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Any Hamiltonian System Is Locally Equivalent to a Free Particle

¹ Facultad de Ciencias Físico Matemáticas de la Benemérita Universidad Autónoma de Puebla, Apartado Postal, Puebla, Pue., México
² Facultad de Ciencias Físico Matemáticas de la Benemérita Universidad Autónoma de Puebla, Apartado Postal, Puebla, Pue., México
³ Facultad de Ciencias Físico Matemáticas de la Benemérita Universidad Autónoma de Puebla, Apartado Postal, Puebla, Pue., México
⁴ Facultad de Ciencias Físico Matemáticas de la Benemérita Universidad Autónoma de Puebla, Apartado Postal, Puebla, Pue., México
⁵ CIDETEC-IPN. Departamento de Posgrado. área de Mecatrónica. Unidad Profesional Adolfo López Mateos. C.P., México, D.F., México
⁶ CIDETEC-IPN. Departamento de Posgrado. área de Mecatrónica. Unidad Profesional Adolfo López Mateos. C.P., México, D.F., México
⁷ Universidad del Istmo, Departamento de Matemáticas Aplicadas, Cd. Universitaria, Tehuantepec, Oax., México

World Journal of Mechanics·Volume 02 (2012)·Pages 246–252·Published 31 October 2012·DOI10.4236/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.

KeywordsHamiltonian SystemCanonical Transformation

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Any Hamiltonian System Is Locally Equivalent to a Free Particle

Abstract

References (14)