Eight Lectures on Theoretical Physics/VII
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General Dynamics. Principle of Least Action.
General Dynamics. Principle of Least Action.
Since I began three weeks ago today to depict for you the present status of the system of theoretical physics and its probable future development, I have continually sought to bring out that in the theoretical physics of the future the most important and the final division of all physical processes would likely be into reversible and irreversible processes. In succeeding lectures, with the aid of the calculus of probability and with the introduction of the hypothesis of elementary disorder, we have seen that all irreversible processes may be considered as reversible elementary processes: in other words, that irreversibility does not depend upon an elementary property of a physical process, but rather depends upon the ensemble of numerous disordered elementary processes of the same kind, each one of which individually is completely reversible, and upon the introduction of the macroscopic method of treatment. From this standpoint one can say quite correctly that in the final analysis all processes in nature are reversible. That there is herein contained no contradiction to the principle regarding the irreversibility of processes expressed in terms of the mean values of elementary processes of macroscopic changes of state, I have demonstrated fully in the third lecture. Perhaps it will be appropriate at this place to interject a more general statement. We are accustomed in physics to seek the explanation of a natural process by the method of division of the process into elements. We regard each complicated process as composed of simple elementary processes, and seek to analyse it through thinking of the whole as the sum of the parts. This method, however, presupposes that through this division the character of the whole is not changed; in somewhat similar manner each measurement of a physical process presupposes that the progress of the phenomena is not influenced by the introduction of the measuring instrument. We have here a case in which that supposition is not warranted, and where a direct conclusion with regard to the parts applied to the whole leads to quite false results. If we divide an irreversible process into its elementary constituents, the disorder and along with it the irreversibility vanishes; an irreversible process must remain beyond the understanding of anyone who relies upon the fundamental law: that all properties of the whole must also be recognizable in the parts. It appears to me as though a similar difficulty presents itself in most of the problems of intellectual life.
Now after all the irreversibility in nature thus appears in a certain sense eliminated, it is an illuminating fact that general elementary dynamics has only to do with reversible processes. Therefore we shall occupy ourselves in what follows with reversible processes exclusively. That which makes this procedure so valuable for the theory is the circumstance that all known reversible processes, be they mechanical, electrodynamical or thermal, may be brought together under a single principle which answers unambiguously all questions regarding their behavior. This principle is not that of conservation of energy; this holds, it is true, for all these processes, but does not determine unambiguously their behavior; it is the more comprehensive principle of least action.
The principle of least action has grown upon the ground of mechanics where it enjoys equal rank and regard with numerous other principles; the principle of d'Alembert, the principle of virtual displacement, Gauss's principle of least constraint, the Lagrangian Equations of the first and second kind. All these principles are equivalent to one another and therefore at bottom are only different formularizations of the same laws; sometimes one and sometimes another is the most convenient to use. But the principle of least action has the decided advantage over all the other principles mentioned in that it connects together in a single equation the relations between quantities which possess, not only for mechanics, but also for electrodynamics and for thermodynamics, direct significance, namely, the quantities: space, time and potential. This is the reason why one may directly apply the principle of least action to processes other than mechanical, and the result has shown that such applications, as well in electrodynamics as in thermodynamics, lead to the appropriate laws holding in these subjects. Since a representation of a unified system of theoretical physics such as we have here in mind must lay the chief emphasis upon as general an interpretation as possible of physical laws, it is self evident that in our treatment the principle of least action will be called upon to play the principal rôle. I desire now to show how it is applied in simple individual cases.
The general formularization of the principle of least action in the interpretation given to it by Helmholz is as follows: among all processes which may carry a certain arbitrarily given physical system subject to given external actions from a given initial position into a given final position in a given time, the process which actually takes place in nature is that which is distinguished by the condition that the integral
wherein an arbitrary displacement of the independent coordinates (and velocities) is denoted by the sign , and denotes the infinitely small increase in energy (external work) which the system experiences in the displacement . The function is the kinetic potential. When we speak here of the positions, the coordinates, and the velocities of the configuration, we understand thereby, not only those special ones corresponding to mechanical ideas, but also all the so-called generalized coordinates with the quantities derived therefrom; and these may represent equally well quantities of electricity, volumes, and the like.
In the applications which we shall now make of the principle of least action, we must first decide as to whether the generalized coordinates which determine the state of the system considered are present in finite number or form a continuous infinite manifold. We shall distinguish the examples here considered in accordance with this viewpoint.
1. The Position (Configuration) is Determined by a Finite Number of Coordinates.
In ordinary mechanics this is actually the case in every system of a finite number of material points or rigid bodies among whose coordinates there exist arbitrary fixed equations of condition. If we call the independent coordinates , , , then the external work is:
wherein , , are the “external force components” which correspond to the individual coordinates, and denotes the energy of the system. Then the principle of least action is expressed by:
From this follow the equations of motion:
and so on for all the indices, , , . Through multiplication of the individual equations by , , addition and integration with respect to time, there results the equation of conservation of energy, whereby the energy is given by the expression:
In ordinary mechanics , if denote the kinetic and the potential energy. Since is a homogeneous function of the second degree with respect to the 's, it follows from that:
But this expression holds by no means in general.
