Lagrangian and Hamiltonian Mechanics

The most basic entity in mechanics is the mass point. It is one of the earliest and most fruitful abstractions in physics. The mass point is an extrapolation from real slabs of matter to something that has no form 

The total time derivatives of the generalized coordinates dqa/dt are called the generalized velocities. They are denoted by qa. Again, q specifies the set qa, oc = 1,2,. ..,n. 

Now that we have introduced coordinates and velocities, the next question is how to predict the time evolution of a mechanical system. This is accomplished by solving a set of ordinary differential equations, the equations of motion, which can be derived from the principle of least action. It was discovered by Maupertuis and was further developed by Euler, Lagrange and Hamilton (d Abro (1951)). 

In order to apply the principle of least action we first assign a function L, the Lagrangian function, to a mechanical system M 

the principle of least action states that the path a mechanical system takes from (qi,ti) to (921 2) is such as to minimize the action 

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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. 

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 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... 

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). 

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],... 

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. 

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. 

Any conservative mechanical system which is either free or subject to holonomic constraints and whose potential does not depend on the generalized velocities is described by standard equations of motion (either Lagrangian or Hamiltonian). The kinetic energy of the iV-particIe system is ... 

Legendre transforms are also used in mechanics to obtain more convenient independent variables (Goldstein, 1980). The Lagrangian L is a function of coordinates and velocities, but it is often more convenient to define the Hamiltonian H with a Legendre transform because the Hamiltonian is a function of coordinates and momenta. Quantum mechanics is based on the Hamiltonian rather than the Lagrangian. 

More complete descriptions of Legendre transforms are provided by Callen (6) and Alberty (15). There is an lUPAC Technical Report on Legendre transforms (18). It is interesting to note that a Legendre transform is used in defining the Hamiltonian for a mechanical system on the basis of the Lagrangian (19). 

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