function that vanishes at the zeros of f is the fiberwise contraction of f with (a non-unique) smooth section of a -vector bundle where is the dual vector space to the constraint vector space V. This is called the regularity condition. Constrained Hamiltonian dynamics from a Lagrangian gauge theory First of all, we will assume the action is the integral of a local Lagrangian that only depends up to the first derivative of the fields. The analysis of more general cases, while possible is more complicated. When going over to the Hamiltonian formalism, we find there are constraints. Recall that in the action formalism, there are on shell and off shell configurations. The constraints that hold off shell are called primary constraints while those that only hold on shell are called secondary constraints. Examples Consider the dynamics of a single point particle of mass with no internal degrees of freedom moving in a pseudo-Riemannian spacetime manifold with metric g. Assume also that the parameter describing the trajectory of the particle is arbitrary (i.e. we insist upon reparametrization invariance). Then, its symplectic space is the cotangent bundle T*S with the canonical symplectic form . If we coordinatize T * S by its position in the base manifold and its position within the cotangent space p, then we have a constraint f = m2 −g(x)−1(p,p) = 0 . The Hamiltonian is, surprisingly enough, = 0. In light of the observation that the Hamiltonian is only defined up to the equivalence class of smooth functions agreeing on the constrained subspace, we can use a new Hamiltonian '= instead. Then, we have the interesting case where the Hamiltonian is the same as a constraint! See Hamiltonian constraint for more details. Consider now the case of a Yang–Mills theory for a real simple Lie algebra (with a negative definite Killing form ) minimally coupled to a real scalar field , which transforms as an orthogonal representation with the underlying vector space under in ( − 1) + 1 Minkowski spacetime. For in , we write as for simplicity. Let A be the -valued connection form of the theory. Note that the A here differs from the A used by physicists by a factor of and . This agrees with the mathematician's convention. The action is given by where g is the Minkowski metric, F is the curvature form (no s or s!) where the second