Problems in the Foundations of Physics pp 17-30 | Cite as

# Reflections on Variational Principles and Invariance Theory

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## 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 C^{1} 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## Preview

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## References

- 1.Yang, K.: The theory of Lie derivatives and its applications. Amsterdam: North-Holland 1957.Google Scholar
- 2.Hestenes, M.E.: Calculus of variations and optimal control theory. New York: John Wiley 1966.zbMATHGoogle Scholar
- 3.Edelen, D.G.B.: Internat. J. Engrg. Sci. 7, 373–389 (1969).MathSciNetzbMATHCrossRefGoogle Scholar
- 4.Edelen, D.G.B.: Internat. J. Engrg Sci. 7, 391–399 (1969).MathSciNetCrossRefGoogle Scholar
- 5.Edelen, D.G.B.: Nonlocal variations and local invariance of fields. New York: American Elsevier 1969.zbMATHGoogle Scholar
- 6.Schouten, J. A.: Ricci calculus (2nd edit). Berlin-Gottingen-Heidelberg: Springer 1954.zbMATHGoogle Scholar
- 7.
- 8.