Notes on Symmetries , Conservation Laws , and Virial RelationsR

These notes describe some aspects of partial diierential equations from the view point of the variational calculus. We begin by showing how a vari-ational problem (the study of critical points of an action functional) can be associated to a diierential equation. Here the idea is to deene the action in such a way that the associated Euler-Lagrange equation is the diierential equation of interest. Then we study how the action and the the Euler-Lagrange equation are aaected by transformation groups that act on the space of functions where the action is deened. This leads to integral identities (virial relations) that are satissed by solutions of the Euler-Lagrange equations. The notion of a symmetry group of an action functional and of a diierential equation is deened in Section III. We describe how symmetry groups lead to conservation laws for the Euler-Lagrange equation (Noether's theorem). The nal section discusses virial relations in more detail and their relation to conservation laws. The references DFN], GF] and HC-I] (listed at the end) provide a complete account of most of the topics covered here. Many diierential equations (both ordinary and partial) that occur in physics can be formulated as a variational problem. There are several advantages in representing the equation this way. For example, it allows us to generalize the notion of a solution of a diierential equation and it leads to methods for nding solutions. In addition, certain features of the equation become more apparent, like conservation laws. We will be letting K(') = 0 denote a partial diierential equation. For instance, if the equation of interest is @ 2 t ' ? ' + ' 2 = 0, then K(') = @ 2 t ' ? ' + ' 2. For arbitrary functions ', the formula K(') may still make sense. Therefore, you can also think of K as a (not necessarily linear) operator, taking the function ' to the function where = K('). In the case = 0 we say that ' is a solution.