Download Taste of Perturbation Theory - Classical Mechanics - Lecture Notes | PHYS 608 and more Study notes Physics in PDF only on Docsity! Lecture 26 Outline - Chaos • Chaos: intro, KAM (Section 11.1, 11.2) • Attractors, Trajectories, Liapunov (Section 11.3, 11.4) • Poincaré Maps, Hénon-Heiles (Section 11.5, 11.6) Torus don’t bore-us • For uncoupled oscillator with ω1 ω2 • Phase-space motion is confined to a surface of a torus • If ratio of ω’s is integer: closed trajectory ω2 ω1 = n thus τ1 = 2π ω1 • Also closed: for commensurate frequencies eg. n = 2 3 • if n is irrational: never repeats but will cover surface of torus (called a dense, periodic orbit) Attractors in general • Stable H0 orbits are slightly modified but maintained given perturbation H = H0 + ∆H • An important case is when init.conds. have particle not on a stable path but system evolves towards attractors: (a) stable phase-space orbit called a Poincaré limit-cycle (b) a particular fixed point in phase-space • attractor is a set of phase-space points towards which eqn soln evolves after transients die out • attractor dimension: point has dA = 0 , trajectory (limit-cycle) dA = 1 , torus dA = 2 • strange attractors are dispersed over phase-space (ie. have fractional dimension dA ) van der Pol equation • 1926 or so, is a limit-cycle attractor dA = 1 and • equation describes mechanical/electrical/cardiac systems m d2 dx2 − (1 − x2) dx dt + mω20x = F cos(ωDt) • with F = 0, = 0 simple harmonic oscillator • with = 0 simple driven ωD oscillator (with ωD close to ω0 motion at ωD ) (a) with slight damping 6= 0 motion drawn to circle (b) with more damping still maintains limit-cycle dA = 1 Onset of chaos • KAM theorem says that increase ∆H then may get • chaotic motion. Eg. consider spaceship in orbit: • Small boost, change to nearby orbit. • Large boost, successive orbits may move further away from each other, if each iteration involves greater change than previous, then orbits move apart exponentially • Liapunov exponent measures this divergence s(n) ≈ s0e nλ OR s(t) = s0e λt • If λ > 0 chaotic motion on time-scale τ ≈ 1/λ • For τ λ cannot reconstruct the distant past. • If λ < 0 characterises spiral into stable orbit. Poincaré Maps • Consider elliptical motion in 2-D plane with precession • But take stroboscopic slice through x = 0 • (we only look at motion one-way through x = 0) Hénon-Heiles Ham • Simple model of star motion about Galactic center • Constants of motion: angular momentum, energy H = p2x 2m + p2y 2m + 1 2 k(x2 + y2) + λ(x2y − 1 3 y3) • with λ = 1 via Lagrange/Hamilton’s eqn’s of motion: ẍ = −x(x + 2y) ÿ = −y − x2 + y2 • Equipotentials of V (x, y) look like: Hénon-Heiles Maps • slices through x = 0 for one trajectory • Goldstein Figure 11.8 consider different starting points • for three different (conserved) E = 1 12 , 1 8 , 1 6 • note that larger E means larger x, y cubic perturbation
