Act~ lVIechaniea 50, 163-- 175 (1984) ACTA MECHANICA 9by Springer-Verlag 1984 An Extremum Variational 1Jriaciple for Classical tIamiltonian Systems* By Dj. S. Djukic, Novi Sad, Yugoslavia, and T. M. Atanackovic, Berlin (Received .February 1, 1983; revised July 18, 1983) A variational principle for a mechanical systems with n-degrees of freedom, which is subject to conservative generalized forces, is formulated. I~ecessary conditions for its extremum are given in detail. Two possibilities for ~0onstructing the functional are estab- lished. The first one is applicable if the canonical equations have certain simple algebraic properties, the second one is applicable in the general case. The theory is based on the concept of a mechanical system, but the results obtained can be applied to all problems in mathematical physics admitting Hamiltonian description. Finally, the theory is used for obtaining approximate solution of nonlinear mechanical problem with two-degrees of freedom. 1. Introduction It is well known that variational description of mechanical systems with n-degrees of freedom have been subject of many investigations. At the present time here are two main problems in this area. The first problem is faced if we want to include in variational description classical nonconservative mechanics with purely noneonservative generalized forces. The second problem is concerning extremality of the variational principles. An extremum principle is one which establishes the equivalence between an equation and the fact that some functional attains an extremum value, either a maximum or a minimum. In the Classical mechanics there are few treatments of the extrema] properties .which are different, in details, than those usually given in variational calculus (see for example [1]). One proof of the extremality, [2, p. 650], is valid for a small time interval of motion. Another, which is due to Bobylev, [3, p. 504] and [2, p. 653], needs our knowledge of the Cauchy integral of the differential equations of motion. Morse [4] has developed a method for dealing with necessary conditions of extremum of the classical action integrM. He transformed the sufficient condition ~ problem into a boundary value problem. The method is applied [5] to harmonic and anharmonie oscillator in one dimension, and proof is given that the action integral is minimum for certain time subinterval during the motion. Dual principles (called complementary when they are extremal) or reciprocal, * Research supported by the U.S. NSI~, Grant No.Yor 82/062.