Classical Mechanics Project #3

by Nadina Gheorghiu
November 12, 1997

On page 64, Goldstein begins his comments on Rund's book by acknowledging that a good deal of what has been written about non-holonomic systems is wrong, including some of what he himself wrote in the first edition of his textbook. He goes on to draw attention to an interesting discussion of "pathological problems encountered in the actual physical world". Discuss these pathological problems without going into mathematical detail.

According to the Merriam-Webster Dictionary, one meaning of "pathological" is an abnormality of structure and function, characteristic of a disease.

We can deal with two kinds of structures:

  1. Physical structures for which, in some cases, a cure could be found (cure = a way to restore the health of the system).
  2. Abstract structures or theories for which always a cure has been finally found.
"Recovery from the disease" is not possible until some minimum but required fundamental modifications of the structures are done. The new theory contains the old one as a limiting case and we recognize this as the Bohr Correspondence Principle. The idea is the following: when a theory is unable to explain something in its domain, the solution is not just to discard the theory, but rather to enrich it with new tools (or "eyes") in order to see what was previously hidden. But the most important result is that each theoretical advance implies a simplification of the problems involved. Take an example: theories about light propagation have been developed one after the other by Huygens, Newton, Fresnel, Maxwell, Hertz and Lorentz. But the question: "Is the ether stationary or is it carried along by moving bodies?" still was unsolved, and became more puzzling than ever after the negative result of Michelson's experiment to prove earth's motion through the ether. There was a difficulty and there seemed to be no solution -- just like it would be to stay at a sick-bed and attempt to help the patient in all possible ways, but with no positive result. In 1905 Einstein was the new "physician", far more radical than his predecessors. He not only has achieved a "cure" for the patient, which means a solution to the problem, but most of all, changed the fundamental Newtonian concepts of space and time. Although the author of the famous expression: "Hypotheses non fingo" (in latin) meaning "I do not make up hypotheses", the great Newton was convinced about the absolute character of space and time. But in his time he was right. There's no need for a new tool if there's nothing that needs fixing.

The main idea of Rund's book is that the calculus of variations is the central and unifying discipline in both mathematics and theoretical physics and this is essentially due to the Hamilton-Jacobi theory. In what way is this book related with the topic debated previously? Rund points out an old and much-disputed difficulty (a kind of pathological case), namely whether or not the dynamics of a classical nonholonomic system can be treated by Lagrange's method. For a nonholonomic system, there exist non-integrable relations between the coordinates. This problem has been re-examined a few times and always from a new point of view, amongst others by Caratheodory, Morse and McShane. A simple example illustrates that the difficulty is not trivial at all: what happens if two given points can be joined by only one curve satisfying the equations of constraint? In this case, the treatment of the problem in Lagrange's framework is meaningless. A natural follow-up question is: does the motion of nonholonomic systems conform to Hamilton's Principles?

For holonomic constraints, there are integrable relations between the coordinates or the conditions of constraint can be expressed by certain equations. In this case, it is possible to obtain Lagrange's equations from Hamilton's Integral Principle, in which consideration is given to the entire motion of the system between two specified moments as well as small variations of the entire motion from the actual motion. Rund's conclusion is that "The Hamilton's Principle is applicable to a dynamical system subject to constraints if and only if these constraints are holonomic".

As the author states on page 363, instead of Hamilton's Principle, it is possible to formulate the fundamental dynamical principle of the extremal curvature: the trajectory in configuration space (the space of the generalized coordinates) of a mechanical system subject to holonomic or nonholonomic constraints is such that the square of its curvature assumes at each point a minimum or alternatively, a maximum value consistent with the constraints.

In a special case (see page 46 in Goldstein), the analytical application of this principle offers the possibility to define quantities named Lagrange's multipliers, with which the motion of some types of nonholonomic system can also be treated by a variational principle. Furthermore, the forces of constraint are obtained by using Lagrange's multiplier method. And thus, the motion of a body constrained to roll without sliding on a given fixed surface is solved. Furthermore, the new contains the old, namely the holonomic case is contained as a special case in the Lagrange multipliers method. Rund also states: "In the absence of constraints, the trajectories of the dynamical system in the configuration space are such that their curvature vanishes everywhere". This is an alternative way of stating Hamilton's Principle.

And thus, at least for some particular cases, the "cure" has been established. The "patient" is saved!

References (besides Goldstein's textbook):

  1. Hanno Rund, "The Hamilton-Jacobi theory in the calculus of variations. Its role in mathematics and physics".
  2. Anton Reiser, "Albert Einstein. A biographical portrait"
  3. Harald Fritzsch, "An equation that changed the world"