The Legendre transformation on singular Lagrangians, e.g. Lagrangians
representing gauge theories, fails due to the presence of constraints. The
Faddeev-Jackiw technique, which offers an alternative to that of Dirac, is a
symplectic approach to calculating a Hamiltonian paired with a well-defined
initial value problem when working with a singular Lagrangian. This phase space
coordinate reduction was generalized by Barcelos-Neto and Wotzasek to simplify
its application. We present an extension of the Faddeev-Jackiw technique for
constraint reduction in gauge field theories and non-gauge field theories that
are coupled to a curved spacetime that is described by General Relativity. A
major difference from previous formulations is that we do not explicitly
construct the symplectic matrix, as that is not necessary. We find that the
technique is a useful tool that avoids some of the subtle complications of the
Dirac approach to constraints. We apply this formulation to the Ginzburg-Landau
action and provide a calculation of its Hamiltonian and Poisson brackets in a
curved spacetime.
