term is a formal shorthand for pretending the Lie bracket is a commutator, is the covariant derivative Dσ = dσ − A[σ] and is the orthogonal form for . What is the Hamiltonian version of this model? Well, first, we have to split A noncovariantly into a time component and a spatial part . Then, the resulting symplectic space has the conjugate variables , (taking values in the underlying vector space of , the dual rep of ), , A, φ and πφ. For each spatial point, we have the constraints, πφ=0 and the Gaussian constraint where since is an intertwiner , ' is the dualized intertwiner ( is self-dual via ). The Hamiltonian, The last two terms are a linear combination of the Gaussian constraints and we have a whole family of (gauge equivalent)Hamiltonians parametrized by . In fact, since the last three terms vanish for the constrained states, we may drop them. Second class constraints 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. See Dirac brackets for diverse illustrations. An example: a particle confined to a sphere Before going on to the general theory, consider a specific example step by step to motivate the general analysis. Start with the action describing a Newtonian particle of mass constrained to a spherical surface of radius within a uniform gravitational field . 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 while retaining the redundant coordinates so constrained. In this case, the particle is constrained to a sphere, therefore the natural solution would be to use angular coordinates to describe the position of the particle instead of Cartesian and solve (automatically eliminate) the constraint in that way (the first choice). For pedagogical reasons, instead, consider the problem in (redundant) Cartesian coordinates, with a Lagrange multiplier term enforcing the constraint. The action is given by where the last term is the Lagrange multiplier term enforcing the constraint. Of course, as indicated, we could have just used different, non-redundant, spherical coordinates and written it as instead, without extra constraints; but we are considering the former coordinatization