Energy and time as conjugate dynamical variables

The energy and time variables of the elementary classical dynamical systems are described geometrically, as canonically conjugate coordinates of an extended phasespace. It is shown that the Galilei action of the inertial equivalence group on this space is canonical, but not Hamiltonian equivariant. Although it has no effect al a classical level, the lack of equivariance makes the Galilei action inconsistent with the canonical quantization. A Hamiltonian equivariani action can be obtained by assuming that the inertial parameter in the extended phase-space is quasi-isotropic. This condition leads naturally to the Lorentz transformations between moving frames as a particular case of symplectic transforrnations. The limit speed appears as a constant factor relating the two additional canonical coordinates to Athe energy and time. Its value is identified with the speed of light by using the relationship Abetween the electromagnetic potentials and the symplectic form of the extended phase-space.