The Variational Principles of Mechanics

INTRODUCTION

1.   The variational approach to mechanics. Ever since Newton laid the solid foundation of dynamics by formulating the laws of motion, the science of mechanics developed along two main lines. One branch, which we shall call vectorial mechanics,¹ starts directly from Newton’s laws of motion. It aims at recognizing all the forces acting on any given particle, its motion being uniquely determined by the known forces acting on it at every instant. The analysis and synthesis of forces and moments is thus the basic concern of vectorial mechanics.

While in Newton’s mechanics the action of a force is measured by the momentum produced by that force, the great philosopher and universalist Leibniz, a contemporary of Newton, advocated another quantity, the vis viva (living force), as the proper gauge for the dynamical action of a force. This vis viva of Leibniz coincides—apart from the unessential factor 2—with the quantity we call today kinetic energy. Thus Leibniz replaced the momentum of Newton by the kinetic energy. At the same time he replaced the force of Newton by the work of the force. This work of the force was later replaced by a still more basic quantity, the work function. Leibniz is thus the originator of that second branch of mechanics, usually called analytical mechanics,² which bases the entire study of equilibrium and motion on two fundamental scalar quantities, the kinetic energy and the work function, the latter frequently replaceable by the potential energy.

Since motion is by its very nature a directed phenomenon, it seems puzzling that two scalar quantities should be sufficient to determine the motion. The energy theorem, which states that the sum of the kinetic and potential energies remains unchanged during the motion, yields only one equation, while the motion of a single particle in space requires three equations; in the case of mechanical systems composed of two or more particles the discrepancy becomes even greater. And yet it is a fact that these two fundamental scalars contain the complete dynamics of even the most complicated material system, provided they are used as the basis of a principle rather than of an equation.

2.   The procedure of Euler and Lagrange. In order to see how this occurs, let us think of a particle which is at a point P1 at a time t1. Let us assume that we know its velocity at that time. Let us further assume that we know that the particle will be at a point P2 after a given time has elapsed. Although we do not know the path taken by the particle, it is possible to establish that path completely by mathematical experimentation, provided that the kinetic and potential energies of the particle are given for any possible velocity and any possible position.

Euler and Lagrange, the first discoverers of the exact principle of least action, proceed as follows. Let us connect the two points P1 and P2 by any tentative path. In all probability this path, which can be chosen as an arbitrary continuous curve, will not coincide with the actual path that nature has chosen for the motion. However, we can gradually correct our tentative solution and eventually arrive at a curve which can be designated as the actual path of motion.

For this purpose we let the particle move along the tentative path in accordance with the energy principle. The sum of the kinetic and potential energies is kept constant and always equal to that value E which the actual motion has revealed at time t1. This restriction assigns a definite velocity to any point of our path and thus determines the motion. We can choose our path freely, but once this is done the conservation of energy determines the motion uniquely.

In particular, we can calculate the time at which the particle will pass an arbitrarily given point of our fictitious path and hence the time-integral of the vis viva i.e., of double the kinetic energy, extended over the entire motion from P1 to P2. This time integral is called action. It has a definite value for our tentative path and likewise for any other tentative path, these paths being always drawn between the same two end-points P1, P2 and always traversed with the same given energy constant E.

The value of this action will vary from path to path. For some paths it will come out larger, for others smaller. Mathematically we can imagine that all possible paths have been tried. There must exist one definite path (at least if P1 and P2 are not too far apart) for which the action assumes a minimum value. The principle of least action asserts that this particular path is the one chosen by nature as the actual path of motion.

We have explained the operation of the principle for one single particle. It can be generalized, however, to any number of particles and any arbitrarily complicated mechanical system.

3.   Hamilton’s procedure. We encounter problems of mechanics for which the work function is a function not only of the position of the particle but also of the time. For such systems the law of the conservation of energy does not hold, and the principle of Euler and Lagrange is not applicable, but that of Hamilton is.

In Hamilton’s procedure we again start with the given initial point P1 and the given end-point P2. But now we do not restrict the trial motion in any way. Not only can the path be chosen arbitrarily—save for natural continuity conditions—but also the motion in time is at our disposal. All that we require now is that our tentative motion shall start at the observed time h of the actual motion and end at the observed time h. (This condition is not satisfied in the procedure of Euler-Lagrange, because there the energy theorem restricts the motion, and the time taken to go from P1 to P2 in the tentative motion will generally differ from the time taken in the actual motion.)

