The principle of least action

Using such an extremal principle, we can obtain also the equation of motion. Lagrange could show further that the use of the laws of Newton and the use of the principle of the least action are two equivalent formulations of the same problem. The expansion of these concepts on quantum mechanics is naturally possible. We become aware due to these considerations that symmetry obviously is related to extremal principles in the mechanical sense. On the other hand, obviously a system tries to achieve a symmetrical condition, when disturbances are missing, which tend to produce as3nmnetry. 

For example, even in classical mechanics we have equations (like Newton s) and the variational principle of the least action. If we introduced something similar to varying constants into the Lagrangian we will change the equations. Similar... 

Based on the dual (dynamic and static) interpretations of equilibria and PVW (equations (1)) Lagrange formulated the integral extreme principle, i.e. the principle of the least action... 

However, the ptractical apphcation of the second law in the analysis of equilibrium irreversible trajectories faced great difficulties. Clausius and then Helmholtz, Boltzmann, J. Thomson, Planck and other researchers tried to harmonize the second law of thermodynamics with the principle of the least action and derive the equation that meets this painciple similar to the equations (3) or (4) for dissipative macroscopic systems (in which the organized energy forms turn into a non-organized form, i.e. heat, due to friction). As is known their attempts were unsuccessful and resulted in understanding the necessity to statistically substantiate thermodynamics (Polak, 2010). 

Taking account of stationary process on the negligibly small trajectory section we can make sure that the product G-r (r - time) is minimum, i.e. the principle of the least action is observed. Relations between the principles of conservative and dissipative systems were considered by the authors in (Kaganovich, 2011 Kaganovich et al., 2007, 2010). Below they will be additionally discussed in brief on the example of the models of hydraulic circuits. 

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... 

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... 

Consider a plane-parallel condenser of capacitance C whose plates are a p-type semiconductor (e.g., a CP) and a metal, and polarize the latter negatively. Excess positive charges (i.e., holes) appear at the surface of the semiconductor, and since its conductivity is low, they are in fact distributed over a certain thickness within the material. These excess holes, or at least part of them, should take part in the conduction. Applying a voltage to an external electrode not in contact with the semiconductor modulates its conductivity. This is the principle of the field effect, and clearly this control of the current through a gate electrode opens the possibility of transistor action without requiring the existence of p-n junctions. 

We call the fields (3.114)-(3.116) fulfilling the balances of mass (3.63), (3.65), momentum (3.76), moment of momentum (3.93), and energy (3.107) a thermodynamic process, because only these are of practical interest. Then we denote the fields (3.114) as the thermokinetic process and the fields (3.115) as the responses (we limit to the models with symmetric T (3.93) in more general models we must introduce also the torque M into responses (3.115), cf. Rems. 17, 32). The fields (3.116) are controlled from the outside (at least in principle). Just constitutive equations, which express the difference among materials, represent the missing equations and are relations between (3.114) and (3.115) [6, 7, 9, 10, 23, 34, 38, 40, 41, 44, 45], Referring to Sect. 2.1 we briefiy recall that constitutive equations are definitions of ideal materials which approximate real materials in the circumstances studied (i.e at chosen time and space scales). Constitutive equations may be proposed in rational thermodynamics using the constitutive principles of determinism, local action, memory, equipresence, objectivity, symmetry, and admissibility. 

Before moving on to the Schrodinger equation, let us briefly review the relevant analytical mechanics. The most significant aspects of analytical mechanics are the least-action principle and the conservation laws based on it. In 1753, L. Euler arranged P.-L.M. de Maupertuis s thoughts in his paper entitled On the least-action principle and proved that the kinetics of mechanical systems obey the least-action principle, to apply this principle to general problems (Ekeland 2009). 

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. 

The complexity of the new insecticidal chemicals brings many other problems. Synthetic organic chemicals are not effective against all pests. There is a marked selectiveness in action even between closely related species of insects. Some insects have already developed resistance to some of the newer materials. The idea of insects developing resistance to certain chemicals is not new. The over-all principle is well established in a few cases. The early development of flies resistant to DDT, a chemical which had been highly and universally effective for fly control, came as a surprise. Other cases of resistance to DDT are being indicated, and at least one kind of mite has developed resistance against another of the newer chemicals—parathion. 

If nonproliferation considerations have not led to official opposition to nuclear power, their effect on fuel cycle policy has been profound. Although, its rhetoric and many of its implementating actions have been more restrained, the Clinton Administration has, in principle, adopted the Carter policy of opposition to reprocessing and plutonium recycle, hr at least one important area, however, it has inexplicably out-Cartered earlier policy by terminating work on proliferation-resistant firel cycles that involve recycle of still highly radioactive plutonium. 

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... 

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