We pass now to the consideration of the quasi-stationary motion of a system of linear conductors carrying simple closed galvanic currents. The state of the system is given by the position and the velocities of the conductors and by the current densities in each of the same. The coordinates referring to the position of the first conductor may be represented by , , , , corresponding designations holding for the remaining conductors. We inquire now as to the increase of energy or the external work, , which corresponds to a virtual displacement of all coordinates. Energy may be conveyed to the system through mechanical actions and through electromagnetic induction as well. The former corresponds to mechanical work, the latter to electromotive work. The former will be of the familiar form:
If we denote by , , the electromotive forces which are induced in the individual conductors through external agencies (e. g., moving magnets which do not belong to the system), then the electromotive work done from outside upon the currents in the conductors of the system is:
if , , denote the quantities of electricity which pass through cross sections of the conductors due to infinitely small virtual currents. The finite current densities will then be denoted by , , . The electrical state of the first conductor is thus determined in general by the current density , the mechanical state (position and velocity) by the coordinates , , , and the corresponding velocities , , , . The coordinates , , are so-called “cyclical” coordinates, since the state does not depend upon their momentary values, but only upon their differential quotients with respect to time, just as, for example, the state of a body rotatable about an axis of symmetry depends only upon the angular velocity, and not upon the angle of rotation. The scheme of notation adopted permits of the direct application of the above formularization of the principle of least action to the case here considered. Thus , where , the mechanical potential, depends only upon the 's and 's, while the electrokinetic potential takes the following form:
The quantities , , , the coefficients of self induction and mutual induction depend, however, in a definite manner upon the coordinates of position , , , , , , , .
In accordance with , we have for the motion of the first conductor:
with corresponding equations for , , , and for the electric current in it:
The laws for the mechanical (ponderomotive) actions may be condensed into the statement that, in addition to the ordinary force upon the first conductor expressed by , there is a mechanical force
which is composed of an action of the current upon itself (first term) and of the actions of the remaining currents upon it (following terms).
The laws of electrical action, on the other hand, are expressed by the statement, that to the external electromotive force in the first conductor there is added the electromotive force
which likewise is composed of an action of the current upon itself (self induction) and of the inducing actions of the remaining currents, and that these two forces compensate each other.
The galvanic conductance or the galvanic resistance is not contained in these equations because the corresponding energy, Joule heat, is produced in an irreversible manner, and irreversible processes are not represented by the principle of least action. One can formally include this action, likewise any other irreversible action, in accordance with the procedure of Helmholz, by introducing it as an external force, in the present case as the electromotive force due to the resistance , which operates to cause a diminution in the energy of the system. For an infinitely small element of time, the amount of this energy change is:
Consequently, since the external work now includes the Joule heat, the external force components , , in the electromotive equations must be increased by the additional terms , , .
The application of the principle of least action to thermodynamic processes is of special interest, because the importance of the question relating to the fixing of the generalized coordinates, which determine the state of the system, here becomes prominent. From the standpoint of pure thermodynamics, the variables which determine the state of a body can certainly be quite arbitrarily chosen, e. g., in the case of a gas of invariable constitution any two of the following quantities may be chosen as independent variables and all others expressed through them: volume , temperature , pressure , energy , entropy . In the present case, the matter is quite different. If we inquire, in order to apply the principle of least action, with regard to the energy change or the total work which will be done upon the gas from without in an infinitely small virtual displacement, it may be written in the form:
is the heat added from without, the mechanical work furnished from without. In order to bring this into agreement with the general formula for external work :
it becomes necessary now to choose and as the generalized coordinates of state and, therefore, to identify with them the previously employed quantities and . Then and are the generalized force components and . Now, since in thermodynamics every reversible change of state proceeds with infinite slowness, the velocity components and , and in general all differential coefficients with respect to time, are to be placed equal to zero, and the principle of least action reduces to:
and, therefore, in our case:
Further, in accordance with :
Now these equations are actually valid, since they only present other forms of the relation
The view here presented is fundamentally that which is given in the energetics of Mach, Ostwald, Helm, and Wiedeburg. The generalized coordinates and are in this theory the “capacity factors,” and the “intensity factors.” So long as one limits himself to an irreversible process, nothing stands in the way of carrying out this method completely, nor of a generalization to include chemical processes.
In opposition to it there is an essentially different method of regarding thermodynamic processes, which in its complete generality was first introduced into physics by Helmholtz. In accordance with this method, one generalized coordinate is , and the other is not , but a certain cyclical coordinate—we shall denote it, as in the previous example, by —which does not appear itself in the expression for the kinetic potential and only appears through its differential coefficient, ; and this differential coefficient is the temperature . Accordingly, is dependent only upon and . The equation for the total external work, in accordance with , is:
and agreement with thermodynamics is obviously found if we set:
The equations for the principle of least action become:
or by integration:
to an additive constant, which we may set equal to . For the energy there results, in accordance with :
is therefore equal to the negative of the function which Helmholz has called the “free energy” of the system, and the above equations are known from thermodynamics.