The characteristic quantity that we now use as the measure of action—there is unfortunately no standard name adopted for this quantity—is the time-integral of the difference between the kinetic and potential energies. The Hamiltonian formulation of the principle of least action asserts that the actual motion realized in nature is that particular motion for which this action assumes its smallest value.

One can show that in the case of conservative systems, i.e. systems which satisfy the law of the conservation of energy, the principle of Euler-Lagrange is a consequence of Hamilton’s principle, but the latter principle remains valid even for nonconservative systems.

4.   The calculus of variations. The mathematical problem of minimizing an integral is dealt with in a special branch of the calculus, called calculus of variations. The mathematical theory shows that our final results can be established without taking into account the infinity of tentatively possible paths. We can restrict our mathematical experiment to such paths as are infinitely near to the actual path. A tentative path which differs from the actual path in an arbitrary but still infinitesimal degree, is called a variation of the actual path, and the calculus of variations investigates the changes in the value of an integral caused by such infinitesimal variations of the path.

5.   Comparison between the vectorial and the variational treatments of mechanics. The vectorial and the variational theories of mechanics are two different mathematical descriptions of the same realm of natural phenomena. Newton’s theory bases everything on two fundamental vectors: momentum and force; the variational theory, founded by Euler and Lagrange, bases everything on two scalar quantities: kinetic energy and work function. Apart from mathematical expediency, the question as to the equivalence of these two theories can be raised. In the case of free particles, i.e. particles whose motion is not restricted by given constraints, the two forms of description lead to equivalent results. But for systems with constraints the analytical treatment is simpler and more economical. The given constraints are taken into account in a natural way by letting the system move along all the tentative paths in harmony with them. The vectorial treatment has to take account of the forces which maintain the constraints and has to make definite hypotheses concerning them. Newton’s third law of motion, action equals reaction, does not embrace all cases. It suffices only for the dynamics of rigid bodies.

On the other hand, Newton’s approach does not restrict the nature of a force, while the variational approach assumes that the acting forces are derivable from a scalar quantity, the work function. Forces of a frictional nature, which have no work function, are outside the realm of variational principles, while the Newtonian scheme has no difficulty in including them.

Such forces originate from inter-molecular phenomena which are neglected in the macroscopic description of motion. If the macroscopic parameters of a mechanical system are completed by the addition of microscopic parameters, forces not derivable from a work function would in all probability not occur.

6.   Mathematical evaluation of the variational principles. Many elementary problems of physics and engineering are solvable by vectorial mechanics and do not require the application of variational methods. But in all more complicated problems the superiority of the variational treatment becomes conspicuous. This superiority is due to the complete freedom we have in choosing the appropriate coordinates for our problem. The problems which are well suited to the vectorial treatment are essentially those which can be handled with a rectangular frame of reference, since the decomposition of vectors in curvilinear coordinates is a cumbersome procedure if not guided by the advanced principles of tensor calculus. Although the fundamental importance of invariants and covariants for all phenomena of nature has been discovered only recently and so was not known in the time of Euler and Lagrange, the variational approach to mechanics happened to anticipate this development by satisfying the principle of invariance automatically. We are allowed sovereign freedom in choosing our coordinates, since our processes and resulting equations remain valid for an arbitrary choice of coordinates. The mathematical and philosophical value of the variational method is firmly anchored in this freedom of choice and the corresponding freedom of arbitrary coordinate transformations. It greatly facilitates the formulation of the differential equations of motion, and likewise their solution. If we hit on a certain type of coordinates, called cyclic or ignorable, a partial integration of the basic differential equations is at once accomplished. If all our coordinates are ignorable, our problem is completely solved. Hence, we can formulate the entire problem of solving the differential equations of motion as a problem of coordinate transformation. Instead of trying to integrate the differential equations of motion directly, we try to produce more and more ignorable coordinates. In the Euler-Lagrangian form of mechanics it is more or less accidental if we hit on the right coordinates, because we have no systematic way of producing ignorable coordinates. But the later developments of the theory by Hamilton and Jacobi broadened the original procedures immensely by introducing the canonical equations, with their much wider transformation properties. Here we are able to produce a complete set of ignorable coordinates by solving one single partial differential equation.

