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Search Results (2)

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Keywords = Legendrian and Lagrangian manifolds

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41 pages, 526 KiB  
Article
Contact Dynamics: Legendrian and Lagrangian Submanifolds
by Oğul Esen, Manuel Lainz Valcázar, Manuel de León and Juan Carlos Marrero
Mathematics 2021, 9(21), 2704; https://doi.org/10.3390/math9212704 - 25 Oct 2021
Cited by 13 | Viewed by 3121
Abstract
We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a [...] Read more.
We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew’s triple is constructed for evolution contact dynamics. Full article
(This article belongs to the Special Issue New Trends in Hamilton-Jacobi Theory: Conservative and Dissipative Dynamics)
8 pages, 878 KiB  
Article
On Higher Order Structures in Thermodynamics
by Valentin Lychagin and Mikhail Roop
Entropy 2020, 22(10), 1147; https://doi.org/10.3390/e22101147 - 12 Oct 2020
Cited by 2 | Viewed by 2123
Abstract
We present the development of the approach to thermodynamics based on measurement. First of all, we recall that considering classical thermodynamics as a theory of measurement of extensive variables one gets the description of thermodynamic states as Legendrian or Lagrangian manifolds representing the [...] Read more.
We present the development of the approach to thermodynamics based on measurement. First of all, we recall that considering classical thermodynamics as a theory of measurement of extensive variables one gets the description of thermodynamic states as Legendrian or Lagrangian manifolds representing the average of measurable quantities and extremal measures. Secondly, the variance of random vectors induces the Riemannian structures on the corresponding manifolds. Computing higher order central moments, one drives to the corresponding higher order structures, namely the cubic and the fourth order forms. The cubic form is responsible for the skewness of the extremal distribution. The condition for it to be zero gives us so-called symmetric processes. The positivity of the fourth order structure gives us an additional requirement to thermodynamic state. Full article
(This article belongs to the Special Issue Thermodynamics, Geometry and Control Theory)
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