A new look at Hamilton's principle

Hamilton's principle and Hamilton's law are discussed. Hamilton's law is then applied to achieve direct solutions to time-dependent, nonconservative, initial value problems without the use of the theory of differential or integral equations. A major question has always plagued competent investigators who use “energy methods,” viz., “Why is it that one can derive the differential equations for a system from Hamilton's principle and then solve these equations (at least in principle) subject to applicable initial and boundary conditions; but one cannot obtain a solution directly from Hamilton's principle except in very special cases?” This paper provides the answer to that question. In Hamilton's own words, “... the peculiar combination it [i.e., Hamilton's law] involves, of the principles of variations with those of partial differentials, for the determination and use of an important class of integrals, may constitute, when it shall be matured by the future labors of mathematicians, a separate branch of analysis.”