Least action

As he gi ew older, Helmholtz became more and more interested in the mathematical side of physics and made noteworthy theoretical contributions to classical mechanics, fluid mechanics, thermodynamics and electrodynamics. He devoted the last decade of his life to an attempt to unify all of physics under one fundamental principle, the principle of least action. This attempt, while evidence of Helmholtz s philosphical bent, was no more successtul than was Albert Einstein s later quest for a unified field theory. Helmholtz died m 1894 as the result of a fall suffered on board ship while on his way back to Germany from the United States, after representing Germany at the Electrical Congress m Chicago in August, 1893. 

The most celebrated textual embodiment of the science of energy was Thomson and Tait s Treatise on Natural Philosophy (1867). Originally intending to treat all branches of natural philosophy, Thomson and Tait in fact produced only the first volume of the Treatise. Taking statics to be derivative from dynamics, they reinterpreted Newton s third law (action-reaction) as conservation of energy, with action viewed as rate of working. Fundamental to the new energy physics was the move to make extremum (maximum or minimum) conditions, rather than point forces, the theoretical foundation of dynamics. The tendency of an entire system to move from one place to another in the most economical way would determine the forces and motions of the various parts of the system. Variational principles (especially least action) thus played a central role in the new dynamics. 

A very exhaustive investigation was carried out by Helmholtz (1884), in which an attempt was made to interpret the second law, as applied to reversible processes, on the basis of the fundamental theorem of dynamics— the principle of Least Action. 

This is an example of the application of a very general theorem, formulated somewhat imperfectly by Maupertius, and called the Principle of Least Action. We can state it in the form that, if the system is in stable equilibrium, and if anything is done so as to alter this state, then something occurs in the system itself which tends to resist the change, by partially annulling the action imposed on the system. 

Laplace s equation, 146 Least action, principle of, 69, 304 Line of heterogeneous states, 172 Liquefaction of gases, 167, 173 of mixtures, 428... 

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... 

By analogy with Hamilton s principle of least action, the simplest proposition that could solve the thermodynamic problem is that equilibrium also depends on an extremum principle. In other words, the extensive parameters in the equilibrium state either maximize or minimize some function. 

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. 

In formulating QED a least-action principle involving a Lagrangian is often used [9,18,20]. This involves the potentials in various forms. Not only is relativistic invariance (Lorenz potentials) desired, but also gauge invariance. At least in the current state of QED, gauge invariance is included as a fundamental part [21,22]. 

The principle of least effort.—The principle of least action underlies all these rules, and it is of great service, and of wide application. P. L. M. Maupertius foreshadowed the idea in 1747 All natural changes take place in such a way that the existing state of things suffers the least possible change or, as W. D. Bancroft (1911) expressed it A system tends to change so as to minimize the effects of an external disturbing force. This has been called the principle of the... 

The physical expression of this principle is that nature is economical when a process can occur in several alternate ways, the one requiring the least expenditure of energy is the one used. Apparent exceptions generally turn out to conform to the principle and to have seemed exceptional because they were viewed in isolation when considered as a part of a larger system, the principle of least action is, in fact, followed. 

A claim made in many spiritual writings, supported by some experiential data from various d-ASCs, is that, with effort, we can become more and more conscious of exactly what we are doing, whether we can become conscious of everything we are doing psychologically at a given moment is unknown. Thus it is unclear whether we can ever be in a position adequately to assess whether the law of least action applies to psychological phenomena. But it may be profitable to postulate that the fifth principle does apply and then proceed to look for mani festations. 

For JT problems of higher dimensions such as that for the T (e t2) problem, the adiabatic potential V is complicated and cannot be written down in an analytical form. However, in such problems, the least action path can be approximated by the minimum energy path (or path of steepest descent) on the adiabatic potential surface. It is the path for which the tangent to it is parallel to the gradient of the APES. 

Fig. 1. The path and the contour plot of the lowest potential energy surface in the T t2 system. By symmetry considerations, the 3D problem is reduced to a calculation in 2D (X, Y) coordinate space. The five paths have been computed with different initial conditions and the one (in the middle) that reaches the zero-point energy (the innermost ring) is the least action path.

Variational principles for classical mechanics originated in modem times with the principle of least action, formulated first imprecisely by Maupertuis and then as an example of the new calculus of variations by Euler (1744) [436], Although not stated explicitly by either Maupertuis or Euler, stationary action is valid only for motion in which energy is conserved. With this proviso, in modem notation for generalized coordinates,... 

Euler s proof of the least action principle for a single particle (mass point in motion) was extended by Lagrange (c. 1760) to the general case of mutually interacting particles, appropriate to celestial mechanics. In Lagrange s derivation [436], action along a system path from initial coordinates P to final coordinates Q is defined by... 

If the end-points are fixed, the integrated term vanishes, and A is stationary if and only if the final integral vanishes. Since Sxa is arbitrary, the integrand must vanish, which is Newton s law of motion. Hence Lagrange s derivation proves that the principle of least action is equivalent to Newtonian mechanics if energy is conserved and end-point coordinates are specified. 

Hamilton s principle exploits the power of generalized coordinates in problems with static or dynamical constraints. Going beyond the principle of least action, it can also treat dissipative forces, not being restricted to conservative systems. If energy loss... 

Since time here is an ignorable variable, it can be eliminated from the dynamics by subtracting ptt from the modified Lagrangian and by solving H = E for t as a function of the spatial coordinates and momenta. This produces Jacobi s version of the principle of least action as a dynamical theory of trajectories, from which time dependence has been removed. The modified Lagrangian is... 

SA = 0 subject to the energy constraint restates the principle of least action. When the external potential function is constant, the definition of ds as a path element implies that the system trajectory is a geodesic in the Riemann space defined by the mass tensor m . This anticipates the profound geometrization of dynamics introduced by Einstein in the general theory of relativity. 

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

Mechanics emerged as a science studying reversible processes that are symmetrical relative to time. Euler, in his "thesis" on the least action... 

AN INTRODUCTION TO THE CALCULUS OF VARIATIONS, Charles Fox. Graduate-level text covers variations of an integral, isoperimetrical problems, least action, special relativity, approximations, more. References. 279pp. 5b x 8b. 

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