Although the actual solution of this differential equation is possible only for a restricted class of problems, it so happens that many important problems of theoretical physics belong precisely to this class. And thus the most advanced form of analytical mechanics turns out to be not only esthetically and logically most satisfactory, but at the same time very practical by providing a tool for the solution of many dynamical problems which are not accessible to elementary methods.³

7.   Philosophical evaluation of the variational approach to mechanics. Although it is tacitly agreed nowadays that scientific treatises should avoid philosophical discussions, in the case of the variational principles of mechanics an exception to the rule may be tolerated, partly because these principles are rooted in a century which was philosophically oriented to a very high degree, and partly because the variational method has often been the focus of philosophical controversies and misinterpretations.

Indeed, the idea of enlarging reality by including tentative possibilities and then selecting one of these by the condition that it minimizes a certain quantity, seems to bring a purpose to the flow of natural events. This is in contradiction to the usual causal description of things. Yet we must not be surprised that for the more universal approach which was current in the 17th and 18th centuries, the two forms of thinking did not necessarily appear contradictory. The keynote of that entire period was the seemingly pre-established harmony between reason and world. The great discoveries of higher mathematics and their immediate application to nature imbued the philosophers of those days with an unbounded confidence in the fundamentally intellectual structure of the world. Thus deus intellectualis was the basic theme of the philosophy of Leibniz, no less than that of Spinoza. At the same time Leibniz had strong teleological tendencies, and his activities had no small influence on the development of variational methods.⁴ But this is not surprising if we observe how the purely esthetic and logical interest in maximum-minimum problems gave one of the strongest impulses to the development of infinitesimal calculus, and how Fermat’s derivation of the laws of geometrical optics on the basis of his principle of quickest arrival could not fail to impress the philosophically-oriented scientists of those days. That the dilettante misuse of these tendencies by Maupertuis and others for theological purposes has put the entire trend into disrepute, is not the fault of the great philosophers.

The sober, practical, matter-of-fact nineteenth century—which carries over into our day—suspected all speculative and interpretative tendencies as metaphysical and limited its programme to the pure description of natural events. In this philosophy mathematics plays the role of a shorthand method, a conveniently economical language for expressing involved relations. Hence, it is not surprising, but quite consistent with the positivistic spirit of the nineteenth century, to meet with the following appraisal of analytical mechanics by one of the leading figures of that trend, E. Mach, in The Science of Mechanics (Open Court, 1893, p. 480: "No fundamental light can be expected from this branch of mechanics. On the contrary, the discovery of matters of principle must be substantially completed before we can think of framing analytical mechanics the sole aim of which is a perfect practical mastery of problems. Whosoever mistakes this situation will never comprehend Lagrange’s great performance, which here too is essentially of an economical character." (Italics in the original.) According to this philosophy the variational principles of mechanics are not more than alternative mathematical formulations of the fundamental laws of Newton, without any primary importance.

However, philosophical trends float back and forth and the last word is never spoken. In our own day we have witnessed at least one fundamental discovery of unprecedented magnitude, namely Einstein’s Theory of General Relativity, which was obtained by mathematical and philosophical speculation of the highest order. Here was a discovery made by a kind of reasoning that a positivist cannot fail to call metaphysical, and yet it provided an insight into the heart of things that mere experimentation and sober registration of facts could never have revealed. The Theory of General Relativity brought once again to the fore the spirit of the great cosmic theorists of Greece and the eighteenth century.

In the light of the discoveries of relativity, the variational foundation of mechanics deserves more than purely formalistic appraisal. Far from being nothing but an alternative formulation of the Newtonian laws of motion, the following points suggest the supremacy of the variational method:

1. The Principle of Relativity requires that the laws of nature shall be formulated in an invariant fashion, i.e. independently of any special frame of reference. The methods of the calculus of variations automatically satisfy this principle, because the minimum of a scalar quantity does not depend on the coordinates in which that quantity is measured. While the Newtonian equations of motion did not satisfy the principle of relativity, the principle of least action remained valid, with the only modification that the basic action quantity had to be brought into harmony with the requirement of invariance.

2. The Theory of General Relativity has shown that matter cannot be separated from field and is in fact an outgrowth of the field. Hence, the basic equations of physics must be formulated as partial rather than ordinary differential equations. While Newton’s particle picture can hardly be brought into harmony with the field concept, the variational methods are not restricted to the mechanics of particles but can be extended to the mechanics of continua.

