Section 2

# Systems of one degree of freedom

by Niels Bohr

Systems of a single degree of freedom shows the simplest illustration of the principles in section 1.

, in which case it has been possible to establish a general theory of stationary states.

This is due to the fact that the motion will be simply periodic, provided the distance between the parts of the system will

) P. Ehrenfest, Phys. Zeitschr. XV p. 660 (1914). The above interpretation of this relation is not stated explicitly by Ehrenfest, but it presents itself directly if the quantum theory is taken in the form corresponding to the fundamental assumption I.

not increase infinitely with the time, a case which for obvious reasons cannot represent a stationary state in the sense defined above.

On account of this, the discussion of the mechanical transformability of the stationary states can, as pointed out by Ehrenfest,

) for systems of one degree of freedom be based on a mechanical theorem about periodic systems due to Boltzmann and originally applied by this author in a discussion of the bearing of mechanics on the explanation of the laws of thermodynamics. For the sake of the considerations in the following sections it will be convenient here to give the proof in a form which differs slightly from that given by Ehrenfest, and which takes also regard to the modifications in the ordinary laws of mechanics claimed by the theory of relativity. Consider for the sake of generality a conservative mechanical system of s degrees of freedom, the motion of which is governed by Hamilton’s equations: dpk dt = − ∂E ∂qk , dqk dt = ∂E ∂pk , (k = 1, . . . , s) (4) where E is the total energy considered as a function of the generalised positional coordinates q1, . . . , qs and the corresponding canonically conjugated momenta p1, . . . , ps. If the velocities are so small that the variation in the mass of the particles due to their velocities can be neglected, the p’s are 1 ) P. Ehrenfest, loc. cit. Proc. Acad. Amsterdam, XVI, p. 591 (1914). 16 defined in the usual way by pk = ∂T ∂qk , (k = 1, . . . , s) where T is the kinetic energy of the system considered as a function of the generalised velocities ˙q1, . . . , ˙qs  q˙k = dqk dt  and of q1, . . . , qs. If the relativity modifications are taken into account the p’s are defined by a similar set of expressions in which the kinetic energy is replaced by T 0 = Xm0c 2

1 − p 1 − v 2/c2  , where the summation is to be extended over all the particles of the system, and v is the velocity of one of the particles and m0 its mass for zero velocity, while c is the velocity of light. Let us now assume that the system performs a periodic motion with the period σ, and let us form the expression

``````I = ∫σ0 ∑s1 pkqkdt (5)
``````

which is easily seen to be independent of the special choice of coordinates q1, . . . , qs used to describe the motion of the system.

In fact, if the variation of the mass with the velocity is neglected we get

``````I = 2∫σ0 Tdt
``````

if the relativity modifications are included, we get a quite analogous expression in which the kinetic energy is replaced by

``````T'' = ∑ 1/2 m0v2 √ 1-v2/c2
``````

Consider next some new periodic motion of the system formed by a small variation of the first motion, but which may need the presence of external forces in order to be a mechanically possible motion. For the variation in I we get then

``````δI = ∫σ0 ∑s1 (qkσpk + pkσpk) dt + ∑s1 pkqk δt|δ0
``````

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