A Treatise on Electricity and Magnetism/Part IV/Chapter V
ON THE EQUATIONS OF MOTION OF A CONNECTED SYSTEM.
553.] In the fourth section of the second part of his Mécanique Analytique, Lagrange has given a method of reducing the ordinary dynamical equations of the motion of the parts of a connected system to a number equal to that of the degrees of freedom of the system.
The equations of motion of a connected system have been given in a different form by Hamilton, and have led to a great extension of the higher part of pure dynamics.
As we shall find it necessary, in our endeavours to bring electrical phenomena within the province of dynamics, to have our dynamical ideas in a state fit for direct application to physical questions, we shall devote this chapter to an exposition of these dynamical ideas from a physical point of view.
554.] The aim of Lagrange was to bring dynamics under the power of the calculus. He began by expressing the elementary dynamical relations in terms of the corresponding relations of pure algebraical quantities, and from the equations thus obtained he deduced his final equations by a purely algebraical process. Certain quantities (expressing the reactions between the parts of the system called into play by its physical connexions) appear in the equations of motion of the component parts of the system, and Lagrange's investigation, as seen from a mathematical point of view, is a method of eliminating these quantities from the final equations.
In following the steps of this elimination the mind is exercised in calculation, and should therefore be kept free from the intrusion of dynamical ideas. Our aim, on the other hand, is to cultivate our dynamical ideas. We therefore avail ourselves of the labours of the mathematicians, and retranslate their results from the language of the calculus into the language of dynamics, so that our words may call up the mental image, not of some algebraical process, but of some property of moving bodies.
The language of dynamics has been considerably extended by those who have expounded in popular terms the doctrine of the Conservation of Energy, and it will be seen that much of the following statement is suggested by the investigation in Thomson and Tait's Natural Philosophy, especially the method of beginning with the theory of impulsive forces.
I have applied this method so as to avoid the explicit consideration of the motion of any part of the system except the coordinates or variables, on which the motion of the whole depends. It is doubtless important that the student should be able to trace the connexion of the motion of each part of the system with that of the variables, but it is by no means necessary to do this in the process of obtaining the final equations, which are independent of the particular form of these connexions.
555.] The number of degrees of freedom of a system is the number of data which must be given in order completely to determine its position. Different forms may be given to these data, but their number depends on the nature of the system itself, and cannot be altered.
To fix our ideas we may conceive the system connected by means of suitable mechanism with a number of moveable pieces, each capable of motion along a straight line, and of no other kind of motion. The imaginary mechanism which connects each of these pieces with the system must be conceived to be free from friction, destitute of inertia, and incapable of being strained by the action of the applied forces. The use of this mechanism is merely to assist the imagination in ascribing position, velocity, and momentum to what appear, in Lagrange's investigation, as pure algebraical quantities.
Let q denote the position of one of the moveable pieces as defined by its distance from a fixed point in its line of motion. We shall distinguish the values of q corresponding to the different pieces by the suffixes 1, 2, &c. When we are dealing with a set of quantities belonging to one piece only we may omit the suffix.
When the values of all the variables (q) are given, the position of each of the moveable pieces is known, and, in virtue of the imaginary mechanism, the configuration of the entire system is determined.
556.] During the motion of the system the configuration changes in some definite manner, and since the configuration at each instant is fully defined by the values of the variables (q), the velocity of every part of the system, as well as its configuration, will be completely defined if we know the values of the variables (q), together with their velocities (, or, according to Newton's notation, ).
557.] By a proper regulation of the motion of the variables, any motion of the system, consistent with the nature of the connexions, may be produced. In order to produce this motion by moving the variable pieces, forces must be applied to these pieces.
We shall denote the force which must be applied to any variable qr by Fr. The system of forces (F) is mechanically equivalent (in virtue of the connexions of the system) to the system of forces, whatever it may be, which really produces the motion.
558.] When a body moves in such a way that its configuration, with respect to the force which acts on it, remains always the same, (as, for instance, in the case of a force acting on a single particle in the line of its motion,) the moving force is measured by the rate of increase of .the momentum. If F is the moving force, and p the momentum,
The time-integral of a force is called the Impulse of the force; so that we may assert that the momentum is the impulse of the force which would bring the body from a state of rest into the given state of motion.
In the case of a connected system in motion, the configuration is continually changing at a rate depending on the velocities so that we can no longer assume that the momentum is the time-integral of the force which acts on it.
But the increment δq of any variable cannot be greater than , where δt is the time during which the increment takes place, and is the greatest value of the velocity during that time. In the case of a system moving from rest under the action of forces always in the same direction, this is evidently the final velocity.
If the final velocity and configuration of the system are given, we may conceive the velocity to be communicated to the system in a very small time δt, the original configuration differing from the final configuration by quantities δq1, δq2, &c., which are less than , , &c., respectively.
