Canonical coordinates

The generalized impulse (also generalized, canonical, canonical conjugated or conjugated pulse) occurs both in the Hamiltonian mechanics as well as in the Lagrangian mechanics. Together with the location he identifies the current state of the system, which varies with time according to the Hamiltonian equations of motion.

As a function of position and velocity of the generalized momentum is the derivative of the Lagrangian on the speed:

In the transition from classical physics to quantum mechanics, the canonical momentum is ( in contrast to the kinetic momentum) is replaced by the momentum operator:

Examples

Classical motion

  • Upon movement of the mass of a particle in a potential unconstrained in Cartesian coordinates
  • Upon movement of the mass of a particle in a potential
  • When moving a point charge with mass in the electromagnetic field

Relativistic motion

  • In the relativistic motion of a particle of rest mass in a potential without constraints in Cartesian coordinates
  • In relativistic motion of a point charge with rest mass in the electromagnetic field
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