# Non-holonomic constraints in field theory

1. Dec 7, 2013

### Bobhawke

My question is the following:

In field theory, if I have a constraint $$\chi(q_a, p_a, \partial_i q_a)$$ that depends on the generalised coordinates q_a, momenta p_a and spatial derivatives only of the q_a $$\partial_i q_a$$ does this count as a non-holonomic constraint? Or is it only non-holonomic if it depends on the time derivatives of q_a?

And related to this, if there is a non-holonomic constraint, can the dynamics of the system still be derived from a stationary action principle with Lagrange multipliers added in to enforce the constraint? And does Dirac's procedure for constructing the Hamiltonian for constrained Hamiltonian systems produce the correct dynamics, or does it fail if there are non-holonomic constraints? If so, why?