Difference between configuration space and phase space

  • #1
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Lagrangian Mechanics uses generalized coordinates and generalized velocities in configuration space.
Hamiltonian Mechanics uses coordinates and corresponding momenta in phase space.

Could anyone please explain the difference between configuration space and phase space.

Thank you in advance for your help...

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  • #2
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Phase space is the cotangent bundle of configuration space. The momenta are part of the phase space, unlike the generalised velocities that are tangent vectors to configuration space. Thus, the phase space of a physical system has a dimension that is double that of configuration space for the same system.

In Lagrangian mechanics, you deal with finding the stationary paths of the action, which is an integral of the Lagrangian, which in turn is a function on the tangent bundle of configuration space (but note that the configuration space itself just describes the configuration of the system!). In Hamiltonian mechanics, you deal with the flows of a vector field in phase space, i.e., a vector field on the cotangent bundle of configuration space (thus, phase space includes both the configuration and the generalised momenta!), related to a function on phase space (the Hamiltonian).
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