Friday, 12 September 2008
Classical Mechanics is Lagrangian; It is not Hamiltonian
12:05 pm, 817R Cathedral of Learning
Abstract: One can (for the most part) formulate a model of a classical system in either the Lagrangian or the Hamiltonian framework. Though it is often thought that those two formulations are equivalent in all important ways, this is not true: the underlying geometrical structures one uses in each case to express the equations of motion are not isomorphic. This raises the question whether one of the two is a more natural framework for the representation of classical systems. If one restricts oneself to the family of classical system whose possible interactions with the environment satisfy a weak condition (which all known classical systems in fact do), the answer is yes: I state and sketch the proof of a theorem, in the context of the intrinsic geometry of tangent bundles, to the effect that Lagrangian mechanics is the more fundamental representation of any such system. I conclude with a brief discussion of the result’s relevance for several problems of philosophical interest, including: the type and amount of knowledge of a physical system one must have in order to determine structures of intrinsic physical significance it manifests; the seeming inevitability of the presence of arbitrary elements in every model we can construct of a physical system; and the cogency of structural realist accounts of physical systems.