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Canonical Transformations Preserve Phase-Space Volume

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Canonical Transformations Preserve Phase-Space Volume Canonical transformations preserve phase-space volume p1 z(t) z(t1) q1 q ... p 2 n One of the most important properties of canonical transformations is that their Jacobian determinants det(∂Zi/∂zj) are always unity. That is, a region of phase space has the same volume whether represented in the old or new canonical coordinates. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Proof for continuous transformations This is most easily seen (in the case of transformations continuously connected with the identity at least) by showing that Hamiltonian flows are incompressible (see figure). To do this we calculate the divergence of z˙ = (∂H/∂p, −∂H/∂q): ∂ ∂H ∂ ∂H div z˙ = · − · ≡ 0 . (1) ∂q ∂p ∂p ∂q Then the conservation of the phase-space volume follows from the ap- plication of Gauss’ theorem since the surface integral of z˙ is the rate of change of the volume within the surface. This result obviously applies to any flows built up from infinitesimal canonical transformations, not only to Hamiltonian time evolution. (In fact the conservation of phase-space volume holds for all canonical transformations.) This result has profound implications both in statistical mechanics and in dynamical systems theory — it implies that there are no attractors in Hamiltonian dynamics: •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Invariant sets in dynamical systems An invariant set of phase space points is mapped onto itself by the dynamical evolution. We can have: • a fixed point • a periodic orbit (cycle) • an invariant surface • an invariant volume . or something much weirder In a dissipative dynamical system the phase space volume shrinks with time, and stable invariant sets are attractors (like “black holes”!). In Hamiltonian systems, phase-space points cannot be attracted to invariant sets because that would violate volume conservation—if an attracting set in a Hamiltonian system is stable, then points initially in its neighbourhood remain in the neighbourhood, but don’t get “sucked in”. If a Hamiltonian invariant set is unstable, then some points in its neig- bourhood will be repelled, but in other directions points must also be attracted so there is no net increase in volume—unstable invariant sets are hyperbolic. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Transformation to cyclic normal form It is clear from the equations of motion that if the Hamiltonian does not depend on one of the generalized coordinates qi then the canonical momentum conjugate to it, pi, is conserved. Such a coordinate is called cyclic (ignorable). This suggests a general strategy for solving Hamiltonian dynamics problems—make a canonical transformation to new canonical coor- dinates Q, P such that the new Hamiltonian is cyclic in all the new generalized coordinates K = K(P ) (we consider the autonomous case only here). Then the dynamics is very simple since all the new mo- menta are constants of the motion and thus the generalized velocities ∂K/∂Pi are also constant, so all the new generalized coordinates evolve linearly in time ∂K Qi = Qi0 + t . (2) ∂Pi If a such a well-behaved transformation exists, the system is known as integrable. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Invariant tori in integrable systems If the problem is such that the orbits remain in a bounded region of phase space, then the old coordinates have to oscillate in a quasiperiodic fashion. But eq. (2) shows that the new coordinates, if they exist, increase secularly like the angle in a rotating pendulum. In fact by suitably normalizing the Qi we have “action-angle” coordinates: K = K(J) ⇒ θ = θ0 + ωt , (3) where the new generalized coordinates θ are angles (e.g. θ + 2πm represents topologically the same point in phase space as θ, with m an array of integers). The new momenta, which we have denoted J, are called the actions. The angular velocity is defined by ω(J) ≡ ∂K/∂J. Then any set of points J = const is an invariant set, because J re- tains its initial value under the dynamics: J˙ = ∂K/∂θ ≡ 0 (it is a constant of the motion). Such a set is an n-torus (embedded in the 2n-dimensional phase space) because specifying θ uniquely determines where we are on the invariant set—it is an invariant torus. The phase space is filled (foliated) with these invariant tori. