Inadequacy of The Standard Hamiltonian Procedure
The standard development of Hamiltonian mechanics is inadequate in several specific situations:
- When the Lagrangian is at most linear in the velocity of at least one coordinate; in which case, the definition of the canonical momentum leads to a constraint. This is the most frequent reason to resort to Dirac brackets. For instance, the Lagrangian (density) for any fermion is of this form.
- When there are gauge (or other unphysical) degrees of freedom which need to be fixed.
- When there are any other constraints that one wishes to impose in phase space.
Read more about this topic: Dirac Bracket
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