Application of Hamilton's law of varying action

The law of varying action enunciated by Hamilton in 1834-1835 permits the direct analytical solution of the problems of mechanics, both stationary and nonstationary, without consideration of force equilibrium and the theory of differential equations associated therewith. It has not been possible to obtain direct analytical solutions to nonstationary systems through the use of energy theory, which has been limited for 140 years to the principle of least action and to Hamilton's principle. It is shown here that Hamilton's law permits the direct analytical solution to nonstationary, initial value systems in the mechanics of solids without any knowledge or use of the theory of differential equations. Solutions are demonstrated for nonconservative, nonstationary particle motion, both linear and nonlinear.