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Invariant tori vs. chaos Survival of invariant tori—KAM Theorem Suppose we perturb an integrable system with Hamiltonian H0: H = H0 + H1. Then we can try to find new action-angle coordi- nates, but in general this procedure breaks down on resonant tori, where there exists an integer array m such that m·ω = 0. (We shall see later what happens at resonance in the example of the kicked rotor.) However, if we choose to study invariant tori that are “sufficiently far” from resonance, then Kolmogorov, Arnol’d and Moser (KAM) showed that some invariant tori survive, even in nonintegrable systems, provided is sufficiently small. Destruction of invariant tori—chaos When exceeds a critical value, the KAM torus breaks up and chaotic orbits can diffuse through the remnant invariant set (a Cantor set). The chaotic orbits fill an invariant volume and separate exponentially fast, so memory of initial conditions is rapidly lost. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Area-preserving map: kicked rotor Consider the period map of a periodically forced 1-degree-of-freedom 1 system (sometimes called a 1 2 degree of freedom Hamiltonian system). Then the phase-space is only 2-dimensional and the conserved “vol- ume” is in fact an area. I.e. the period map is area preserving. Consider a pendulum (rotor): in which gravity is applied purely im- z pulsively in a periodic fashion: ∞ X g(t) = g ∆t δ(t − t ) (4) 0 n θ l n=−∞ ( 1−cosθ ) l where tn ≡ n∆t, with n an integer, δ t δ 0 and ( ) is the Dirac function, so mg that g0 is the time average of g(t): •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Trial function The Hamiltonian is p2 H(θ, p, t) = + mg(t)l(1 − cos θ) . (5) 2ml2 Between the kicks the motion is one of uniform rotation with constant angular velocity p/ml. p(t) That is, θ is a continuous, piecewise pn linear function of t and p a piecewise θ(t) θ ε n constant step function with only the g(t) ∆t values θn at, and pn immediately be- tn−1 tn tn+1 t fore, t = tn as yet unknown: 1 θ(t) = [(t − t)θ + (t − t )θ ] , ∆t n+1 n n n+1 p(t) = pn+1 (6) for t in each range tn ≡ n∆t < t < tn+1 ≡ (n + 1)∆t. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Variational solution Inserting this trial function in the phase-space action integral over the interval t−N + ε ≤ t ≤ tN + ε, N ≥ 1, ε → 0+, we get N−1 2 X pn0+1∆t S = p 0 (θ 0 − θ 0) − − mg l∆t (1 − cos θ 0 ) . ph n +1 n +1 n 2ml2 0 n +1 n0=−N (7) The conditions for Sph to be stationary under variations of θn and pn+1, −N < n < N, (noting that θn occurs in the terms of the sum for both n0 = n and n0 = n − 1) are ∂S p ∆t ph = θ − θ − n+1 = 0 , (8) n+1 n 2 ∂pn+1 ml and ∂Sph = pn − pn+1 − mg0l∆t sin θn = 0 . (9) ∂θn These equations define an area-preserving map known as the Standard Map: •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Standard Map Commonly the map is re-expressed in terms of an angle coordinate x with period normalized to unity and a nondimensionalized momentum y, which is p expressed in units of 2πml2/∆t, θ = 2πx , 2πml2 p = y . (10) ∆t Then the map becomes k y = y − sin 2πx , n+1 n 2π n xn+1 = xn + yn+1 , (11) where the chaos parameter k is defined by g (∆t)2 k ≡ 0 . (12) l •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Poincar´eplot Some typical iterated orbits of the map, showing both chaotic and 1 regular regions. The remnants of the librating pendulum orbits around the O points are seen as large “islands” around (x, y) = (0, 0) and (0, 1) [also (1, 0) and (1, 1) because the phase space is periodic in y as well as x], but y the region around the pendulum X point is highly chaotic. There are also new islands due to reso- nances not present in the physical pendulum problem (in fact an in- finite number of them), each with 0 chaotic separatrices: 01x •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Green’s critical k The value of k used in the Poincar´eis kc = 0.971635406 ..., which is the value at which the last KAM invariant curves with the topology of a rotating pendulum orbit become unstable and break up into invariant Cantor sets. Two of these KAM curves are shown, being the only curves completely crossing the figure from left boundary to right. For k < kc the motion is bounded in y, while for k greater than this value a phase-space point can diffuse without bound in the positive or negative y-direction. From eq. (12) we see that the integrable limit ∆t → 0 corresponds to k → 0 [noting that eq.
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