Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 343All one–DOF Hamiltonian systems are integrable and all the solutions lie onlevel curves of the Hamiltonian function, which are topologically equivalentwith the circle S 1 . This is actually a general characteristic of all n−DOFintegrable Hamiltonian systems: their bounded motions lie on nD invarianttori T n = S 1 × · · · × S 1 , or homoclinic orbits. The homoclinic orbitis sometimes called a separatrix because it is the boundary between twodistinctly different types of motion.For example, in case of a damped Duffing oscillator (3.168) with δ ≠ 0,we have∂ q f + ∂ p g = −δ,and according to the Bendixon criterion for δ > 0 it has no closed orbits.The vector–field X given by equations (3.168) has three fixed pointsgiven by (q, p) = (0, 0), (±1, 0). The eigenvalues λ√ 1,2 of the associatedlinearized vector–field are given by λ 1,2 = −δ/2 ± 1 2δ 2 + 4, for the fixed√point (0, 0), and by λ 1,2 = −δ/2 ± 1 2δ 2 − 8, for the fixed point (±1, 0).Hence, for δ > 0, (0, 0) is unstable and (±1, 0) are asymptotically stable;for δ = 0, (±1, 0) are stable in the linear approximation (see, e.g., [Wiggins(1990)]).Another example of one–DOF Hamiltonian systems is a simple pendulum(again, all physical constants are scaled to unity), given by Hamiltonianfunction H = p22− cos q. This is the first integral of the cylindricalHamiltonian vector–field (q, p) ∈ S 1 × R, defined by canonical equations˙q = p, ṗ = − sin q.This vector–field has fixed points at (0, 0), which is a center (i.e., the eigenvaluesare purely imaginary), and at (±π, 0), which are saddles, but sincethe phase–space manifold is the cylinder, these are really the same point.The basis of human arm and leg dynamics represents the coupling oftwo uniaxial, SO(2)−joints. The study of two DOF Hamiltonian dynamicswe shall start with the most simple case of two linearly coupled linearundamped oscillators with parameters scaled to unity. Under general conditionswe can perform a change of variables to canonical coordinates (the‘normal modes’) (q i , p i ), i = 1, 2, so that the vector–field X H is given by˙q 1 = p 1 , ˙q 2 = p 2 , ṗ 1 = −ω 2 1q 1 , ṗ 2 = −ω 2 2q 2 .This system is integrable, since we have two independent functions of