An underlying geometrical manifold for Hamiltonian mechanics

L. P. Horwitz, A. Yahalom, J. Levitan, M. Lewkowicz

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

We show that there exists an underlying manifold with a conformal metric and compatible connection form, and a metric type Hamiltonian (which we call the geometrical picture), that can be put into correspondence with the usual Hamilton–Lagrange mechanics. The requirement of dynamical equivalence of the two types of Hamiltonians, that the momenta generated by the two pictures be equal for all times, is sufficient to determine an expansion of the conformal factor, defined on the geometrical coordinate representation, in its domain of analyticity with coefficients to all orders determined by functions of the potential of the Hamiltonian–Lagrange picture, defined on the Hamilton–Lagrange coordinate representation, and its derivatives. Conversely, if the conformal function is known, the potential of a Hamilton–Lagrange picture can be determined in a similar way. We show that arbitrary local variations of the orbits in the Hamilton–Lagrange picture can be generated by variations along geodesics in the geometrical picture and establish a correspondence which provides a basis for understanding how the instability in the geometrical picture is manifested in the instability of the the original Hamiltonian motion.

Original languageEnglish
Article number124501
JournalFrontiers of Physics
Volume12
Issue number1
DOIs
StatePublished - 1 Feb 2017

Keywords

  • Riemann–Euclidean Hamiltonian equivalence
  • coordinate mapping
  • geometrical Hamiltonian
  • stability analysis

Fingerprint

Dive into the research topics of 'An underlying geometrical manifold for Hamiltonian mechanics'. Together they form a unique fingerprint.

Cite this