Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 365Action–Angle VariablesUnder the hypothesis of the Liouville Theorem, we can find symplecticcoordinates (I i , ϕ i ) such that the first integrals F i depend only on I i and ϕ i(for i = 1, ..., n) are angular coordinates on the n−torus T n ≃ M f = {x :F i (x) = f i }, which is invariant with respect to the phase–flow. We chooseangular coordinates ϕ i on M f so that the phase–flow with Hamiltonianfunction H = F 1 takes an especially simple form [Arnold (1989)]:˙ϕ i = ω i (f i ), ϕ i (t) = ϕ i (0) + ω i t.Now we look at a neighborhood of the n−manifold M f = T n in the system’s2nD phase–space P .In the coordinates (F i , ϕ i ) the phase–flow with Hamiltonian functionH = F 1 can be written in the form of the simple system of 2n ODEs˙ F i = 0, ˙ϕ i = ω i (F i ), (i = 1, ..., n), (3.180)which is easily integrated: F i (t) = F i (0), ϕ i (t) = ϕ i (0) + ω i (F i (0)) t.Thus, in order to integrate explicitly the original canonical system ofODEs, it is sufficient to find the variables ϕ i in explicit form. It turnsout that this can be done using only quadratures. A construction of thevariables ϕ i is given below [Arnold (1989)].In general, the variables (F i , ϕ i ) are not symplectic coordinates. However,there are functions of F i , which we denote by I i = I i (F i ), (i = 1, ..., n),such that the variables (I i , ϕ i ) are symplectic coordinates: the original symplecticstructure dp i ∧dq i is expressed in them as dI i ∧dϕ i . The variables I ihave physical dimensions of action and are called action variables; togetherwith the angle variables ϕ i they form the action–angle system of canonicalcoordinates in a neighborhood of M f = T n .The quantities I i are first integrals of the system with Hamiltonianfunction H = F 1 , since they are functions of the first integrals F i . Inturn, the variables F i can be expressed in terms of I i and, in particular,H = F 1 = H(I i ). In action–angle variables, the ODEs of our flow (3.180)have the form˙ I i = 0, ˙ϕ i = ω i (I i ), (i = 1, ..., n).A system with one DOF in the phase plane (p, q) is given by the Hamiltonianfunction H(p, q). In order to construct the action–angle variables,we look for a canonical transformation (p, q) → (I, ϕ) satisfying the two