Abstract
In this paper, the invariances of the equation of motion of a classical particle, to coordinate and time translations, to scale transformations and inversions, and to Galilean transformations, are considered individually. Resultant conditions on the equation of motion are given, and, for invariance to the one-parameter continuous transformations, it is shown that the equation of motion can be reduced from second to first order. Associated with each such reduction is a conservation law. The implications of the invariance of the system Lagrangian to these transformations are indicated, and the conservation laws, if any, associated with them. Some requirements on the Lagrangian for invariant equations of motion are also presented, and it is shown that the invariance of an equation of motion derived from a Lagrangian does not imply the invariance of that Lagrangian to the transformation. It is also shown that time-translation invariance of the equation of motion does not always require conservation of the Hamiltonian.
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