Hamilton's Principle Of Least Action Research Articles

Analytical and experimental investigations of an autoparametric beam structure

This paper discusses theoretical and experimental investigations of vibrations of an autoparametric system composed of two beams with rectangular cross sections. Different flexibilities in the two orthogonal directions are the specific features of the structure. Differential equations of motion and associated boundary conditions, up to third-order approximation, are derived by application of the Hamilton principle of least action. Experimental response of the system, tuned for the 1:4 internal resonance condition, are performed for random and harmonic excitations. The most important vibration modes are extracted from a real mechanical system. It is shown that certain modes in the stiff and flexible directions of both beams may interact, and, intuitively unexpected out-of-plane motion may appear. Preliminary numerical calculations, based on the mathematical model, are also presented.

Variational mechanics in one and two dimensions

We develop heuristic derivations of two alternative principles of least action. A particle moving in one dimension can reverse direction at will if energy conservation is the only criterion. Such arbitrary changes in the direction of motion are eliminated by demanding that the Maupertuis–Euler abbreviated action, equal to the area under the momentum versus position curve in phase space, has the smallest possible value consistent with conservation of energy. Minimizing the abbreviated action predicts particle trajectories in two and three dimensions and leads to the more powerful Hamilton principle of least action, which not only generates conservation of energy, but also predicts motion even when the potential energy changes with time. Introducing action early in the physics program requires modernizing the current obscure and confusing terminology of variational mechanics.

Derivation of the Evans Wave Equation from the Lagrangian and Action: Origin of the Planck Constant in General Relativity

The Evans wave equation is derived from the appropriate Lagrangian and action, identifying the origin of the Planck constant ħ in general relativity. The classical Fermat principle of least time, and the classical Hamilton principle of least action, are expressed in terms of a tetrad multiplied by a phase factor exp(iS/ħ), where S is the action in general relativity. Wave (or quantum) mechanics emerges from these classical principles of general relativity for all matter and radiation fields, giving a unified theory of quantum mechanics based on differential geometry and general relativity. The phase factor exp(iS/ħ) is an eigenfunction of the Evans wave equation and is the origin in general relativity and geometry of topological phase effects in physics, including the Aharonov-Bohm class of effects, the Berry phase, the Sagnac effect, related interferometric effects, and all physical optical effects through the Evans spin field B(3) and the Stokes theorem in differential geometry. The Planck constant ħ is thus identified as the least amount possible of action or angular momentum or spin in the universe. This is also the origin of the fundamental Evans spin field B(3), which is always observed in any physical optical effect. It originates in torsion, spin and the second (or spin) Casimir invariant of the Einstein group. Mass originates in the first Casimir invariant of the Einstein group. These two invariants define any particle.