Ivancevic_Applied-Diff-Geom

340 Applied Differential Geometry: A Modern IntroductionReal 1−DOF Hamiltonian DynamicsA vector–field X(t) on the momentum phase–space manifold M can begiven by a system of canonical equations of motion˙q = f(q, p, t, µ), ṗ = g(q, p, t, µ), (3.165)where t is time, µ is a parameter, q ∈ S 1 , p ∈ R×S 1 are coordinates andmomenta, respectively, while f and g are smooth functions on the phase–space R×S 1 .If time t does not explicitly appear in the functions f and g, the vector–field X is called autonomous. In this case equation (3.165) simplifies as˙q = f(q, p, µ), ṗ = g(q, p, µ). (3.166)By a solution curve of the vector–field X we mean a map x = (q, p),from some interval I ⊂ R into the phase–space manifold M, such thatt ↦→ x(t). The map x(t) = (q(t), p(t)) geometrically represents a curve inM, and equations (3.165) or (3.166) give the tangent vector at each pointof the curve.To specify an initial condition on the vector–field X, byx(t, t 0 , x 0 ) = (q(t, t 0 , q 0 ), p(t, t 0 , p 0 )),geometrically means to distinguish a solution curve by a particular pointx(t 0 ) = x 0 in the phase–space manifold M. Similarly, it may be usefulto explicitly display the parametric dependence of solution curves,as x(t, t 0 , x 0 , µ) = (q(t, t 0 , q 0 , µ q ), p(t, t 0 , p 0 , µ p )), where µ q , µ p denoteq−depen-dent and p−dependent parameters, respectively.The solution curve x(t, t 0 , x 0 ) of the vector–field X, may be also referredas the phase trajectory through the point x 0 at t = t 0 . Its graph over t isrefereed to as an integral curve; more precisely, graphx(t, t 0 , x 0 ) ≡ {(x, t) ∈ M × R : x = x(t, t 0 , x 0 ), t ∈ I ⊂ R}.Let x 0 = (q 0 , p 0 ) be a point on M. By the orbit through x 0 , denotedO(x 0 ), we mean the set of points in M that lie on a trajectory passingthrough x 0 ; more precisely, for x 0 ∈ U, U open in M, the orbit through x 0is given by O(x 0 ) = {x ∈ R×S 1 : x = x(t, t 0 , x 0 ), t ∈ I ⊂ R}.Consider a general autonomous vector–field X on the phase–space manifoldM, given by equation ẋ = f(x), x = (q, p) ∈ M. An equilibrium solution,singularity, or fixed point of X is a point ¯x ∈ M such that f(¯x) = 0,i.e., a solution which does not change in time.