Variational Principle, Conserved Quantity and Stability

In physics there exist a number of variational principles such as the Lagrangian approach, Fermat’s least time principle for light/ Maupertuis’ variational principle for particles and the maximization of entropy subject to a constraint. These approaches are associated with global type solutions and appear as independent approaches. For example, the Lagrangian approach considers various paths from the same initial and final points and chooses the one which extremizes the action: Integral dt L. Fermat’s principle is associated with various different paths from a starting point to an endpoint and chooses the one associated with least overall (global) time. Entropy is proportional to the ln of all arrangements and so one thinks of globally maximizing these. Thus taken independently the focus of variational principles is to choose an extreme solution from a large number of possible solutions which immediately begs the question: How does the system (nature) evaluate all of these different paths/possibilities? In (1) we suggested that the above variational principles may be associated with underlying local conservation laws. For example, for the Lagrangian case there is an associated conservation of energy. Thus v(x) must change locally so that .5mv(x)v(x) + V(x)= E. For Fermat’s principle there is conservation of momentum in the x direction (if the mirror or medium surface also lies in this direction). Thus light may move locally such that its (px) remains constant. (py) only changes abruptly when there is an interface change or mirror in its direction. Mathematically one may construct the above variational principles from the given conservation laws. As a result, there is no reason to necessarily consider a variational principle independently of its locally conserved quantity as argued in (1). In other words, one does not need to consider many different paths. Conservation leads locally to a single path. In this note we ask: If conservation leads locally to a solution, why consider a variational principle at all? Is it simply a convenient computational tool? We argue that there seems to be the idea of stable equilibrium present. In classical mechanics one has the notion of a stable equilibrium (static problem) which is associated with extremization. This is often associated with a stationary object which when slightly perturbed undergoes periodic motion. Here we argue that dynamical systems such as a Lagrangian system, or light or gas particles may also be seen as being in a stable equilibrium if one considers a variational principle together with its conserved quantity. The conserved quantity (px or E etc) characterizes the equilibrium while the variational principle shows that this is a stable equilibrium, even if light or a particle is traveling and interacting in space