Classical Mechanics Is Lagrangian; It Is Not Hamiltonian; The Semantics of Physical Theory Is Not Semantical

Curiel, Erik (2009) Classical Mechanics Is Lagrangian; It Is Not Hamiltonian; The Semantics of Physical Theory Is Not Semantical.

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 to formulate each theory are not isomorphic.
This raises the question whether one of the two is a more natural
framework for the representation of classical systems. In the
event, the answer is yes: I state and prove two technical results,
inspired by simple physical arguments about the generic properties
of classical systems, to the effect that, in a precise sense,
classical systems evince exactly the geometric structure Lagrangian
mechanics provides for the representation of systems, and none that
Hamiltonian mechanics does. The argument not only clarifies the
conceptual structure of the two systems of mechanics, their
relations to each other, and their respective mechanisms for
representing physical systems. It also provides a decisive
counter-example to the semantical view of physical theories, and
one, moreover, that shows its crucial deficiency: a theory must be,
or at least be founded on, more than its collection of models (in
the sense of Tarski), for a complete semantics requires that one
take account of global structures defined by relations among the
individual models. The example also shows why naively structural
accounts of theory cannot work: simple isomorphism of theoretical
and empirical structures is not rich enough a relation to ground a
semantics.