# Canonical Transformations

# Canonical Transformations

This chapter considers a broader class of transformations known as canonical transformations, which transform the whole of the extended phase space in a more general way. Each new canonical coordinate or momentum is allowed to be a function of all of the previous phase-space coordinates, including the previous canonical momenta. Thus the new position and time variables may depend, through their dependence on the momenta, on the old velocities as well as the old positions. The Lagrange equations will not in general be form invariant under such transformations. Canonical transformations are the most general phase-space transformations that preserve the extended Hamilton equations. There are several equivalent definitions of canonical transformations, three of which are the Poisson bracket condition, the direct condition, and the Lagrange bracket condition. The definition of canonical transformation includes the Lorentz transformation of special relativity. Also discussed are symplectic coordinates, form invariance of Poisson brackets, and form invariance of the Hamilton equations.

*Keywords:*
canonical transformations, Poisson bracket condition, direct condition, Lagrange bracket condition, Hamilton equations, Poisson brackets, special relativity, Lorentz transformation, extended phase space, symplectic coordinates

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