Natural Boundary and Initial Conditions from a Modification of Hamilton's Principle

Abstract

In a Hamiltonian variational formulation of a field theory, certain boundary conditions arise naturally whereas others arise as constraints on the admissible variations, and no initial conditions arise naturally. When the Hamiltonian formulation is used to obtain approximate solutions to boundary-value problems, the approximating functions need not satisfy the natural conditions but must satisfy the constraint conditions. Although, in many instances, it is desirable for the approximating functions to satisfy the constraint—and even the natural—conditions, in other instances it is imperative that the approximating functions do not satisfy certain constraint conditions. A procedure is introduced for transforming Hamilton's principle so that the initial conditions and all conditions at boundaries and internal surfaces of discontinuity arise naturally and no constraint conditions are required. The transformation is effected by modifying the principle slightly, using Lagrange multipliers in the classical manner, and adding an appropriate initial-value term to the Lagrangian. A particularly useful approximation technique is applied to a problem with an internal surface of discontinuity, and it is shown that the transformed principle can be used whereas the usual form of Hamilton's principle cannot. It is noted that the transformed principle has an important advantage over the method of least squares.