Reflections on Variational Principles and Invariance Theory

Abstract

At this juncture, a minimal requirement for respectability of a physical theory seems to be that it admit a variational principle. This being the case, one eventually becomes aware of a degree of uneasiness, for a variational principle requires the specification of a Lagrangian function and a Lagrangian function is not something that is readily available from knowledge of the physically primitive variables of a phenomena. This follows from the fact that a Lagrangian function for a variational principle must be defined on a space of functions, while the physical primitive variables of a phenomena are a collection of quantities that are quantified by those specific functions that agree with the values of the physical quantities. In general, a Lagrangian function must be defined over a much larger domain of functions than just those that represent the phenomena that we wish to quantify. To be specific, if there is only one variable φ of interest, in addition to the space-time variables, it is usual to consider the Lagrangian function as a function of φ and its first partial derivatives for all functions φ that are C¹ functions on a given domain of the space-time variables. The question thus arises even in this simple problem: how do we determine the Lagrangian function when it can only be sampled for those functions φ that actually arise in physical phenomena. It is obvious that additional assumptions must be made about the phenomena before an identification of the Lagrangian function for a phenomena can be made.