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Symmetry in linear physical systems

Arjan van der Schaft, Rodolphe Sepulchre, Tom Chaffey

TL;DR

This work develops a unified framework for linear physical systems endowed with symmetry, focusing on reciprocal and input-output Hamiltonian (IOH) systems. It provides complementary input-output and state-space characterizations, grounded in signature matrices, symmetric/antisymmetric structures, and generating functionals tied to Lagrangian subspaces, with a geometric perspective via Volterra and Hankel operators. Time-reversibility is analyzed and linked to reciprocity and IOH through adjoint relations and compatible inner products $G$ and $\Omega$, yielding conditions for cyclo-passivity, losslessness, and port-Hamiltonian representations. The results support scalable, physically interpretable design of network components and outline paths toward nonlinear generalizations and neuromorphic applications.

Abstract

Physical systems with symmetry arise abundantly in applications, and are endowed with interesting mathematical structures. The present paper focusses on linear reciprocal and input-output Hamiltonian systems. Their characterization is studied from an input-output as well as from a state point of view. Geometrically, it turns out that they both define Lagrangian subspaces with corresponding generating functionals. Furthermore, the relations with time reversibility are analyzed. The system classes under consideration are expected to admit scalable control laws, and to be important building blocks in design.

Symmetry in linear physical systems

TL;DR

This work develops a unified framework for linear physical systems endowed with symmetry, focusing on reciprocal and input-output Hamiltonian (IOH) systems. It provides complementary input-output and state-space characterizations, grounded in signature matrices, symmetric/antisymmetric structures, and generating functionals tied to Lagrangian subspaces, with a geometric perspective via Volterra and Hankel operators. Time-reversibility is analyzed and linked to reciprocity and IOH through adjoint relations and compatible inner products and , yielding conditions for cyclo-passivity, losslessness, and port-Hamiltonian representations. The results support scalable, physically interpretable design of network components and outline paths toward nonlinear generalizations and neuromorphic applications.

Abstract

Physical systems with symmetry arise abundantly in applications, and are endowed with interesting mathematical structures. The present paper focusses on linear reciprocal and input-output Hamiltonian systems. Their characterization is studied from an input-output as well as from a state point of view. Geometrically, it turns out that they both define Lagrangian subspaces with corresponding generating functionals. Furthermore, the relations with time reversibility are analyzed. The system classes under consideration are expected to admit scalable control laws, and to be important building blocks in design.

Paper Structure

This paper contains 17 sections, 10 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Reciprocal systems
  3. Basic definitions and properties
  4. Geometric formulation
  5. Reciprocity and passivity; relaxation systems
  6. Port-Hamiltonian formulation of passive reciprocal systems
  7. Input-output Hamiltonian systems
  8. Definitions, basic properties and geometric formulation
  9. Volterra operators of input-output Hamiltonian systems
  10. Nonnegative input-output Hamiltonian systems
  11. Time-reversibility, and its connections to reciprocal and IO Hamiltonian systems
  12. Reciprocal systems and signed time-reversibility
  13. IO Hamiltonian systems and time-reversibility
  14. Conclusions and outlook
  15. Lagrangian subspaces and Dirac structures
  16. ...and 2 more sections

Key Result

Proposition 2

A system $\Sigma$ is reciprocal with respect to $\sigma$ if and only if is a Lagrangian subspace. Furthermore, the generating functional of $\mathcal{L}$ is the functional $\mathfrak{H}$ defined in quad.

Theorems & Definitions (26)

  • Definition 1
  • Remark 1
  • Remark 2
  • Proposition 2
  • Proposition 3
  • Definition 2
  • Remark 3
  • Definition 3
  • Example 1
  • Proposition 4
  • ...and 16 more