Least action, principle

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

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]. 

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... 

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). 

The least-action principle leads to the Euler-Lagrange equation determining motion paths. Suppose that a mechanical system is located at two different points with coordinates, qi and q2, at different times, t = t and t = t2, respectively. Then, the system transfers between these points under the condition that the action... 

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