Classical Lagrangian and Hamiltonian

We can rewrite equations (3.212) and (3.213) so that they represent the vector and scalar potentials at a point R due to a moving electron at R  

Now suppose that at the instant in time to which these expressions refer, we have another electron at Rx. The Tagrangian for this electron is 

If we substitute for A and fi using (3.214) and (3.215) we obtain the Lagrangian for electron 1 when the motion of electron 2 is regarded as known, 

---

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... 

Classical Mechanics of Constrained Systems within Lagrangian and Hamiltonian Formalisms... 

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 ... 

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. 

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. 

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. 

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. 

We next give a brief derivation of the G( )-terin in the RBU Hamiltonian, which has previously not been done. The general procedure is to derive the classical Lagrangian for the constrained motions of the model and then transforming this to the Hamiltonian forin[40, 41]. Let... 

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],... 

Earlier in this section it was commented on how the minimal-coupling QED Hamiltonian is obtained from fhe classical Lagrangian function. A few words are in order regarding the derivation of the multipolar Hamiltonian (6). One method involves the application of a canonical transformation to the minimal-coupling Hamiltonian [32]. In classical mechanics, such a transformation renders the Poisson bracket and Hamilton s canonical equations of motion invariant. In quantum mechanics, a canonical transformation preserves both the commutator and Heisenberg s operator equation of motion. The appropriate generating function that converts H uit is propor-... 

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. 

To set up the Hamiltonian, we must return to the axiomatic foundations of quantum mechanics and find a set of canonically conjugate momenta and coordinates in the classical problem, in order to set up a quantum-mechanical Hamiltonian operator. To find the classical momenta and coordinates, we must find the Lagrangian L appropriate to charged particles moving in the field. The momentum conjugate to is then... 

Crowell discovered a variety of effects numerically, including modified Rabi flopping, which has an inverse frequency dependence similar to that observed in the solid state in reciprocal noise [73]. The latter is also explained by Crowell [17] using a non-Abelian model. A variety of other effects of RFR on the quantum electrodynamical level was also reported numerically [17]. The overall result is that the occurrence, classically, of the B V> field means that there is a quantum electrodynamical Hamiltonian generated by the classical term proportional to 3 2. This induces transitional behavior because it contributes to the dynamics of probability amplitudes [17]. The Hamiltonian is a quartic potential where the value of determines the value of the potential. The latter has two minima one where B = 0 and the other for a finite value of the B i) field, corresponding to states that are invariants of the Lagrangian but not of the vacuum. 

In accordance with the classical definition of the Hamiltonian, eqn (8.52), and recalling that the derivative of the Lagrangian with respect to a velocity (eqn (8.51)) is the momentum conjugate to the corresponding coordinate, the Hamiltonian operator is defined as... 

---