The Law of Varying Action enunciated by Hamilton in 1834-1835 permits the direct solution of the problems of Mechanics, stationary or non-stationary.All problems of statics but only certain problems of dynamics fall under the classification of "stationary" to which direct solutions have heretofore been pos- solutions are demonstrated for conservative and non-conservative, stationary and/or non-stationary particle motion. The generality of this theory of mechanics, which is free of the constraints imposed by the mathematics of differential equations, will be demonstrated in subsequent papers on stationary and non-stationary motion of beams and plates. All of these papers will stress three.major points: 1) simplicity, 2) generality, and 3) accuracy.2.
The theory of Ritz is applied to the equation that Hamilton called the “Law of Varying Action.” Direct analytical solutions are obtained for the transient motion of beams, both conservative and nonconservative. The results achieved are compared to exact solutions obtained by the use of rigorously exact free-vibration modes in the differential equations of Lagrange and to an approximate solution obtained through the application of Gurtin’s principles for linear elastodynamics. A brief discussion of Hamilton’s law and Hamilton’s principle is followed by examples of results for both free-free and cantilever beams with various loadings.
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