3. The Principle of General Relativity is automatically satisfied if the fundamental action of the variational principle is chosen as an invariant under any coordinate transformation. Since the differential geometry of Riemann furnishes us such invariants, we have no difficulty in setting up the required field equations. Apart from this, our present knowledge of mathematics does not give us any clue to the formulation of a co-variant, and at the same time consistent, system of field equations. Hence, in the light of relativity the application of the calculus of variations to the laws of nature assumes more than accidental significance.

¹This use of the term does not necessarily imply that vectorial methods are used.

²See note on terminology at end of chap. I, section 1.

³The present book does not discuss other integration methods which are not based on the transformation theory. Concerning such methods the reader is referred to the advanced text-books mentioned in the Bibliography.

⁴See the attractive historical study of A. Kneser, Das Prinzip der kleinsten Wirkung von Leibniz bis zur Gegenwart (Leipzig: Teubner, 1928).

THE VARIATIONAL PRINCIPLES OF MECHANICS

CHAPTER 1

THE BASIC CONCEPTS OF ANALYTICAL MECHANICS

1.   The principal viewpoints of analytical mechanics.  The analytical form of mechanics, as introduced by Euler and Lagrange, differs considerably in its method and viewpoint from vectorial mechanics. The fundamental law of mechanics as stated by Newton: mass times acceleration equals moving force holds in the first instance for a single particle only. It was deduced from the motion of a particle in the field of gravity on the earth and was then applied to the motion of planets under the action of the sun. In both problems the moving body could be idealized as a mass point or a particle, i.e. a single point to which a mass is attached, and thus the dynamical problem presented itself in this form: A particle can move freely in space and is acted upon by a given force. Describe the motion at any time. The law of Newton gave the differential equation of motion, and the dynamical problem was reduced to the integration of that equation.

If the particle is not free but associated with other particles, as for example in a solid body, or a fluid, the Newtonian equation is still applicable if the proper precautions are observed. One has to isolate the particle from all other particles and determine the force which is exerted on it by the surrounding particles. Each particle is an independent unit which follows the law of motion of a free particle.

This force-analysis sometimes becomes cumbersome. The unknown nature of the interaction forces makes it necessary to introduce additional postulates, and Newton thought that the principle action equals reaction, stated as his third law of motion, would take care of all dynamical problems. This, however, is not the case, and even for the dynamics of a rigid body the additional hypothesis that the inner forces of the body are of the nature of central forces had to be made. In more complicated situations the Newtonian approach fails to give a unique answer to the problem.

The analytical approach to the problem of motion is quite different. The particle is no longer an isolated unit but part of a system. A mechanical system signifies an assembly of particles which interact with each other. The single particle has no significance; it is the system as a whole which counts. For example, in the planetary problem one may be interested in the motion of one particular planet. Yet the problem is unsolvable in this restricted form. The force acting on that planet has its source principally in the sun, but to a smaller extent also in the other planets, and cannot be given without knowing the motion of the other members of the system as well. And thus it is reasonable to consider the dynamical problem of the entire system, without breaking it into parts.

But even more decisive is the advantage of a unified treatment of force-analysis. In the vectorial treatment each point requires special attention and the force acting has to be determined independently for each particle. In the analytical treatment it is enough to know one single function, depending on the positions of the moving particles; this work function contains implicitly all the forces acting on the particles of the system. They can be obtained from that function by mere differentiation.

Another fundamental difference between the two methods concerns the matter of auxiliary conditions. It frequently happens that certain kinematical conditions exist between the particles of a moving system which can be stated a priori. For example, the particles of a solid body may move as if the body were rigid, which means that the distance between any two points cannot change. Such kinematical conditions do not actually exist on a priori grounds. They are maintained by strong forces. It is of great advantage, however, that the analytical treatment does not require the knowledge of these forces, but can take the given kinematical conditions for granted. We can develop the dynamical equations of a rigid body without knowing what forces produce the rigidity of the body. Similarly we need not know in detail what forces act between the particles of a fluid. It is enough to know the empirical fact that a fluid opposes by very strong forces any change in its volume, while the forces which oppose a change in shape of the fluid without changing the volume are slight. Hence, we can discard the unknown inner forces of a fluid and replace them by the kinematical conditions that during the motion of a fluid the volume of any portion must be preserved. If one considers how much simpler such an a priori kinematical condition is than a detailed knowledge of the forces which are required to maintain that condition, the great superiority of the analytical treatment over the vectorial treatment becomes apparent.

