Some Further Remarks on Hamilton ’ s Principle

The development of the equations of motion for a mechanical ystem from Hamilton’s principle can be viewed as a problem in he calculus of variations when the constraints on the system are olonomic and the forces are derivable from a potential function. endering stationary the integral of the Lagrangian over a fixed nterval of time taken between two fixed points in configuration pace, then, yields the equations of motion for the system. Howver, it is interesting to investigate the types of forces that can be ngendered through the use of an appropriately chosen function integrand whose integral when rendered stationary, yields the roper equations of motion even when the forces acting on a ystem do not arise from a potential, that is, when they are nononservative. In 1931, Bolza 1 gave a general procedure for finding such an ntegrand for a single degree-of-freedom system. This was folowed by Douglas 2,3 who obtained the necessary and sufficient onditions for the existence of an integrand for multidegree-ofreedom systems. However, it is difficult to obtain the integrands or given, specific forces. In a 1963 note, Leitmann 4 provided ome examples of such forces and the corresponding integrands or which a variational principle exists. A single degree-ofreedom system was considered and two examples were provided. ecently, the so-called semi-inverse method 5 has attracted uch attention due to its simplicity and applicability to certain ases. However, this method assumes a specific form for the funcion f , which needs to be obtained from experience, intuition, or oth, and utilizes a Lagrange multiplier type approach. In this paper, we extend the results in Ref. 4 to some more eneral nonpotential systems and provide a more systematic way f handling the inverse problem of the calculus of variations. The ain difficulties lie in performing the necessary integrations exlicitly, as will be seen. The examples given in Ref. 4 arise as pecial cases of the results provided herein. Finally, we apply the eneral results to some specific systems to indicate the nature of he nonpotential forces, which the results encompass.