Second Class Constraints
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In a constrained Hamiltonian system, a dynamical quantity is second class if its Poisson bracket with at least one constraint is nonvanishing. A constraint that has a nonzero Poisson bracket with at least one other constraint, then, is a second class constraint. Before going on to the general theory, let's look at a specific example step by step to motivate the general analysis. Let's start with the action describing a Newtonian particle of mass m constrained to a surface of radius R within a uniform gravitational field g. When one works in Lagrangian mechanics, there are several ways to…mehr

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In a constrained Hamiltonian system, a dynamical quantity is second class if its Poisson bracket with at least one constraint is nonvanishing. A constraint that has a nonzero Poisson bracket with at least one other constraint, then, is a second class constraint. Before going on to the general theory, let's look at a specific example step by step to motivate the general analysis. Let's start with the action describing a Newtonian particle of mass m constrained to a surface of radius R within a uniform gravitational field g. When one works in Lagrangian mechanics, there are several ways to implement a constraint: one can switch to generalized coordinates that manifestly solve the constraint or one can use a Lagrange multiplier.