However, more fundamental than all the previous features is the unifying principle in which the analytical approach culminates. The equations of motion of a complicated mechanical system form a large number—even an infinite number—of separate differential equations. The variational principles of analytical mechanics discover the unifying basis from which all these equations follow. There is a principle behind all these equations which expresses the meaning of the entire set. Given one fundamental quantity, action, the principle that this action be stationary leads to the entire set of differential equations. Moreover, the statement of this principle is independent of any special system of coordinates. Hence, the analytical equations of motion are also invariant with respect to any coordinate transformations.

Note on terminology.   The word analytical in the expression analytical mechanics has nothing to do with the philosophical process of analyzing, but comes from the mathematical term analysis, referring to the application of the principles of infinitesimal calculus to problems of pure and applied mathematics. While the French and German literature reserves the term analytical mechanics for the abstract mathematical treatment of mechanical problems by the methods of Euler, Lagrange, and Hamilton, the English and particularly the American literature frequently calls even very elementary applications of the calculus to problems of simple vectorial mechanics by the same name. The term mechanics includes statics and dynamics, the first dealing with the equilibrium of particles and systems of particles, the second with their motion. (A separate application of mechanics deals with the mechanics of continua—which includes fluid mechanics and elasticity— based on partial, rather than ordinary, differential equations. These problems are not included in the present book.)

Summary.   These, then, are the four principal viewpoints in which vectorial and analytical mechanics differ:

1.   Vectorial mechanics isolates the particle and considers it as an individual; analytical mechanics considers the system as a whole.

2.   Vectorial mechanics constructs a separate acting force for each moving particle; analytical mechanics considers one single function: the work function (or potential energy). This one function contains all the necessary information concerning forces.

3.   If strong forces maintain a definite relation between the coordinates of a system, and that relation is empirically given, the vectorial treatment has to consider the forces necessary to maintain it. The analytical treatment takes the given relation for granted, without requiring knowledge of the forces which maintain it.

4.   In the analytical method, the entire set of equations of motion can be developed from one unified principle which implicitly includes all these equations. This principle takes the form of minimizing a certain quantity, the action. Since a minimum principle is independent of any special reference system, the equations of analytical mechanics hold for any set of coordinates. This permits one to adjust the coordinates employed to the specific nature of each problem.

2.   Generalized coordinates.  In the elementary vectorial treatment of mechanics the abstract concept of a coordinate does not enter the picture. The method is essentially geometrical in character.

Vector methods are eminently useful in problems of statics. However, when it comes to problems of motion, the number of such problems which can be solved by pure vector methods is relatively small. For the solution of more involved problems, the geometrical methods of vectorial mechanics cease to be adequate and have to give way to a more abstract analytical treatment. In this new analytical foundation of mechanics the coordinate concept in its most general aspect occupies a central position.

Analytical mechanics is a completely mathematical science. Everything is done by calculations in the abstract realm of quantities. The physical world is translated into mathematical relations. This translation occurs with the help of coordinates. The coordinates establish a one-to-one correspondence between the points of physical space and numbers. After establishing this correspondence, we can operate with the coordinates as algebraic quantities and forget about their physical meaning. The end result of our calculations is then finally translated back into the world of physical realities.

During the century from Fermat and Descartes to Euler and Lagrange tremendous developments in the methods of higher mathematics took place. One of the most important of these was the generalization of the original coordinate idea of Descartes. If the purpose of coordinates is to establish a one-to-one correspondence between the points of space and numbers, the setting up of three perpendicular axes and the determination of length, width, and height relative to those axes is but one way of establishing that correspondence. Other methods can serve equally well. For example, the polar coordinates r, θ, ϕ may take the place of the rectangular coordinates x, y, z. It is one of the characteristic features of the analytical treatment of mechanics that we do not specify the nature of the coordinates which translate a given physical situation into an abstract mathematical situation.

Let us first consider a mechanical system which is composed of N free particles, free in the sense that they are not restricted by any kinematical conditions. The rectangular coordinates of these particles:

characterize the position of the mechanical system, and the problem of motion is obviously solved if xi,yi, zi are given as functions of the time t.

The same problem is likewise solved, however, if the xi,yi, zi are expressed in terms of some other quantities

and then these quantities qk are determined as functions of the time t.

