Path integrals for actions that are not quadratic in their time derivatives

The standard way to construct a path integral is to use a Legendre transformation to find the hamiltonian, to repeatedly insert complete sets of states into the time-evolution operator, and then to integrate over the momenta. This procedure is simple when the action is quadratic in its time derivatives, but in most other cases Legendre's transformation is intractable, and the hamiltonian is unknown. This paper shows how to construct path integrals when one can't find the hamiltonian because the first time derivatives of the fields occur in ways that make a Legendre transformation intractable; it focuses on scalar fields and does not discuss higher-derivative theories or those in which some fields lack time derivatives.