Trieste Quantum Seminars:

Scope of the action principle

Dr. Ward Struyve
Centre for Logic and Philosophy of Science, KU Leuven, Belgium

Abstract: Laws of physics are often presented in the form of an action principle where the physically allowed histories correspond to the extrema of an action, defined as the time integral of a Lagrangian. The advantages of such an action principle are manifold. A natural question, then, is to what extent such an action principle can be constructed. More specifically, given a set of differential equations that are local in time, can these equations be obtained as the Euler-Lagrange equations of some Lagrangian? This question is known as the inverse problem of Lagrangian mechanics. In general, the answer depends on the specific structure of the equations: some systems admit a Lagrangian formulation, while others do not. A key assumption in the standard formulation of the inverse problem is that no auxiliary variables are introduced. Relaxing this assumption, makes the problem trivial. I will suggest two different ways of constructing a Lagrangian using auxiliary variables. In the first, the equations of motion enforce that the auxiliary variables vanish. In the second, the auxiliary variables remain completely unconstrained by the equations of motion and thus play the role of gauge degrees of freedom. In either case, the auxiliary variables are trivial and the resulting theory is arguably a mere reformulation of the original theory.