where the components of the forces Fk are obtained from U by differentiation.
2. In Einstein’s relativistic mechanics we haveL = T× − U
where
(4)
and m0k is the rest-mass, vk the magnitude of the velocity, and c the velocity of light. This T× is different from the kinetic energy
(5)
In this case the equations of motion can also be written in the form (3) if the mass depends on the velocity according to the law:
(6)
3. If magnetic forces are acting on the system we have,
(7)
where e k is the charge of the particle and Ak the vector potential of the magnetic field for that configuration of the system.
The equations of motion are of the second order with respect to time and it is often convenient to transform them into twice as many equations of the first order. This has been done by Hamilton in the following symmetrical way:
Introducing as unknown functions besides the coördinates the momenta
(8)
and using instead of L(q1, q1, q2, q2...) the function
(9)
the Principle of Least Action can be written
(10)
and the equations of Euler-Lagrange take the symmetrical form
(11)
Equations (11) are also true if the Hamilton function H depends explicitly on the time t. If this is not the case we find
or
(12)
We shall now discuss the physical meaning of H in the same three cases considered above:
1. In the mechanics of Galileo and Newton, where T is a homogeneous quadratic function of the components of the velocities we have, according to Euler’s theorem,
and therefore, according to (9) and since L = T−U,H = T+U.
H is therefore the total energy and (12) is the law of conservation of energy. This holds only for “inertial systems” and not for accelerated systems of coördinates. In such a system, for instance in a rotating system, H is constant, but does not represent the energy.
2. In reltitivistic mechanics we find by a simple calculation
whence H is also the total energy.
If it is desired to express this in terms of the momenta, we find, combining the momenta corresponding to the components of velocity with a vector momentum
and by elimination of vk
(13)
3. In a magnetic field we do not have a simple proportionality between velocity and momentum, but
and even in this case H is the total energyH = T + U.
Introducing the momenta we find a rather complicated expression
(14)
Before beginning with the general integration of the canonical equations, we shall consider some simple examples. If the Hamiltonian function, H, is independent of one coördinate, for example of q1,H = H(p1, q2, p2 · · ·t),
we get from the canonical equationsṗ1 = 0
therefore,p1 = const.,
and so we have found an integral of these equations. Such is the case, for example, if q1 is the angle of rotation about an axis passing through the center of gravity of a solid body, therefore the coördinate is called a “cyclic” variable. In this case it is easily shown that p 1 is the moment of momentum of the system about the axis.
It may happen that H is independent of all the qk’sH(p1 p2 · · ·t)
then the canonical equations are completely integrated by the formulas
(15)
where the ωk’s are characteristic constants of the system, α k and βk constants of integration.
We see that the mechanical problem is solved if we can find such coördinates that H depends only on the momenta. This is the method of integration which we shall use in the following. The difficulty now is that variables of this type cannot be found by means of a simple point transformation of qk, but only by a simultaneous transformation of qk and pk.
We shall now find all the transformations of pk and qk which do not change the form of the canonical equations. Such transformations are called “canonical transformations.” This condition is evidently fulfilled if the Principle of Least Action (1) does not change its form by a transformationpk = pk(q1, q2··· p1, p2···t)
andqk = qk(q1, q2, ···p1, p2,···t);
in other words, if the sum
differs from the corresponding expression in the new coördinates by a quantity which is a total differential of the time. We must set, therefore,
(16)
This equation is easily satisfied. Let us choose for V an arbitrary function of the new and old coördinates and of the time
We obtain by comparison of the coefficients of and :
(17)
Expressing qk, pk in terms of qk, pk we obtain the desired equations of transformation. But we can give to these canonical transformations several other forms, by using, instead of qk, , other independent variables. There are in all four such combinations possible from which we select the common case where qk , pk are used as independent variables. To do this we write instead of V
which is evidently, like V, an arbitrary function, and consider here V as a function of qk, pk. Then we obtain
and therefore, by comparison of the coefficients,
(18)
We illustrate this equation by a few simple examples:
The functionV = q1p1 + q2p2
gives the identical transformationq1 = q1, p1 = p1, q2 = q2, p2 = p2.
