The Lagrangian and Hamiltonian

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 

It is clear that 4 and A are not uniquely determined by these equations, as one can modify both potentials through the addition of the scalar z(l t), i-C. 

If the physical results are to remain unchanged under a unitary transformation, it is necessary to transform the operators as well as the state functions (eqns (8.28) and (8.29)). A gauge transformation has no effect on a coordinate operator but the momentum operator p is changed into p + [e/c) x- Thus, to maintain gauge invariance in properties determined by the momentum operator, p must be replaced by a new operator. In the presence of an electromagnetic field or just a magnetic field, the momentum operator is replaced by the expression 

Variation of the action integral with respect to T, subject to the natural boundary condition that V, I -n = 0 on the surface at infinity and the condition that the variations in T vanish at the time end-points, yields as the equation of motion the Euler-Lagrange equation of the variation. This statement of Schrodinger s equation, the one appropriate for use when the system is in the presence of electric and magnetic fields, is 

This result may be re-expressed in terms of H°, the usual field-free Hamiltonian, and H an interaction Hamiltonian, as 

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A familiar example of Legendre transformation is the relationship that exists between the Lagrangian and Hamiltonian functions of classical mechanics [17]. In thermodynamics the simplest application is to the internal energy function for constant mole number U(S, V), with the differentials... 

Finally, the advantages of respectively, the Lagrangian and Hamiltonian methods, are compared below. Two points should be emphasized ... 

The Lagrangian-based functional [i, 2] or 2] derives directly from the Lagrangian as employed in the quantum action principle. For a total system, both the Lagrangian- and Hamiltonian-based functionals yield identical variational results. This equivalence in variational behaviour is maintained for the corresponding subsystem functionals only if the subsystem is bounded by a zero-flux surface. Only an atomic region ensures an equivalence in both the values and the variational properties of the two types of functionals (eqns (E5.6a,b)) thereby preserving the properties obtained for a total system. 

A Legendre transformation also connects the Lagrangian and Hamiltonian functions in classical mechanics. For a particle moving in one dimension, the Lagrangian L = T—V can be written as... 

The force and velocity are vectors, whose direction and magnitude are both of importance. In complex problems it is often preferable to reformulate classical mechanics in terms of a scalar, such as the energy, which is characterized only by its magnitude. This gives rise to the Lagrangian and Hamiltonian equations of motion. The latter equations are of most interest here and are dqi d K dt dpi ... 

So far we have only considered elementary relativistic mechanics based on the equation of motion given by Eq. (3.124). Similarly to the nonrelativistic discussion in chapter 2 we will now derive the Lagrangian and Hamiltonian formulation of relativistic mechanics. 

References to more detailed discussions and to derivations of the Lagrangian and Hamiltonian formulations of classical mechanics can be found in the Farther Reading section. 

A course in classical mechanics is an essential requirement of any first degree course in physics. In this volume Dr Brian Cowan provides a clear, concise and self-contained introduction to the subject and covers all the material needed by a student taking such a course. The author treats the material from a modern viewpoint, culminating in a final chapter showing how the Lagrangian and Hamiltonian formulations lend themselves particularly well to the more modem areas of physics such as quantum mechanics. Worked examples are included in. the text and there are exercises, with answers, for the student. 

Energy is usually written as the volume integral over the Hamiltonian, and not the Lagrangian, and Eq. (382) may be transformed into a volume integral over a Hamiltonian if we define the effective potential energy... 

The purpose of this section is to illustrate the methods of Lagrangian and Hamiltonian mechanics with the help of a simple mechanical system the double pendulum. It is shown that although the equations of motion for this system look very simple, the double pendulum is a chaotic system. 

T. Kimura, T. Ohtani, and R. Sugarno, On the consistency between Lagrangian and Hamiltonian formalisms in quantum mechanics III. Prog. Theor. Phys. 48, 1395-1407 (1972). 

All phenomena of classical nonrelativistic mechanics are solely based on Newton s laws of motion, which are valid in any inertial frame of reference. The natural symmetry operations of classical mechanics are the Galilean transformations, mediating the transition from one inertial coordinate system to another. The fundamental laws of classical mechanics can equally well be formulated applying the elegant Lagrangian and Hamiltonian descriptions based on Hamilton s action principle. Maxwell s equations for electric and magnetic fields are introduced as the basic laws of classical electrodynamics. 

The equation of motion for a nonrelativistic charged particle with charge cj and mass m subject to time-dependent electromagnetic fields will now be derived within the Lagrangian and also within the Hamiltonian formalism. The... 

Newton s formulation is not the only way in which classical equations of motion can be formulated. Lagrange (Joseph Louis Lagrange, France, 1736-1813), Hamilton (William Rowan Hamilton, Ireland, 1805-1865), and others developed different means, and it is the formulation of Hamilton that has proven the most useful framework for developing the mechanics of quantum systems. It is important to realize that Newtonian, Lagrangian, and Hamiltonian mechanics offer equivalent descriptions of classical systems. 

The Lagrangian equations can be turned into another useful form involving generalized coordinates and momenta and by defining the Hamiltonian function... 

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