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On Certain Class of Lagrange Functions with Common Equations of Motion but Various Poisson Brackets
Author(s) -
Cisło J.,
Łopuszański J.,
Stichel P. C.
Publication year - 1995
Publication title -
fortschritte der physik/progress of physics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/prop.2190430805
Subject(s) - mathematics , invariant (physics) , euclidean space , euclidean geometry , mathematical analysis , lagrange multiplier , geometry , mathematical physics , mathematical optimization
A set of one particle Lagrange functions in 3‐dimensional Euclidean space, which all yield the same set of solutions of the corresponding Euler‐Lagrange Equations, is examined. The only imposed restriction is the existence of a rotationally invariant Lagrange function of a standard structure (in the sense of the classical one‐particle mechanics) among the above mentioned Lagrange functions. The general form of Lagrange functions of this set is presented. This set decomposes into subsets each of which differs from each other by the form of the Poisson Brackets for the position and velocity coordinates of the particle. These inequivalent subsets are discussed. The results can be immediately generalized to n = 2, 4, 5, … dimensional Euclidean spaces.