Application of Hamilton's law of varying action

Abstract

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.

Finite element formulations via the theorem of expended power in the Lagrangian, Hamiltonian and total energy frameworks

“…For convenience, we consider hyperelastic-plastic material behavior [Belytschko et al 2000;Simo and Hughes 1998], because we have used the total Lagrangian description [Hibbitt et al 1970] in this work. From a phenomenological viewpoint, the multiplicative decomposition of the deformation gradient tensor [Lee and Liu 1967;Lee 1969; Simo and Hughes 1998; Nemat-Nasser 1982; Asaro and Lubarda 2006] has been used in the context of not only phenomenological polycrystalline plasticity, but also single crystal metal plasticity as follows:…”

“…Noether's theorem relates the invariance properties of the Lagrangian to the conservation laws of physics [Noether 1918;Byers 1996;1999]. Thus, the spatial translational invariance of the Lagrangian in configuration space gives rise to conservation of linear momentum, its rotational invariance to conservation of angular momentum, and the autonomous Lagrangian has invariance in time.…”

“…At the same time, it is known that Hamilton's principle has a logical inconsistency for handling transient initial value problems [Gurtin 1964a;1964b;Tonti 1973;1984;Bailey 1975;Simkins 1981;Carini and Genna 1998]. Hamilton's law of varying action, which is not a variational principle, circumvents this deficiency and is more appropriate to use.…”

Simulation of a Moving, Elastic Beam Using Hamilton's Weak Principle

“…Bailey [5,6] was the first to demonstrate that Hamilton's Law can be used to obtain direct solutions to non-stationary, non-conservative, initial value problems. He proved that this can be accomplished "without any reference to or knowledge of differential equations" [6].…”

“…Substituting a basic truncated power series for q, he integrated equation (2.1) to obtain a system of algebraic equations. Using this technique, he obtained solutions for various dynamics problems, including particle motion [5] and problems with rigid bodies [7]. Applying linear constitutive and kinematic equations, Bailey further extended this concept to problems of beam vibrations [8].…”