The Virial Theorem for Action-governed Theories

We describe a simple derivation of virial relations, for arbitrary physical systems that are governed by an action. The virial theorem may be derived directly from the action, A, with no need to go via the equations of motion, and is simply a statement of the stationarity of the action with respect to certain variations in the degrees of freedom. Only a sub-class of solutions that obey appropriate boundary conditions satisfy each virial relation. There is a set of basic virial relations (one for each degree of freedom) of the form ∂A i c ∂c c=1 = 0, where A i c is the action in which one degree of freedom has been scaled by a constant factor c. Linear combinations of these may be put in simple forms, taking advantage of homogeneity properties of the action, and of dimensional considerations. When some of the degrees of freedom are of the same type a tensor virial theorem presents itself; it may be obtained in a similar way by considering the variation of the action under linear transformations among these degrees of freedom. Further generalizations are discussed. Symmetries of the action may lead to identities involving the virial relations. Beside pointing to a unified provenance of the virial relations, and affording general systematics of them, our method is a simple prescription for deriving such relations. It is particularly useful for treating high-derivative and non-local theories. We bring several examples to show that indeed the usual virial relations are obtained by this procedure, and also to produce some new virial relations. 1. Introduction In default of a general definition of the virial theorem we describe it, drawing from known examples (see, for some of many,[1-9], and, in particular,[10]): It consists of a set of global (integral) relations that are satisfied by a subclass of solutions of the equations of motion. This sub-class is defined by requiring that the solutions obey certain boundary requirements. The derivation of these relations proceeds as follows: One contracts the equations of motion with functions of the degrees of freedom and integrate over the variables on which the latter depend. One then integrates by parts so as to reduce the order of derivatives, as many times as possible–discarding boundary terms, as one proceeds, by imposing requirements on the boundary behavior of the solutions. One ends up with a virial theorem consisting of (1) the set of integral relations, …