Furthermore, the method of Helmholz permits of being carried through consistently, and so long as one limits himself to the consideration of reversible processes, it is in general quite impossible to decide in favor of the one method or the other. However, the method of Helmholz possesses a distinct advantage over the other which I desire to emphasize here. It lends itself better to the furtherance of our endeavor toward the unification of the system of physics. In accordance with the purely energetic method, the independent variables and have absolutely nothing to do with each other; heat is a form of energy which is distinguished in nature from mechanical energy and which in no way can be referred back to it. In accordance with Helmholz, heat energy is reduced to motion, and this certainly indicates an advance which is to be placed, perhaps, upon exactly the same footing as the advance which is involved in the consideration of light waves as electromagnetic waves.
To be sure, the view of Helmholz is not broad enough to include irreversible processes; with regard to this, as we have earlier stated in detail, the introduction of the calculus of probability is necessary in order to throw light on the question. At the same time, this is also the real reason that the exponents of energetics will have nothing to do with the strict observance of irreversible processes, and they either declare them as doubtful or ignore them completely. In reality, the facts of the case are quite the reverse; irreversible processes are the only processes occurring in nature. Reversible processes form only an ideal abstraction, which is very valuable for the theory, but which is never completely realized in nature.
II. The Generalized Coordinates of State Form a Continuous Manifold.
The laws of infinitely small motions of perfectly elastic bodies furnish us with the simplest example. The coordinates of state are then the displacement components, , , , of a material point from its position of equilibrium , considered as a function of the coordinates , , . The external work is given by a surface integral:
(, surface element; , inner normal). The kinetic potential is again given by the difference of the kinetic energy and the potential energy :
The kinetic energy is:
wherein denotes a volume element, the volume density. The potential energy is likewise a space integral of a homogeneous quadratic function which specifies the potential energy of a volume element. This depends, as is seen from purely geometrical considerations, only upon the “strain coefficients:”
In general, therefore, the function contains independent constants, which characterize the whole elastic behavior of the substance. For isotropic substances these reduce on grounds of symmetry to . Substituting these values in the expression for the principle of least action we obtain:
If we put for brevity:
it turns out, as the result of purely mathematical operations in which the variations , , and likewise the variations , , are reduced through suitable partial integration with respect to the variations , , , that the conditions within the body are expressed by:
and at the surface, by:
as is known from the theory of elasticity. The mechanical significance of the quantities , , as surface forces follows from the surface conditions.
For the last application of the principle of least action we will take a special case of electrodynamics, namely, electrodynamic processes in a homogeneous isotropic non-conductor at rest, e. g., a vacuum. The treatment is analogous to that carried out in the foregoing example. The only difference lies in the fact that in electrodynamics the dependence of the potential energy upon the generalized coordinate is somewhat different than in elastic phenomena.
We therefore again put for the external work:
and for the kinetic potential:
On the other hand, we write here:
Through these assumptions the dynamical equations including the boundary conditions are now completely determined. The principle of least action furnishes:
From this follow, in quite an analogous way to that employed above in the theory of elasticity, first, for the interior of the non-conductor:
or more briefly
and secondly, for the surface:
These equations are identical with the known electrodynamical equations, if we identify with the electric, and with the magnetic energy (or conversely). If we put
( and , the field strengths, , the dielectric constant, , the permeability) and compare these values with the above expressions for and we may write:
It follows then, by elimination of , that:
and further, by substitution of and in equation found above for the interior of the non-conductor, that:
Comparison with the known electrodynamical equations expressed in Gaussian units:
(, velocity of light in vacuum) results in a complete agreement, if we put:
From either of these two equations it follows that:
the square of the velocity of propagation.
We obtain from for the energy entering the system from without:
or, taking account of the surface equation :
an expression which, upon substitution of the values of and from , turns out to be identical with the Poynting energy current.
We have thus by an application of the principle of least action with a suitably chosen expression for the kinetic potential arrived at the known Maxwellian field equations.
Are, then, the electromagnetic processes thus referred back to mechanical processes? By no means; for the vector employed here is certainly not a mechanical quantity. It is moreover not possible in general to interpret as a mechanical quantity, for instance, as a displacement, as a velocity, as a rotation. Thus, e. g., in an electrostatic field is constant. Therefore, increases with the time beyond all limits, and can no longer signify a rotation. While from these considerations the possibility of a mechanical explanation of electrical phenomena is not proven, it does appear, on the other hand, to be undoubtedly true that the significance of the principle of least action may be essentially extended beyond ordinary mechanics and that this principle can therefore also be utilized as the foundation for general dynamics, since it governs all known reversible processes.
- The breaking up of the energy differentials into two factors by the exponents of energetics is by no means associated with a special property of energy, but is simply an expression for the elementary law that the differential of a function is equal to the product of the differential by the derivative .
- With regard to the impossibility of interpreting electrodynamic processes in terms of the motions of a continuous medium, cf. particularly, H. Witte: “Über den gegenwärtigen Stand der Frage nach einer mechanischen Erklärung der elektrischen Erscheinungen” Berlin, 1906 (E. Ebering).