The smaller we suppose the increment of time δt, the greater must be the impressed forces, but the time-integral, or impulse, of each force will remain finite. The limiting value of the impulse, when the time is diminished and ultimately vanishes, is defined as the instantaneous impulse, and the momentum p, corresponding to any variable q, is defined as the impulse corresponding to that variable, when the system is brought instantaneously from a state of rest into the given state of motion.
This conception, that the momenta are capable of being produced by instantaneous impulses on the system at rest, is introduced only as a method of defining the magnitude of the momenta, for the momenta of the system depend only on the instantaneous state of motion of the system, and not on the process by which that state was produced.
In a connected system the momentum corresponding to any variable is in general a linear function of the velocities of all the variables, instead of being, as in the dynamics of a particle, simply proportional to the velocity.
The impulses required to change the velocities of the system suddenly from , , &c. to , , &c., are evidently equal to p1' - p1, 'pp2' - p2, the changes of momentum of the several variables.
Work done by a Small Impulse.
559.] The work done by the force Fl during the impulse is the space-integral of the force, or
If is the greatest and the least value of the velocity during the action of the force, W must be less than
|and greater than|
If we now suppose the impulse to be diminished without limit, the values of and will approach and ultimately coincide with that of and we may write , so that the work done is ultimately
or, the work done by a very small impulse is ultimately the product of the impulse and the velocity.
Increment of the Kinetic Energy.
560.] When work is done in setting a conservative system in motion, energy is communicated to it, and the system becomes capable of doing an equal amount of work against resistances before it is reduced to rest.
The energy which a system possesses in virtue of its motion is called its Kinetic Energy, and is communicated to it in the form of the work done by the forces which set it in motion.
If T be the kinetic energy of the system, and if it becomes T + δT, on account of the action of an infinitesimal impulse whose components are δp1, δp1, &c., the increment δT must be the sum of the quantities of work done by the components of the impulse, or in symbols,
The instantaneous state of the system is completely defined if the variables and the momenta are given. Hence the kinetic energy, which depends on the instantaneous state of the system, can be expressed in terms of the variables (q), and the momenta (p). This is the mode of expressing T introduced by Hamilton. When T is expressed in this way we shall distinguish it by the suffix p, thus, Tp.
The complete variation of Tp is
The last term may be written
which diminishes with δt, and ultimately vanishes with it when the impulse becomes instantaneous.
Hence, equating the coefficients of δp in equations (1) and (2), we obtain
or, the velocity corresponding to the variable q is the differential coefficient of Tp with respect to the corresponding momentum p.
We have arrived at this result by the consideration of impulsive forces. By this method we have avoided the consideration of the change of configuration during the action of the forces. But the instantaneous state of the system is in all respects the same, whether the system was brought from a state of rest to the given state of motion by the transient application of impulsive forces, or whether it arrived at that state in any manner, however gradual.
In other words, the variables, and the corresponding velocities and momenta, depend on the actual state of motion of the system at the given instant, and not on its previous history.
Hence, the equation (3) is equally valid, whether the state of motion of the system is supposed due to impulsive forces, or to forces acting in any manner whatever.
We may now therefore dismiss the consideration of impulsive forces, together with the limitations imposed on their time of action, and on the changes of configuration during their action.
Hamilton's Equations of Motion.
561.] We have already shewn that
Let the system move in any arbitrary way, subject to the conditions imposed by its connexions, then the variations of p and q are
and the complete variation of Tp is
But the increment of the kinetic energy arises from the work done by the impressed forces, or
In these two expressions the variations δq are all independent of each other, so that we are entitled to equate the coefficients of each of them in the two expressions (7) and (8). We thus obtain
where the momentum pr and the force Fr belong to the variable qr.
There are as many equations of this form as there are variables. These equations were given by Hamilton. They shew that the force corresponding to any variable is the sum of two parts. The first part is the rate of increase of the momentum of that variable with respect to the time. The second part is the rate of increase of the kinetic energy per unit of increment of the variable, the other variables and all the momenta being constant.
The Kinetic Energy expressed in Terms of the Momenta and Velocities.
562.] Let , , &c. be the momenta, and , , &c. the velocities at a given instant, and let , , &c., , , &c. be another system of momenta and velocities, such that
It is manifest that the systems will be consistent with each other if the systems, are so.
Now let n vary by δn. The work done by the force F1 is
Let n increase from 0 to 1, then the system is brought from a state of rest into the state of motion and the whole work expended in producing this motion is
and the work spent in producing the motion is equivalent to the kinetic energy. Hence
where denotes the kinetic energy expressed in terms of the momenta and velocities. The variables &c. do not enter into this expression.
The kinetic energy is therefore half the sum of the products of the momenta into their corresponding velocities.
When the kinetic energy is expressed in this way we shall denote it by the symbol . It is a function of the momenta and velocities only, and does not involve the variables themselves.
