Generalized impulse

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The generalized momentum , also generalized , canonical , canonically conjugated , or conjugated momentum , occurs both in Hamiltonian mechanics and in Lagrange mechanics . Together with the location, it characterizes the respective state of the system, which changes over time according to Hamilton's equations of motion .

As a function of location and speed , the generalized momentum is the partial derivative of the Lagrange function with respect to speed:

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

Examples

Classic movement

the generalized momentum is equal to the kinetic momentum:
  • When a particle of mass moves in a potential in cylindrical coordinates
the generalized momentum conjugated to the angle is the component of the angular momentum in the direction of the cylinder axis:
the generalized momentum has a contribution from the vector potential of the field in addition to the kinetic momentum :

Relativistic movement

  • With the relativistic movement of a particle of mass in a potential without constraints in Cartesian coordinates
the generalized momentum is equal to the kinetic momentum:
  • With the relativistic movement of a point charge with the mass in the electromagnetic field
the generalized momentum has a contribution from the vector potential of the field in addition to the kinetic momentum:

literature

  • Wolfgang Nolting: Basic Course Theoretical Physics 2 Analytical Mechanics . 7th edition. Springer, Heidelberg 2006, ISBN 3-540-30660-9 .