Canonical transformations with time as a coordinate

The traditional definition of canonical transformation in classical analytic mechanics does not allow time to transform, and so excludes the Lorentz transformation. To remedy this anachronism, the well-known set of symmetric Lagrangian equations is used, which treat time as just another generalized coordinate and the negative of the traditional Hamiltonian function as just another generalized momentum. Based on this augmented phase space, time-symmetric canonical transformations are defined, which include time as a transformable coordinate. A complete theory is then presented of the invariance of Hamiltonian equations under these time-symmetric canonical transformations. This theory is based on a central fact: the nonexistence of symmetric Hamiltonian equations that would treat all coordinates and momenta equally in the same way that the symmetric Lagrangian theory does. The only available equations of Hamiltonian form are the nonsymmetric Hamiltonian equations, each set of which singles out one particular coordinate and its conjugate momentum for special treatment. It is these nonsymmetric Hamiltonian equations which are form invariant under time-symmetric canonical transformations. Also presented is a new variational principle that varies generalized coordinates and generalized velocities independently and unifies Lagrangian and Hamiltonian mechanics in one principle.