563.] There is a third method of expressing the kinetic energy, which is generally, indeed, regarded as the fundamental one. By solving the equations (3) we may express the momenta in terms of the velocities, and then, introducing these values in (13), we shall have an expression for T involving only the velocities and the variables. When T is expressed in this form we shall indicate it by the symbol . This is the form in which the kinetic energy is expressed in the equations of Lagrange.
564.] It is manifest that, since , and are three different expressions for the same thing,
Hence, if all the quantities and vary,
The variations δp are not independent of the variations δq and , so that we cannot at once assert that the coefficient of each variation in this equation is zero. But we know, from equations (3), that
so that the terms involving the variations δp vanish of themselves.
The remaining variations and δq are now all independent, so that we find, by equating to zero the coefficients of , &c,
or, the components of momentum are the differential coefficients of with respect to the corresponding velocities.
Again, by equating to zero the coefficients of , &c.,
or, the differential coefficient of the kinetic energy with respect to any variable is equal in magnitude but opposite in sign when T is expressed as a function of the velocities instead of as a function of the momenta.
In virtue of equation (18) we may write the equation of motion (9),
which is the form in which the equations of motion were given by Lagrange.
565.] In the preceding investigation we have avoided the consideration of the form of the function which expresses the kinetic energy in terms either of the velocities or of the momenta. The only explicit form which we have assigned to it is
in which it is expressed as half the sum of the products of the momenta each into its corresponding velocity.
We may express the velocities in terms of the differential coefficients of Tp with respect to the momenta, as in equation (3),
This shews that Tp is a homogeneous function of the second degree of the momenta p1, p2, &c.
We may also express the momenta in terms of , and we find
which shews that is a homogeneous function of the second degree with respect to the velocities , , &c.
If we write
then, since both and are functions of the second degree of and of p respectively, both the P's Q's will be functions of the variables q only, and independent of the velocities and the momenta. We thus obtain the expressions for T,
The momenta are expressed in terms of the velocities by the linear equations
and the velocities are expressed in terms of the momenta by the linear equations
In treatises on the dynamics of a rigid body, the coefficients corresponding to P11, in which the suffixes are the same, are called Moments of Inertia, and those corresponding to P12, in which the suffixes are different, are called Products of Inertia. We may extend these names to the more general problem which is now before us, in which these quantities are not, as in the case of a rigid body, absolute constants, but are functions of the variables q1, q2, &c.
In like manner we may call the coefficients of the form Q11 Moments of Mobility, and those of the form Q12, Products of Mobility. It is not often, however, that we shall have occasion to speak of the coefficients of mobility.
566.] The kinetic energy of the system is a quantity essentially positive or zero. Hence, whether it be expressed in terms of the velocities, or in terms of the momenta, the coefficients must be such that no real values of the variables can make T negative.
We thus obtain a set of necessary conditions which the values of the coefficients P must satisfy.
The quantities P11, P22, &c., and all determinants of the symmetrical form
which can be formed from the system of coefficients must be positive or zero. The number of such conditions for n variables is 2n - 1.
The coefficients Q are subject to conditions of the same kind.
567.] In this outline of the fundamental principles of the dynamics of a connected system, we have kept out of view the mechanism by which the parts of the system are connected. We have not even written down a set of equations to indicate how the motion of any part of the system depends on the variation of the variables. We have confined our attention to the variables, their velocities and momenta, and the forces which act on the pieces representing the variables. Our only assumptions are, that the connexions of the system are such that the time is not explicitly contained in the equations of condition, and that the principle of the conservation of energy is applicable to the system.
Such a description of the methods of pure dynamics is not unnecessary, because Lagrange and most of his followers, to whom we are indebted for these methods, have in general confined themselves to a demonstration of them, and, in order to devote their attention to the symbols before them, they have endeavoured to banish all ideas except those of pure quantity, so as not only to dispense with diagrams, but even to get rid of the ideas of velocity, momentum, and energy, after they have been once for all supplanted by symbols in the original equations. In order to be able to refer to the results of this analysis in ordinary dynamical language, we have endeavoured to retranslate the principal equations of the method into language which may be intelligible without the use of symbols.
As the development of the ideas and methods of pure mathematics has rendered it possible, by forming a mathematical theory of dynamics, to bring to light many truths which could not have been discovered without mathematical training, so, if we are to form dynamical theories of other sciences, we must have our minds imbued with these dynamical truths as well as with mathematical methods.
In forming the ideas and words relating to any science, which, like electricity, deals with forces and their effects, we must keep constantly in mind the ideas appropriate to the fundamental science of dynamics, so that we may, during the first development of the science, avoid inconsistency with what is already established, and also that when our views become clearer, the language we have adopted may be a help to us and not a hindrance.
- See Professor Cayley's 'Report on Theoretical Dynamics', British Association, 1857; and Thomson and Taits Natural Philosophy'.