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In the classic & famous book "Electromagnetic fields & Interactions" by Richard Becker (Dover publishing), on page 55 (of volume 2) , author says:

If the system possesses f degrees of freedom, each orbit is specified by f-1 constants in addition to energy.

My question is: From where do these f-1 constants come, and why f-1 ?

Next, on page 56 (of volume 2), author includes these f-1 constants in quantum condition, along with co-ordinates and energy.

$J_r = (1/2\pi) \int p_r(q_r,E,c1,c2,....,c_{f-1})dq_r = n_r(h/2\pi)$

where $ r=1,2,.....,f$

My question is: What is justification for this inclusion ? In other words, why are they included in function $p_r$ ?

Any help will be highly appreciated.

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Let a $d$-dimensional Hamiltonian system (i.e. $2d$-dimensional phase space) be given. Then the existence of $d-1$ observables $c_1,\dots,c_{d-1}$ that are in involution with each other and with the Hamiltonian means that there are $d$ constants of motion along each orbit - the Hamiltonian itself and the value of $c_1,\dots,c_{d-1}$.

Systems for which such a maximal system of observables in involution exists are called Liouville integrable, and the Liouville-Arnold theorem says that for every such system there are action-angle coordinates, in which the phase space is foliated into $d$-dimensional tori whose angular coordinates are the "angles", and each torus is labeled by the values of $H,c_1,\dots,c_{d-1}$ on it.

Now, on a torus labeled by specific values for $H,c_1,\dots,c_{d-1}$, choose any one motion on it and parametrize it by the coordinate $q^k$ (split it into parts if $q^k$ can't injectively parametrize the curve, which will in general be the case). This expresses $p_k$ along the trajectory as a function of $q^k$. It is now a fact that the integral $$ \int p_k(q^k)\mathrm{d}q^k$$ does not depend on the motion chosen, only on the torus we are on. The geometric meaning of this integral may be seen as the area of the projection of the motion onto the $p_k$-$q^k$-plane. Therefore, it would be more prudent to write $J_k = \frac{1}{2\pi}\int p_k(q^k)\mathrm{d}q^k$ as a function of $H$ and $c_1,\dots,c_{d-1}$ instead.

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  • $\begingroup$ I am not mathematician. I did not understand what you wrote. Maybe I have to read about Hamiltonian dynamics first. $\endgroup$
    – atom
    Apr 26, 2016 at 4:04

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