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Jet space extensions of infinite-dimensional Hamiltonian systems

Till Preuster, Manuel Schaller, Bernhard Maschke

Abstract

We analyze infinite-dimensional Hamiltonian systems corresponding to partial differential equations on one-dimensional spatial domains formulated with formally skew-adjoint Hamiltonian operators and nonlinear Hamiltonian density. In various applications, the Hamiltonian density can depend on spatial derivatives of the state such that these systems can not straightforwardly be formulated as boundary port-Hamiltonian system using a Stokes-Dirac structure. In this work, we show that any Hamiltonian system of the above class can be reformulated as a Hamiltonian system on the jet space, in which the Hamiltonian density only depends on the extended state variable itself and not on its derivatives. Consequently, well-known geometric formulations with Stokes- Dirac structures are applicable. Additionally, we provide a similar result for dissipative systems. We illustrate the developed theory by means of the the Boussinesq equation, the dynamics of an elastic rod and the Allen-Cahn equation.

Jet space extensions of infinite-dimensional Hamiltonian systems

Abstract

We analyze infinite-dimensional Hamiltonian systems corresponding to partial differential equations on one-dimensional spatial domains formulated with formally skew-adjoint Hamiltonian operators and nonlinear Hamiltonian density. In various applications, the Hamiltonian density can depend on spatial derivatives of the state such that these systems can not straightforwardly be formulated as boundary port-Hamiltonian system using a Stokes-Dirac structure. In this work, we show that any Hamiltonian system of the above class can be reformulated as a Hamiltonian system on the jet space, in which the Hamiltonian density only depends on the extended state variable itself and not on its derivatives. Consequently, well-known geometric formulations with Stokes- Dirac structures are applicable. Additionally, we provide a similar result for dissipative systems. We illustrate the developed theory by means of the the Boussinesq equation, the dynamics of an elastic rod and the Allen-Cahn equation.
Paper Structure (11 sections, 7 theorems, 60 equations)

This paper contains 11 sections, 7 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Motivation and problem statement
  3. Hamiltonian systems
  4. Boundary port-Hamiltonian systems
  5. Problem statement
  6. Hamiltonian jet space extensions
  7. Definition of the extended state space
  8. Lift of the Hamiltonian system
  9. Dissipative Hamiltonian systems
  10. Conclusion
  11. Acknowledgments

Key Result

Proposition 4

le2005diracvillegas2007portMascvdSc23 Consider the flow space $\mathscr{F}=L^{2}\left( \left[a,\,b\right],\,\mathbb R^n\right)\times \mathbb{R}^{p} \ni \left(f,\,f_{\partial}\right)^{\top}=\bar{f}$ and the effort space $\mathscr{E}=\mathscr{F}^*\ni\left(e ,\, e_{\partial}\right)^{\top}=\bar{e}$. The Consider a Hamiltonian operator $\mathcal{J}$ defined in eq:Hamiltonian_operator. Then there exists

Theorems & Definitions (24)

  • Definition 1: Hamiltonian systems
  • Example 2
  • Example 3
  • Proposition 4
  • Definition 5
  • Example 6
  • Example 7
  • Example 8
  • Proposition 9
  • proof
  • ...and 14 more