Ivancevic_Applied-Diff-Geom

344 Applied Differential Geometry: A Modern Introduction(q i , p i ), i.e., HamiltoniansH 1 = p2 12 + ω2 1(q 1 ) 2, H 2 = p2 222 + ω2 2(q 2 ) 2.2The level curves of these functions are compact sets (topological circles);therefore, the orbits in the 4D phase–space R 4 actually lie on the two–torusT 2 . By making the appropriate change of variables, it can be shown (see,e.g., [Wiggins (1990)]) that the whole dynamics of the two linearly coupledlinear undamped oscillators is actually contained in the equations˙ θ 1 = ω 1 ,˙ θ 2 = ω 2 , (θ 1 , θ 2 ) ∈ S 1 × S 2 ≡ T 2 . (3.169)The flow on the two–torus T 2 , generated by (3.169), is simple to calculateand is given byθ 1 (t) = ω 1 t + θ 10 , θ 1 (t) = ω 1 t + θ 10 , (mod 2π),and θ 1 and θ 2 are called the longitude and latitude. However, orbits underthis flow will depend on how ω 1 and ω 2 are related. If ω 1 and ω 2 arecommensurate (i.e., the equation mω 1 +nω 2 = 0, (n, m) ∈ Z has solutions),then every phase curve of (3.169) is closed. However, if ω 1 and ω 2 areincommensurate i.e., upper equation has no solutions), then every phasecurve of (3.169) is everywhere dense on T 2 .Somewhat deeper understanding of Hamiltonian dynamics is related tothe method of action–angle variables. The easiest way to introduce this ideais to consider again a simple harmonic oscillator (3.167). If we transformequations (3.167) into polar coordinates using q = r sin θ, p = r cos θ,then the equations of the vector–field become ṙ = 0, ˙θ = 1, having theobvious solution r = const, θ = t+θ 0 . For this example polar coordinateswork nicely because the system (3.167) is linear and, therefore, all of theperiodic orbits have the same period.For the general, nonlinear one–DOF Hamiltonian system (3.164) wewill seek a coordinate transformation that has the same effect. Namely, wewill seek a coordinate transformation (q, p) ↦→ (θ(q, p), I(q, p)) with inversetransformation (θ, I) ↦→ (q(I, θ), p(I, θ)) such that the vector–field (3.164)in the action–angle (θ, I) coordinates satisfies the following conditions: (i)I ˙ = 0; (ii) θ changes linearly in time on the closed orbits with ˙θ = Ω(I).We might even think of I and θ heuristically as ’nonlinear polar coordinates’.In such a coordinate system Hamiltonian function takes the formH = H(I), and also, Ω(I) = ∂ I H, i.e., specifying I specifies a periodicorbit.