Canonical coordinates and transformations
Safko, John L. Formerly, Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina.
Stehle, Philip Department of Physics, University of Pittsburgh, Pittsburgh, Pennsylvania.
- General principles
- Some applications
- Poisson brackets
- Transition to quantum theory
- Links to Primary Literature
- Additional Readings
Classical mechanics is the study of the positions and motions of particles or extended bodies. Initially the description used the coordinates and velocities of the objects under consideration and applied Newton's laws to describe their motions. Later the Lagrangian formulation was used to relate the positions, velocities, and resulting accelerations through an action principle. In the Lagrangian method, the velocities are considered as the time derivatives of the positions, and the equations of motion are determined by varying possible paths of the system holding the end points of the variation constant. A more general method is the Hamiltonian formulation, which considers the positions and momenta of the objects in the system under consideration as independent quantities. For the Hamiltonian variation, the momenta and the positions are varied independently subject to a constraint of the relation between the velocities and the momenta.
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