The geometry of the state space of physical systems and the consequences on the definition of Port-Hamiltonian systems

Abstract: In the first part, we recall the geometric structure of the state space of physical systems. Indeed, for Thermodynamical systems, it is well accepted that the system is first defined by its so-called equilibrium properties. These properties are defined by a set of relations among the extensive and intensive variables, the Thermodynamic Phase variables, which should satisfy the Gibbs' equations. Actually Gibbs' equations define a Legendre submanifold of the Thermodynamic Phase Space which is generated by a family of functions, called thermodynamic functions. This Legendre submanifolds actually defines the state space of the system.

A similar construction holds for Hamiltonian systems arising for mechanical systems or electro-mechanical systems' models, when instead of defining a Hamiltonian function, one considers the reciprocal constitutive relations relating the energy and the co-energy variables. These reciprocal relations define a Lagrangian submanifold of the cotangent space of the energy variables (the space of energy and the co-energy variables).

In the second part of the talk, we shall draw the consequence of the definition of the state space Lagrange or Legendre submanifolds for Hamiltonian and port Hamiltonian systems. Indeed, defining the state space as a submanifold of some phase space, corresponds to an implicit definition of the Hamiltonian dynamics. For irreversible Thermodynamic systems, one defines a contact Hamiltonian system on the Thermodynamic Phase Space, leaving invariant some Legendre submanifold. For Hamiltonian systems defined on Lagrange submanifolds, one defines a implicit Hamiltonian system restricted to some Lagrange submanifold.

We shall finally present some ongoing work, how this geometric perspective of the state space of physical systems, leads to define a novel class of Port Hamiltonian systems equipped with a new type of port variables, derived from the definition of Lagrange or Legendre submanifolds. We shall illustrate the work with various simple examples taken from physical and engineering systems.

mathematical physicsanalysis of PDEsdifferential geometrydynamical systemsfunctional analysisnumerical analysisoptimization and controlspectral theory

Audience: researchers in the discipline

( slides )

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Port-Hamiltonian Seminar

Series comments: Slides and recordings can be found here: uni-wuppertal.sciebo.de/s/CQfBsXr9iOI17ZY

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