Action integrals Hamilton

To develop a system of mechanics from here without the introduction of any other concepts, apart from energy, some general principle that predicts the course of a mechanical change is required. This could be like the Maupertuis principle of least action or Fermat s principle of least time. It means that the actual path of the change will have an extreme value e.g. minimum) of either action or time, compared to all other possible paths. Based on considerations like these Hamilton formulated the principle that the action integral... 

As an introduction to relativistic dynamics, it is of interest to treat time as a dynamical variable rather than as a special system parameter distinct from particle coordinates. Introducing a generic global parameter r that increases along any generalized system trajectory, the function t(r) becomes a dynamical variable. In special relativity, this immediately generalizes to A (r) for each independent particle, associated with spatial coordinates x (r). Hamilton s action integral becomes... 

Since dt cannot be singled out for special treatment, the covariant generalization of Hamilton s variational principle for a single particle requires an invariant action integral... 

Of fundamental significance in the development of this theory is Hamilton s principle of least action. It states that the action integral... 

The classical equations of motion in Hamilton s form can be obtained from a modification of Hamilton s principle as outlined above. The Lagrangian in the action integral is expressed in terms of the Hamiltonian using eqn (8.52) to yield the integral... 

The equations of motion for the nuclei are obtained from Hamilton s least action principle. The nuclei total kinetic energy, K, is given by the sum of individual nucleus kinetic energy, (l/2)Mk(dXk/dt)2. The time integral of the Lagrangian L(X,dX /dt,t) = K-V is the action S of the system. For different paths (X=X(t)) the action has different numerical values. 

Following Atkins [68], the propagation of particles follows a path dictated by Newton s laws, equivalent to Hamilton s principle, that particles select paths between two points such that the action associated with the path is a minimum. Therefore, Fermat s principle for light propagation is Hamilton s principle for particles. The formal definition of action is an integral identical in structure with the phase length in physical optics. Therefore, particles are associated with wave motion, the wave-particle dualism. Hamilton s principle of least... 

If the n 1 frequencies O), are nonresonant, then the truncated normal form (at any order) is integrable. That is, the quantities Ji = j(pj + qj), i = 1, and, f = p qn are integrals of the truncated normal form, and the Hamiltonian can be written as a function of these integrals. In these coordinates the quantity 0/i /0J becomes a constant and the qn-Pn component of Hamilton s equations becomes a simple, 2-DOF saddle. This implies that actions remain constant while passing through the saddle region—a situation that has been numerically observed in realistic examples [63,69]. 

On substituting in the Hamilton-Jacobi equation (7), 41, equation (9) would again result but in averaging subsequently over the unperturbed motion, H1(w°, J) would remain dependent on wp°. We cannot therefore apply the method without further consideration. The deeper physical reason for this is that the variables vfi, J°, with which the angle and action variables w, J of the perturbed motion are correlated, are not determined by the unperturbed motion on account of its degenerate character, other degenerate action variables, connected with the Jp° s by linear non-integral relations, could be introduced in place of the Jp° s, by a suitable choice of co-ordinates. 

Such a variation principle may be looked upon as the extension from one to three dimensions of Hamilton s Principle of Stationary Action in dynamics. We will show below that the analogous extension from one to three dimensions of Hamilton s less known Principle of Variable Action " (which regards the integral I in its dependence on both the boundary B of D and on the boundary values of ip) throws significant light on the structure of colloid theory. 

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