Principle of 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. 

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. 

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 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. 

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,... 

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... 

Finally we should ask the question Why is our universe so well-described by group theory My best guess is that it is an important aspect of some general principle of least action. 

Now that we have introduced coordinates and velocities, the next question is how to predict the time evolution of a mechanical system. This is accomplished by solving a set of ordinary differential equations, the equations of motion, which can be derived from the principle of least action. It was discovered by Maupertuis and was further developed by Euler, Lagrange and Hamilton (d Abro (1951)). 

In order to apply the principle of least action we first assign a function L, the Lagrangian function, to a mechanical system M... 

Then, the principle of least action states that the path a mechanical system takes from (qi,ti) to (921 2) is such as to minimize the action... 

Not only mechanics but all of physics can be derived from the principle of least action. There are appropriate Lagrangian functions for electrodynamics, quantum mechanics, hydrodynamics, etc., which all allow us to derive the basic equations of the respective discipline from the principle of least action. In this sense, the principle of least action is the most powerful economy principle known in physics since it is sufficient to know the principle of least action, and the rest can be derived. Nature as a whole seems to be organized according to this principle. The principle of least action can be found under various names in nearly every branch of science. For instance the principle of least cost in economy or Fermat s principle of least time in optics. 

A last question remains How do we choose the function L for a given mechanical system A very important part of the answer is that L cannot, in principle, be derived by pure reasoning. L has to be chosen such that the equations of motion obtained from the principle of least action refiect the physical reality. 

The actual motion of the system does not depend on whether it is described with the help of the old or the new set of coordinates. Therefore, the original principle of least action still holds ... 

For further reading, see Density Functionals Theory and Application s. D. Joubert, Ed.. Springer 1998. G. Arfken, Mathematical Methods for Physicists, Acad. Press. 3rd ed. (1985). For a very readable Introduction by Richard Feynman, see The Principle of Least Action, in. R.P. Feynman. R.B. Leighton emd M. Sands. The Feynman Lectures on Physics. Addison-Wesley (1966). Vol. n chapter 19. The book by H.T. Davis. Statistical Mechanics of Phases. Interfaces and Thin FUms. Wiley (1996). contains a chapter (9) on this matter. 

We shall first find that Lagrangian for a system of charged particles in an electromagnetic field which, through the principle of least action, gives the correct equation of motion. The electric and magnetic fields of the electromagnetic field, d and B, respectively, are related to the scalar and vector potentials, and A, by the equations... 

The necessary condition for S to have a minimum is that the variation of the integral is zero. Hence, the variational principle of least action is written in the form ... 

The principle of least action, 8A = 0 for a particular choice of system state vector li/r), yields the equations that describe the system dynamics. In the case of a state vector that can explore the entire Hilbert space, the stationarity of the action yields the time-dependent Schrodinger equation. For any approximate family of state vectors this procedure yields an equation that approximates the time-dependent Schrodinger equation in a manner that is variationally optimal for the particular choice of state vector form. 

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