Lie groups with Vessiot--Guldberg Lie algebras of right-invariant vector fields and associated with left-invariant contact forms are classified. Our results are illustrated by examples with relevant physical and mathematical applications, e.g. Schwarz equations, Brockett systems, quantum mechanical systems, etc. Finally, a Poisson coalgebra method to derive superposition rules for contact Lie systems of Liouville is developed.

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Bregman dynamics, contact transformations and convex optimization

et al. 2023

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Recent research on accelerated gradient methods of use in optimization has demonstrated that these methods can be derived as discretizations of dynamical systems. This, in turn, has provided a basis for more systematic investigations, especially into the geometric structure of those dynamical systems and their structure-preserving discretizations. In this work, we introduce dynamical systems defined through a contact geometry which are not only naturally suited to the optimization goal but also subsume all previous methods based on geometric dynamical systems. As a consequence, all the deterministic flows used in optimization share an extremely interesting geometric property: they are invariant under contact transformations. In our main result, we exploit this observation to show that the celebrated Bregman Hamiltonian system can always be transformed into an equivalent but separable Hamiltonian by means of a contact transformation. This in turn enables the development of fast and robust discretizations through geometric contact splitting integrators. As an illustration, we propose the Relativistic Bregman algorithm, and show in some paradigmatic examples that it compares favorably with respect to standard optimization algorithms such as classical momentum and Nesterov’s accelerated gradient.

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Symmetries, Conservation and Dissipation in Time‐Dependent Contact Systems

Cited by 5 publications

References 56 publications

Thermodynamic Entropy as a Noether Invariant from Contact Geometry

Thermodynamic Entropy as a Noether Invariant from Contact Geometry

Contact Lie systems: theory and applications

Bregman dynamics, contact transformations and convex optimization

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