in the Poisson bracket then one gets the correct equations of motion. That is, the equations of motion are given by Before analyzing the Hamiltonian, consider the three constraints, Note the nontrivial Poisson bracket structure of the constraints. In particular, The above Poisson bracket does not just fail to vanish off-shell, which might be anticipated, but even on-shell it is nonzero. Therefore, and are second class constraints while is a first class constraint. Note that these constraints satisfy the regularity condition. Here, we have a symplectic space where the Poisson bracket does not have "nice properties" on the constrained subspace. However, Dirac noticed that we can turn the underlying differential manifold of the symplectic space into a Poisson manifold using his eponymous modified bracket, called the Dirac bracket, such that this Dirac bracket of any (smooth) function with any of the second class constraints always vanishes. Effectively, these brackets (illustrated for this spherical surface in the Dirac bracket article) project the system back onto the constraints surface. If one then wished to canonically quantize this system, then one need promote the canonical Dirac brackets, not the canonical Poisson brackets to commutation relations. Examination of the above Hamiltonian shows a number of interesting things happening. One thing to note is that, on-shell when the constraints are satisfied, the extended Hamiltonian is identical to the naive Hamiltonian, as required. Also, note that dropped out of the extended Hamiltonian. Since is a first class primary constraint, it should be interpreted as a generator of a gauge transformation. The gauge freedom is the freedom to choose , which has ceased to have any effect on the particle's dynamics. Therefore, that dropped out of the Hamiltonian, that 1 is undetermined, and that = pλ is first class, are all closely interrelated. Note that it would be more natural not to start with a Lagrangian with a Lagrange multiplier, but instead take as a primary constraint and proceed through the formalism: The result would the elimination of the extraneous dynamical quantity. However, the example is more edifying in its current form. Example: Proca action Another example we will use is the Proca action. The fields are and the action is where and . and are canonical variables. The second class constraints are and . The Hamiltonian is given by . See also Dirac bracket Holonomic constraint Analysis of flows References Further reading Classical mechanics Theoretical physics