Lagrangian equations action integrals

For a system with N degrees of freedom, q, i = 1 to N, this equation is obtained for each of the N coordinates qi. These are Lagrange s equations of motion, the equations of motion for a system obeying classical mechanics. Thus, the Lagrangian, which minimizes the value of the action integral along the true trajectory between the times tj and fj, is also the function which yields the equations of motion when inserted into the Euler equation (8.50). 

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... 

In 1892 Helmholtz inquired whether we...can cast the empirically known laws of electrodynamics, as they are formulated in Maxwell s equations, in the form of a minimal principle [44]. Indeed such a minimal principle exists in the form of the principle of least action For a system of n degrees of freedom there exists a Lagrangian L qi,qi,t) such that the action integral... 

From the correspondence between equations (6) and (1), we conclude that the Lagrangian function, the dissipation function, 0 and the action integral, 5X of this system are given by ... 

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. 

Equation (8.144) is an alternative form of the expression given in eqn (8.125) for the total system. The principle of stationary action for a subsystem can be expressed for an infinitesimal time interval in terms of a variation of the Lagrangian integral, similar to that given in eqn (8.127) for the total system. For the atomic Lagrangian, assuming F to have no explicit time dependence, this statement is... 

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