Conditionally periodic systems

If we consider systems of several degrees of freedom, the motion will be periodic only in singular cases.

The general conditions which determine the stationary states cannot therefore be derived through the same simple considerations of the former section.

Sommerfeld and others have recently succeeded through a generalisation of (10), to obtain conditions for an important class of systems of several degrees of freedom.

In connection with (1), this gave convincing agreement with experimental results about line-spectra.

Subsequently, these conditions have been proved by Ehrenfest and especially by Burgers to be invariant for slow mechanical transformations.

To the generalisation under consideration we are naturally led, if we first consider such systems for which the motions corresponding to the different degrees of freedom are dynamically independent of each other.

This occurs if the expression for the total energy E in Hamilton’s equations (4) for a system of s degrees of freedom can be written as a sum E1 +· · ·+Es, where Ek contains qk and pk only.

An example is a particle moving in a field of force where the force-components normal to 3 mutually perpendicular fixed planes are functions of the distances from these planes respectively.

Since in such a case the motion corresponding to each degree of freedom in general will be periodic, just as for a system of one degree of

…

freedom, we may expect that the condition (10) is here replaced by a set of s conditions:

… (15)

where:

• the integrals are taken over a complete period of the different q’s respectively
• n1, . . . , ns are entire numbers

These conditions are shown to be invariant for any slow transformation of the system for which the independency of the motions corresponding to the different coordinates is maintained.

A more general class of systems with a single degree of freedom, with conditions as (15), is where each of the momenta pk is a function of qk only.

• This is done by a suitable choice of coordinates
• Here, the motions corresponding to the different degrees of freedom are not independent of each other

A simple system of this kind consists of a particle moving in a plane orbit in a central field of force.

Let:

• the length of the radius-vector from the centre of the field to the particle be q1
• the angular distance of this radius-vector from a fixed line in the plane of the orbit be q2

From (4), since E does not contain q2, we get the well known result that during the motion the angular momentum p2 is constant and that the radial motion, given by the variations of p1 and q1 with the time, will be exactly the same as for a system of 1 degree of freedom.

Sommerfeld applied the quantum theory to the spectrum of a non-periodic system.

• He assumed that the stationary states of the above system are given by 2 conditions of the form:

… (16)

The first integral must be taken over a period of the radial motion. But there is a difficulty in fixing the limits of integration of q2.

This disappears if we notice that an integral of the type under consideration will not be changed by a change of coordinates where q is replaced by some function of this variable.

If instead of the angular distance of the radius-vector, we take for q2 some continuous periodic function of this angle with period 2π, then:

• every point in the plane of the orbit will correspond to one set of coordinates only
• the relation between p and q will be exactly of the same type as for a periodic system of one degree of freedom for which the motion is of oscillating type.

It follows that the integration in the second of the conditions (16) has to be taken over a complete revolution of the radius-vector.

Consequently, this condition is equivalent with the simple condition that the angular momentum of the particle round the centre of the field is equal to an entire multiple of h 2π.

Ehrenfest pointed out that the conditions (16) are invariant for such special transformations of the system for which the central symmetry is maintained.

This follows immediately from the fact that the angular momentum in transformations of this type remains invariant, and that the equations of motion for the radial coordinate as long as p2 remains constant are the same as for a system of one degree of freedom.

On the basis of (16), Sommerfeld has, as mentioned in the introduction, obtained a brilliant explanation of the fine structure of the lines in the hydrogen spectrum, due to the change of the mass of the electron with its velocity.

To this theory we shall come back in Part 2.

Epstein and Schwarzschild pointed out that the central systems considered by Sommerfeld form a special case of a more general class of systems for which conditions of the same type as (15) may be applied.

These are the so called “conditionally periodic systems”, to which we are led if the equations of motion are discussed by means of the Hamilton-Jacobi partial differential equation.

In the expression for the total energy E as a function of the q’s and the p’s, let the latter quantities be replaced by the partial differential coefficients of some function S with respect to the corresponding q’s respectively, and consider the partial differential equation:

… (17)

obtained by putting this expression equal to an arbitrary constant α1. If then S = F(q1, . . . , qs, α1, . . . , αs) + C, where α2, . . . , αs, and C are arbitrary constants like α1, is a total integral of (17), we get, as shown by Hamilton and Jacobi, the general solution of the equations of motion (4) by putting

… (18)

where t is the time and β1, . . . , βs a new set of arbitrary constants. By means of (18) the q’s are given as functions of the time t and the 2s constants α1, . . . , αs, β1, . . . , βs which may be determined for instance from the values of the q’s and ¨q’s at a given moment.

Now the class of systems, referred to, is that for which, for a suitable choice of orthogonal coordinates, it is possible to find a total integral of (17) of the form

… (20)

where Sk is a function of the s constants α1, . . . , αs and of qk only.

In this case, in which the equation (17) allows of what is called “separation of variables”, we get from (19) that every p is a function of the α’s and of the corresponding q only.

If during the motion the coordinates do not become infinite in the course of time or converge to fixed limits, every q will, just as for systems of one degree of freedom, oscillate between two fixed values, different for the different q’s and depending on the α’s. Like in the case of a system of one degree of freedom, pk will become zero and change its sign whenever qk passes through one of these limits.

Apart from special cases, the system will during the motion never pass twice through a configuration corresponding to the same set of values for the q’s and p’s, but it will in the course of time pass within any given, however small, distance from any configuration corresponding to a given set of values q1, . . . , qs,

representing a point within a certain closed s-dimensional extension limited by s pairs of (s − 1)-dimensional surfaces corresponding to constant values of the q’s equal to the above mentioned limits of oscillation. A motion of this kind is called “conditionally periodic”. It will be seen that the character of the motion will depend only on the α’s and not on the β’s, which latter constants serve only to fix the exact configuration of the system at a given moment, when the α’s are known. For special systems it may occur that the orbit will not cover the above mentioned s-dimensional extension everywhere dense, but will, for all values of the α’s, be confined to an extension of less dimensions. Such a case we will refer to in the following as a case of “degeneration”.

Since for a conditionally periodic system which allows of separation in the variables q1, . . . , qs the p’s are functions of the corresponding q’s only, we may, just as in the case of independent degrees of freedom or in the case of quasiperiodic motion in a central field, form a set of expressions of the type

… (21)

where the integration is taken over a complete oscillation of qk. As, in general, the orbit will cover everywhere dense an s-dimensional extension limited in the characteristic way

mentioned above, it follows that, except in cases of degeneration, a separation of variables will not be possible for two different sets of coordinates q1, . . . , qs and q

…

), and since a change of coordinates of this type will not affect the values of the expressions (21), it will be seen that the values of the I’s are completely determined for a given motion of the system. By putting

Ik = nkh, (k = l, . . . , s) (22)

where n1, . . . , ns are positive entire numbers, we obtain therefore a set of conditions which form a natural generalisation of condition (10) holding for a system of one degree of freedom.

Since the I’s, as given by (21), depend on the constants α1, . . . , αs only and not on the β’s, the α’s may, in general, inversely be determined from the values of the I’s.

The character of the motion will therefore, in general, be completely determined by the conditions (22), and especially the value for the total energy, which according to (17) is equal to α1, will be fixed by them. In the cases of degeneration referred to above, however, the conditions (22) involve an ambiguity, since in general for such systems there will exist an infinite number of different sets of coordinates which allow of a separation of variables, and which will lead to different motions in the stationary states, when these conditions are applied.

As we shall see below, this ambiguity will not influence the fixation of the total energy in the stationary states, which is the essential factor in the theory of spectra based on (1) and in the applications of the quantum theory to statistical problems.