The functionV = q1p1 ± q1p2 + q2p2
gives
For three pairs of variables the functionV = q1(p1 + p2 + p3) + q2(p1 + p3) + q3p3
gives the transformation
In these examples the coördinates and impulses are transformed among themselves. The general condition is that V shall be a linear function of q and p
Then we have
If βi and γi vanish we have
This transformation is linear, homogeneous and contragredient. To this group belongs the case of orthogonal transformations, for instance the rotation of rectangular coördinates.
We obtain a point transformation, that is a transformation of the qk’s among themselves, when V is linear in p:
that is
and we have corresponding relations for the momenta.
As an example we shall give the transformation of rectangular coördinates into polar coördinates. Here we place,−V = pxr cosφ sinθ + pyr sinφ sinθ + pzr cosθ.
Then we have
and the expression p2x + p2y + p2z is transformed into
As an example of the first form given to the canonical transformation, where V depends on q and q, we choose
Then we have
or
Hence the expression
is transformed into cp.
This example can be used to explain how the canonical transformations are employed in the integration of the equations of motion. For this we consider the harmonic oscillator in which
Therefore
If in the last transformation given we place c2 = mκ, H is transformed to cp/m. This is the solution of the problem. For now q= φ is a cyclic variable and we have
In the original coördinates the motion is represented by
LECTURE 3
The Hamilton-Jacobi partial differential equation—Action and angle variables—The quantum conditions.
In the same way we can now consider the most general case. Let us suppose that H does not depend explicitly on t. We shall denote constant momenta by αk, the new variables which are linear functions of the time by φk, the number of degrees of freedom by f. Then we have to determine a functionS(q1, q2...qf, α1, α2...αf)
so that, by the transformation,
(1)
H becomes a function depending only on the αk’s,w(α1,a2...αf).
Replacing pk by its value W(α1,a2...αf). H(q1,q2... p1, p2...)
we obtain the condition
(2)
This expression can be looked upon as a partial differential equation for the
determination of S. The problem is now to determine a so-called complete integral of this equation, that is, an integral which depends of f−1 arbitrary constants α2... αf, where α1 is to be identified with W, or, if no particular constant α1 is to be privileged in this manner, then we must find an integral which depends on f constants α1...αf, among which there exists a relationW = W(α1...αf).
The motion is then represented by
(3)
We shall call Equation (2) the Hamilton-Jacobi differential equation and S the action-function. An important property of S is the following: We have
Therefore S is a line integral, taken along the orbit, from a fixed point Q0 to a moving point Q.
(4)
In Galilean-Newtonian mechanics this has a simple significance, because in this case,
and we have
(5)
where is the time average of T.
We have seen already that the quantum theory is closely related to the periodic properties of the motion. In fact, Bohr’s theory permits the definition of stationary states only for such motions as can be decomposed by harmonic analysis into periodic components. The astronomers call this class of motions “conditioned periodic.” We prefer to call them “multiple periodic.” These motions are defined in the following way: It is possible to introduce instead of variables qk, pk new variables wk, Ik by means of the canonic transformation
which satisfy the following conditions:
(A) The position of the system depends periodically on wk, with the fundamental period 1. That is, if the qk’s are uniquely determined by the position of the system, then they can be expanded in a Fourier series:
where τ represents a number of integers τ1, τ2...τf and we place(wτ) = w1τ1 + w2τ2 +···+ wfτf.
If one of the qk’s is an angle, it is not uniquely determined by the position of the system, but only within a multiple of a constant, as for instance 2π. Then the above condition of periodicity is also true except for a multiple of that constant.
(B) Hamilton’s function can be transformed into a function W which depends only on the Ik’s.
It follows that the I’s are constants and the w’s are linear functions of the time t,wk = νkt + βk.
The q’s can therefore be represented by trigonometric series in t with the frequenciesν1τ1 + ν2τ2 +···+ νfτf
where, according to the results obtained above,
wk, Ik are not yet uniquely determined by these conditions. For instance we can setwk = wk + f(I1...If)
andĪk = Ik + Ck.
