Generalized conformal Hamiltonian dynamics and the pattern formation equations

Partha Guha, A. Ghose-Choudhury

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

We demonstrate the significance of the Jacobi last multiplier in Hamiltonian theory by explicitly constructing the Hamiltonians of certain well known first-order systems of differential equations arising in the activator–inhibitor (AI) systems. We investigate the generalized Hamiltonian dynamics of the AI systems of Turing pattern formation problems, and demonstrate that various subsystems of AI, depending on the choices of parameters, are described either by conformal or contact Hamiltonian dynamics or both. Both these dynamics are subclasses of another dynamics, known as Jacobi mechanics. Furthermore we show that for non Turing pattern formation, like the Gray–Scott model, may actually be described by generalized conformal Hamiltonian dynamics using two Hamiltonians. Finally, we construct a locally defined dissipative Hamiltonian generating function Hudon et al. (2008) of the original system. This generating function coincides with the “free energy” of the associated system if it is a pure conformal class. Examples of pattern formation equation are presented to illustrate the method.

Original languageBritish English
Pages (from-to)195-208
Number of pages14
JournalJournal of Geometry and Physics
Volume134
DOIs
StatePublished - Dec 2018

Keywords

  • Autocatalysis system
  • Jacobi bracket
  • Jacobi's last multiplier
  • Morphogenesis
  • Singular Lagrangians
  • Turing pattern formation

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