Action integrals Lagrangian

Noether s theorem will be proved here for a classical relativistic theory defined by a generic field , which may have spinor or tensor indices. The Lagrangian density (, 9/x) is assumed to be Lorentz invariant and to depend only on scalar forms defined by spinor or tensor fields. It is assumed that coordinate displacements are described by Jacobi s theorem S(d4x) = d4x 9/xgeneral variation of the action integral, evaluated over a closed space-time region 2, is... 

The Lagrangian L and the action integral S are both functions in the sense that they are prescriptions of how to assign numbers to elements of their respective domains. But while L operates on numbers (the values of g, q and t at any given time t), the function S operates on functions, i.e. on test paths q t). The function 5[g] is thus often referred to as a functional. 

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... 

The first step in applying the principle of stationary action is to generalize the variation of the action integral to include a variation of the time end-points and to retain the variations at the end-points in order to define the generator F t). We shall express the Lagrangian operator in terms of the complete set of... 

The variation of the Lagrangian ojjerator is formally identical to the variation of the classical action integral as developed in eqns (8.45)-(8.48). We may take over the final result in eqn (8.48) completely in its corresponding operator form, retaining in this case the terms involving Sq at the time endpoints as these variations are no longer required to vanish. The addition of this result to the end-point variations in eqn (8.82) yields, for the general... 

That this is the correct Lagrangian density is demonstrated by showing that the variation of the resulting action integral with respect to T and T yields... 

It is this property that is common to the corresponding Lagrangian and action integrals for a quantum subsystem, the chemical atom. 

Since the zero-flux boundary condition (eqn (8.109)) is also satisfied by an atom, that is, by a quantum subsystem, the atomic action and Lagrangian integrals vanish as they do for a total closed system. Indeed, one may view the vanishing of the action integral over some total system as being the result of the action integral vanishing separately over the space of each atom in the system. 

The techniques of the calculus of variations, now familiar to the reader, are applied to the expression for [ T, n] (eqn (8.111)) with the Lagrangian density as defined in eqn (8.100). Included is a variation of the time end-points as detailed in eqns(8.81) and (8.82), and a variation of the surface of the subsystem in the manner previously detailed in eqn (5.74). Variations in P and at the time end-points are also allowed in this generalized variation of the action integral. Following these steps, one obtains the following explicit results for variations in T and the complex conjugates of these (cc) for variations in 4 ... 

Thus, the atomic Lagrangian and action integrals in the presence of an electromagnetic field, like their field-free counterparts, vanish as a consequence of the zero-flux surface condition (eqn (8.109)). These properties are common to the corresponding integrals for the total system and it is a consequence of this equivalence in properties that the action integrals for the total system and each of the atoms which comprise it have similar variational properties. 

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 beauty of the above topological definition of the atom in a molecule lies in the fact that it coincides with the rigorous quantum mechanical definition of an open subsystem [27, 33, 34]. In particular, the atomic action integral, which is defined through the atomic one-particle Lagrangian density, is zero within the atomic volume ... 

Action integral in classical or Lagrangian mechanics Imaginary surface in phase space enclosing J7, used in classical mechanics Sensitivity (projection) matrix Specific entropy of mixture (kJ/kgK)... 

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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