Lagrangian and Hamiltonian Formulation

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. 

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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. 

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. 

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. 

This paper presents an account of the dynamics of electric charges coupled to electromagnetic fields. The main approximation is to use non-relativistic forms for the charge and current density. A quantum theory requires either a Lagrangian or a Hamiltonian formulation of the dynamics in atomic and molecular physics the latter is almost universal so the main thrust of the paper is the development of a general Hamiltonian. It is this Hamiltonian that provides the basis for a recent demonstration that the S-matrix on the energy shell is gauge-invariant to all orders of perturbation theory. 

At the outset it is important to recognize that several mathematical frameworks for the description of dynamic systems are in common use. In this context classical mechanics can be divided into three disciplines denoted by Newtonian mechanics, Lagrangian mechanics and Hamiltonian mechanics reflecting three conceptually different mathematical apparatus of model formulation [35, 52, 2, 61, 38, 95, 60, 4],... 

The Hamiltonian formulation thus provides the view in which the statistical mechanics and modern kinetic theory are constructed. We define the Hamiltonian function by the Legendre transform of the Lagrangian function (2.6) [49, 52] ... 

This is not a big step from the Newtonian form of the equations of motion q = v Mv = F. More generally, for molecular models, the Hamiltonian and Lagrangian formulations are interchangeable, but the use of the Hamiltonian form is preferred for allowing simplified description of the geometric character of the solutions of the system as we discuss in Chaps. 2-4. 

Note that L2 does not explicitly depend on proper time t, since according to Eq. (3.92) r is uniquely determined by the space-time vector x and the 4-velocity u. For a better comparison between the three-dimensional formulation (Li) and the explicitly covariant formulation (L2) we have employed the velocity v = r (instead of r itself) in Eq. (3.139). Both Lagrangians Li and L2 do not represent physical observables and are therefore not uniquely determined. According to the Hamiltonian principle of least action given by Eq. (2.48), 5S = 0, they only have to yield the same equation of motion. This is in particular guaranteed if even the actions themselves are identical, i.e.. 

The equations of motions can be formulated in the Newtonian, Hamiltonian, or Lagrangian approaches, all of which correspond to the microcanonical (NVE) statistical-mechanical ensemble in which the number of atoms, N, volume, V, and total energy, E, are conserved. Simulations in other statistical-mechanical ensembles can... 

This nonconservative (v — dependent) force is not derivable from a potential energy V in the manner of Eq. 1.50. If a Lagrangian function can nevertheless be found that obeys Eq. 1.57, the formulation of the Hamiltonian using Eq. 1.62 will still be valid. Recasting Eq. 1.64 in terms of the vector and scalar potentials (Eqs. 1.39 and 1.42), we obtain... 

The problem for us is therefore to derive the classical Hamiltonian function for an electron in the presence of electromagnetic fields, which is normally done from the classical Lagrangian. Hamilton s and Lagrange s generalizations of classical mechanics are essentially the same theory as Newton s formulation but are more elegant and often computationally easier to use. In our context, their importance lies in the fact that they serve as a springboard to quantum mechanics. 

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