This indirect procedure of solving the problem of motion provides great analytical advantages and is in fact the decisive factor in solving dynamical problems. Mathematically, we call it a coordinate transformation. It is a generalization of the transition from rectangular coordinates x, y, z of a single point in space to polar coordinates r, θ, ϕ. The relations

are generalized so that the old variables are expressed as arbitrary functions of the new variables. The number of variables is not 3 but 3N, since the position of the mechanical system requires 3N coordinates for its characterization. And thus the general form of such a coordinate transformation appears as follows:

We can prescribe these functions, f1, . . . ,f3N, in any way we wish and thus shift the original problem of determining the xi, yi, zi as functions of t to the new problem of determining the q1,.. q3N as functions of t. With proper skill in choosing the right coordinates, we may solve the new problem more easily than the original one. The flexibility of the reference system makes it possible to choose coordinates which are particularly suitable for the given problem. For example, in the planetary problem, i.e. a particle revolving around a fixed attracting centre, polar coordinates are better suited to the problem of motion than rectangular ones.

The advantage of generalized coordinates is even more obvious if mechanical systems with given kinematical conditions are considered. Such conditions find their mathematical expression in certain functional relations between the coordinates. For example, two atoms may form a molecule, the distance between the two atoms being determined by strong forces which are in equilibrium at that distance. Dynamically this system can be considered as composed of two mass points with coordinates x1, y1, z1 and x2 y2, z2 which are kept at a constant distance a from one another. This implies the condition

Because of this condition, the 6 coordinates x1,..., z2 cannot be prescribed arbitrarily. It suffices to give 5 coordinates; the sixth coordinate is then determined by the auxiliary condition (12.5). However, it is obviously inappropriate to designate one of the rectangular coordinates as a dependent variable when the relation (12.5) is symmetrical in all coordinates. It is more natural to prescribe the three rectangular coordinates of the centre of mass of the system and add two angles which characterize the direction of the axis of the diatomic molecule. The 6 rectangular coordinates x1, ... , z2 are expressible in terms of these 5 parameters.

As another example, consider the case of a rigid body, which can be composed of any number of particles. But whatever the number of particles may be, it is sufficient to give the three coordinates of the centre of mass and three angles which define the position of the body relative to the centre of mass. These 6 parameters determine the position of the rigid body completely. The coordinates of each of the component particles can be expressed as functions of these 6 parameters.

In general, if a mechanical system consists of N particles and there are m independent kinematical conditions imposed, it will be possible to characterize the configuration of the mechanical system uniquely by

independent parameters

in such a way that the rectangular coordinates of all the particles are expressible as functions of the variables (12.7):

The number n is a characteristic constant of the given mechanical system which cannot be altered. Less than n parameters are not enough to determine the position of the system. More than n parameters are not required and could not be assigned without satisfying certain conditions. We express the fact that n parameters are necessary and sufficient for a unique characterization of the configuration of the system by saying that it has "n degrees of freedom." Moreover, we call the n parameters q1, q2,...,qn the generalized coordinates of the system. The number N of particles which compose the mechanical system is immaterial for the analytical treatment, as are also the coordinates of these particles. It is the generalized coordinates q1, q2,...,qn and certain basic functions of them which are of importance. A rigid body may be composed of an infinity of mass points, yet for the mechanical treatment it is a system of not more than 6 independent coordinates.

Examples:

One degree of freedom: A piston moving up and down. A rigid body rotating about a fixed axis.

Two degrees of freedom: A particle moving on a given surface.

Three degrees of freedom: A particle moving in space. A rigid body rotating about a fixed point (top).

Four degrees of freedom: Two components of a double star revolving in the same plane.

Five degrees of freedom: Two particles kept at a constant distance from each other.

Six degrees of freedom: Two planets revolving about a fixed sun. A rigid body moving freely in space.

The generalized coordinates q1, q2, . . . , qn may or may not have a geometrical significance. It is necessary, however, that the functions (12.8) shall be finite, single valued, continuous and differentiable, and that the Jacobian of at least one combination of n functions shall be different from zero. These conditions may be violated at certain singular points, which have to be excluded from consideration. For example, the transformation (12.3) from rectangular to polar coordinates satisfies the general regularity conditions, but special care is required at the values r = 0 and θ = 0, for which the Jacobian of the transformation vanishes.

In addition to these restrictions in the small, certain conditions in the large have to be observed. It is necessary that a proper continuous range of the variables q1, q2, ... , qn shall permit a sufficiently wide range of the original rectangular coordinates, without restricting them more than the given kinematical conditions require. For example, the transformation (12.3) guarantees the complete, infinite range of the variables x, y, z if r varies between 0 and ∞, θ between 0 and π, and ϕ between 0 and 2π. However, such conditions