I'm a little confused by the discussion in the last section $\S 50$ of Landau and Lifshitz's (Classical) Mechanics (1960, first English ed.). Here, they consider finite motion of a system whose Hamilatonian is separable (for a Hamilton-Jacobi treatment) and apply the action angle formalism.

As they explain, the abbreviated action can be written as a sum of functions (given the variables' separability):

$$S_0=\sum\limits_i S_i(q_i), \tag{52.1} $$

where each $S_i$ can be written $\int p_i $d$q_i$. They then say

Since the motion is finite, each co-ordinate can take values only in a finite range. When $q_i$ varies 'there and back' in this range, the action increases by $$\Delta S_0=\Delta S_i = 2\pi I_i\tag{52.3} $$ where $$I_i\equiv\oint p_i\text{d}q_i/2\pi,\tag{52.4} $$ the integral being taken over the variation just mentioned.

A footnote here on p. 163 then reads,

It should be emphasised, however, that this refers to the formal variation of the coordintae $q_i$ over the whole possible range of values, not to its variation during the period of the actual motion as in the case of motion in one dimension. An actual finite motion of a system with several degree of freedom not only it not in general periodic as a whole, but does not even involve a periodic time variation of each co-ordinate separately."

They then define the action variables as

$$\omega_i=\frac{\partial S_o}{\partial Ii} \tag{52.5}$$

so that over a 'there and back' variation of the $q_i$ corresponds to a change $$\Delta \omega_i=2\pi. \tag{52.8}$$

Now what I'm having trouble with is understanding why the variation of a $q_i$ across the range of its values has any significance in describing the actual evolution of the system (assuming it isn't periodic).

Note: Perhaps I'm simply misunderstanding what they mean by "finite motion" here. I'm taking this to be applicable in any case where you know a finite range of the $q_i$ for some relevant temporal duration. If what they mean is that the possible values of the $q_i$ are bound in some region for all time, then I can see how things would necessarily be conditionally periodic. (I'm thinking of something like Poincare's recurrence theorem here.) However, in this case my question would be whether one can generalise this to all systems in some temporal evolution, modelling the evolution as some portion of a larger periodicity?


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