# Averaging II: Adiabatic Invariance for Integrable Systems (argued via the Averaging Principle)

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1 Averaging II: Adiabatic Invariance for Integrable Systems (argued via the Averaging Principle In classical mechanics an adiabatic invariant is defined as follows[1]. Consider the Hamiltonian system with slowly varying external parameter λ: H = H(p, q; λ, λ = ɛt (1 ṗ = H q q = H p Definition: The quantity A(p, q; λ is an adiabatic invariant of the system above if for every κ > there is an ɛ > such that if < ɛ < ɛ and < t < 1/ɛ, then A(p(t, q(t; ɛt A(p(, q(; < κ. The principle of adiabatic action invariance is that the values of the actions I for an integrable system are indeed adiabatic invariants. (Note that most systems with N > 1 are not integrable, but one has to start somewhere. huh? Here s what we are saying. Consider a hamiltonian system with N position degrees of freedom (N total degrees of freedom that is a function of some fixed external parameter λ. We require that the system be integrable (have at least N independent constants of motion for any fixed value of λ in some region, say [, 1]. Since the system is integrable, its motion (trajectory is specified by I. That is, we can construct the actions I i = p dq from the i (hollow torus that the trajectory fills out. The torus, (or equivalently, the quantity I, can be considered to be a function of the starting point of the motion (p, q in phase space. It is also a function of λ. Now consider the system where λ = ɛt. I(p(, q(; is the action associated with the torus for a constant λ specified by the initial, instantaneous values of p, q, and λ. Likewise I(p(t, q(t; ɛt is the action associated with the torus (for constant λ specified by the values of p, q, and λ at the instant t. A quantity is an adiabatic invariant if we can make its maximum devation from its initial value arbitrarily small during a process in which we change an external parameter a lot (say from to 1 simply by varying the external parameter slowly enough. (Mentally translate these words into the 1

2 definition above. Our principle is that the I i are adiabatic invariants (see Arnold s quote regarding such principles. It is straightforward to get this principle of adiabatic action invariance from the averaging principle. Following Arnold: For a fixed λ we can canonically transform the system (1 into the action-angle variable system via a generating function that depends on λ. words, there is a generating function 1 F = F (I, q; λ such that via p = F q, In other φ = F I, H new = H old ( we get the new Hamiltonian system where I and φ are the action-angle variables (meaning that I i = p dq and i ( I H = φ I =, φ = ( H ω(i, λ (3 I φ Now let λ = ɛt. Now the canonical transformation generated by (the same function F is time-dependent so that the last part of ( is changed to H new H old = ( F/ t I,q = ɛ F/ λ. Thus, we have the system of equations I = ɛf(i, φ; λ (4 φ = ω(i, λ + ɛg(i, φ; λ (5 λ = ɛ (6 where g = F I λ, f = F φ λ We apply the averaging principle. The averaged system has the form (7 J i = ɛ f i i = 1,,..., N J N+1 = ɛ (8 But f i = (π N π π ( F dφ 1... dφ N φ i λ = (do the φ i integral (9 Therefore, in the averaged system the J i components corresponding to the actions I i do not change. 1 Here we use a type generating function. If we considered the actions I to be a position (Q instead of a momentum, we could use a type 1 generating function. In this case ( would be altered.

4 shells (a N dimensional region and Ω isn t well defined. Again, same entropy (or at least uncertainty idea. II. HARMONIC OSCILLATOR EXAMPLE I = (π 1 p dq (1 H = p /m + (1/mω q = E (11 p = ± me m ω q (1 I = 1 qmax π me m ω q dq q min (it is customary to integrate in sense that I is positive = 1 qmax me 1 mω π E q dq = 1 π I = E ω q min umax=1 me 1 u 1 π/ π/ u min = 1 1 u du cos y dy, sin y = u du mω /(E, u = mω E ( cos y dy, (nd term (13 Therefore adiabatic invariance of actions predicts that as a parameter (m or ω of the HO is changed by a significant amount but very slowly, the ratio of the oscillator s energy to its frequency remains approximately constant. The direct check to see whether or not this is true for the HO involves essentially some of the same averaging and time-scale related arguments as used in our proof of the averaging theorem. Here is a quick version with only one little leap of faith. Say we are changing ω from ω to ω uniformly over a long time τ = 1/ɛ ω(t = ω (1 + ɛt( sec 1 (14 4

5 Now Ė = H t = (1/mq (ω ω = ɛmω ω(t 1Hz (15 Imagine integrating Ė over the long time τ = 1/ɛ. Since Ė is always positive we are going to get a change in energy E that is O(1. So E is definitely not an adiabatic invariant. Now check the time derivative of the action I = E/ω (I ll suppress the extra Hz unit here. d E = dt( Ė ω ω E ω ω = ɛmq ω + ɛ E ω ω (16 = ɛ ω ω (mω q E = ɛ ω (U E ω U T = ɛω (17 ω Now lets integrate d(e/ω/dt over the long time τ to get the change in I I = ɛω τ U T ω dt (18 Is I of O(1 or O(ɛ? If it is O(1, like E then I is not an adiabatic invariant. If it is O(ɛ, then we can make I arbitrarily small by making the rate of change, ɛ, arbitrily small (although the total change in the external parameter is large. Look at the integrand in (18. The 1/ω changes slowly with time. But (U T rapidly oscillates between +E and E, two oscillations per period of the HO motion. So we do not expect the quantity t (U T /ω dt to be assymptotically proportional to t and thus we expect τ (U T /ω dt to not be O(1/ɛ but to be of some smaller order such as O(1 or O(ɛ. Then by (18 we see that I is indeed O(ɛ or smaller, and it is not O(1. Thus I = E/ω is indeed an adiabatic invariant as the principle predicted. [1] V. I. Arnold, Mathematical Methods of Classical Mechanics, nd Ed., Springer-Verlag, New York, [] E. Ott, Phys. Rev. Lett. 4, 168 (1979. [3] C. Jarzynski, Phys. Rev. Lett. 71, 839 (

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Chaotic enhancement of hydrogen atoms excitation in magnetic and microwave fields Giuliano Benenti, Giulio Casati Università di Milano, sede di Como, Via Lucini 3, 22100 Como, Italy arxiv:cond-mat/9612238v1