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Reflections on Variational Principles and Invariance Theory

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Part of the Studies in the Foundations, Methodology and Philosophy of Science book series (FOUNDATION, volume 4)

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.

Keywords

Variational Principle EULER Equation Boundary Data Invariance Theory Lagrangian Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  1. 1.Center for the Application of MathematicsLehigh UniversityBethlehemUSA

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