FOLLOW-UP: How to Tell Whether a Functional is Extremized

This post is a follow-up to an earlier post (link here) about how to tell whether a stationary point of a functional is a maximum, minimum, or saddle point. In particular, as I thought about it more, I realized that using the analogy to discrete vectors could help when formulating a more general expression for the second derivative of the nonrelativistic classical action for a single degree of freedom (i.e. the corresponding Hessian operator). Additionally, I thought of a few other examples of actions whose Hessian operators are positive-definite. Finally, I've thought more about how to express these equations for systems with multiple degrees of freedom (DOFs) as well as for fields and about how these ideas connect to the path integral formulation of quantum mechanics. Follow the jump to see more