Boundary conditions for constraint systems in variational principle

Abstract

Abstract We show the well-posed variational principle in constraint systems. In a naive procedure of the variational principle with constraints, the proper number of boundary conditions does not match that of physical degrees of freedom , which implies that, even in theories with up to first-order derivatives, the minimal (or extremal) value of the action with the boundary terms is not a solution of the equation of motion in the Dirac procedure of constrained systems. We propose specific and concrete steps to solve this problem. These steps utilize the Hamilton formalism, which allows us to separate the physical degrees of freedom from the constraints. This reveals the physical degrees of freedom that are necessary to be fixed on boundaries, and also enables us to specify the variables to be fixed and the surface terms.