Any Hamiltonian System Is Locally Equivalent to a Free Particle

Elizabeth Galindo-Linares, Esperanza Navarro-Morale, Gilberto Silva-Ortigoza, Román Suárez-Xique, Magdalena Marciano-Melchor, Ramón Silva-Ortigoza, Edwin Román-Hernández

CIDETEC-IPN. Departamento de Posgrado. área de Mecatrónica. Unidad Profesional Adolfo López Mateos. C.P., México, D.F., México.
Facultad de Ciencias Físico Matemáticas de la Benemérita Universidad Autónoma de Puebla, Apartado Postal, Puebla, Pue., México.
Universidad del Istmo, Departamento de Matemáticas Aplicadas, Cd. Universitaria, Tehuantepec, Oax., México.

DOI: 10.4236/wjm.2012.25030                

Abstract

In this work we use the Hamilton-Jacobi theory to show that locally all the Hamiltonian systems with n degrees of freedom are equivalent. That is, there is a canonical transformation connecting two arbitrary Hamiltonian systems with the same number of degrees of freedom. This result in particular implies that locally all the Hamiltonian systems are equivalent to that of a free particle. We illustrate our result with two particular examples; first we show that the one-dimensional free particle is locally equivalent to the one-dimensional harmonic oscillator and second that the two-dimensional free particle is locally equivalent to the two-dimensional Kepler problem.

Keywords

Hamiltonian System; Canonical Transformation

Conflicts of Interest

The authors declare no conflicts of interest.

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