These form a canonic transformation, which is evidently compatible with the conditions (A) and (B). In order to exclude this indetermination we further set the condition:(C) The function S× = S −
shall be periodic in wk with the period 1:
The canonic transformation in question can also be expressed by means of the function Sx as follows:
Then indeed we can prove rigorously that wk, Ik , which are called angle and action variables, are essentially uniquely determined by the conditions (A), (B) and (C). “Essentially” expresses the following: If we make a canonic transformation of the form
where the c’s are whole numbers and the determinant |c kl | = ±1, all the conditions (A), (B), (C) are still satisfied. Aside from this indetermination, however, wk, Ik are really uniquely determined in all cases when the mechanical system is not degenerate, that is when there is no identical relation in νk of the formν1τ1 + ν2τ2 + ···+ νfτf = 0
with the τ’s whole numbers.
This theorem was first given by Burgers but his proof is not sufficient. A rigorous proof can be found in my book “Atommechanik”; this proof was given by my associate, F. Hund. This arbitrariness in the determination of the Ik ’s, whereby the latter are determined except for a whole-number transformation of determinant ± 1, is of essential importance for the applications of the quantum theory, for it is just these quantities that are equated to multiples of Planck’s constant h; i.e.,I1 = n1h , I2 = n2h , . . . I f = nfh,
and from these equations it follows that also the Īk’s are multiples of h.
LECTURE 4
Adiabatic invariants—The principle of correspondence.
In order to justify this method of quantization, it must be shown in the first place that the I’s are adiabatic invariants. The general proof of this theorem was first outlined by Burgers and also by Krutkow; later more rigorous proofs were given by von Laue, Dirac and also by Jordan and myself. I shall not give here these rather complicated considerations, but shall only explain the significance of I and its adiabatic invariance using the example of the harmonic resonator. Using the Hamiltonian function,
and then applying a canonic transformation, we have found above a solution of the problem of the resonator, which, although not quite satisfying the conditions (A), (B), (C), is easy to transform into one which fulfils these conditions. It is only necessary to place
Then the transformation is
and the energy-function becomes
whereω = 2π ν
and also
As q is periodic in w with the period 1, and as H depends only on I, therefore the conditions (A) and (B) are fulfilled. To see whether condition (C) is also satisfied, we must only remember that the canonic transformation was found through the function
and then through the formulas,
This V is therefore identical with the Sx introduced above. It can be written in the form
and, since it is periodic, the condition (C) is also fulfilled.
The quantum conditionI = nh
gives therefore the energy levels,W = nhν
FIG. 2
in agreement with Planck’s hypothesis. In order to verify that I = W/ν is really an adiabatic invariant, we represent the resonator by a pendulum swinging with small amplitude. Let m be the mass of the bob, l the length of the wire and g the acceleration of gravity. Suppose now that the length l is changed very slowly: the problem is to calculate how W and ν vary. The forces which stretch the wire for any value of the angle φ are the component of gravity mg cos φ = mg (1 − φ 2 /2) and the centrifugal force mlΦ2. The work done in shortening the wire is therefore
(1)
If the process of shortening is slow enough and has no period comparable with that of the pendulum, then it is permissible to introduce a mean amplitude, and we may write
where the dash denotes average over a period. The work done is now split up in two parts:—mgdl is the work done in lifting the bob, and
is the increase in the energy of oscillation. Now we know that for harmonic oscillations:
and hence
Now, since ν is proportional to 1/√l, therefore dν = − dl 2l and
whence, by integration,
(2)
which proves our theorem. The general proof of adiabatic invariance consists essentially of quite analogous considerations.
As another important example let us consider the rotator, that is, a body which can be rotated about an axis. If A is the moment of inertia with respect to the axis and φ the angle of rotation, then we have
whence it follows, for the momentum p corresponding to φ p = AΦ.
p is the angular momentum and we have
φ is therefore a cyclic variable andp = constant.
If we set φ = 2πw, the position of the system is a periodie function of w of period 1. The canonic transformation(φ, p ) → (w, I)
is evidently characterized by the function S = φI/2π and has the form
whence it follows that Sx = S—wI = 0 is a periodic function. Finally, we obtain
The conditions (A), (B), (C) are fulfilled and we have to setI = hν,
which gives the energy levels
(3)
The Dreams That Stuff is Made of Page 79