Adiabatic Invariants
المؤلف:
Sidney B. Cahn And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 31
25-7-2016
1462
Adiabatic Invariants
a) (Adiabatic Invariants) Consider a system with canonical variables
At the time t = 0 let C0 be an arbitrary closed path in phase space and
Assume that the point p, q of C0 moves in phase space according to Hamilton's equations. At a later time the curve C0 will have become another closed curve Ct. Show that
and, for a harmonic oscillator with Hamiltonian H = (p2/2m) + (mω2q2/2) show that
along a closed curve H = (p, q) = E.
b) (Dissolving Spring) A mass m slides on a horizontal frictionless track. It is connected to a spring fastened to a wall. Initially, the amplitude of the oscillations is A1 and the spring constant of the spring is K1. The spring constant then decreases adiabatically at a constant rate until the value K2 is reached. (For instance, assume that the spring is being dissolved in acid.) What is the new amplitude?
Hint: Use the result of (a).
SOLUTION
a)
For a harmonic oscillator H(p, q) = E
This trajectory in phase space is obviously an ellipse:
With
(1)
The adiabatic invariant
where we transformed the first integral along the curve into phase area integral which is simply I = σ/2π, where σ is the area of an ellipse σ = πAB So, taking A and B from (1) gives
b) The fact that the spring constant decreases adiabatically implies that although the energy is not conserved its rate of change will be proportional to the rate of change in the spring constant: It can be shown that in this approximation the quantity found in (a)—the so-called adiabatic invariant—remains constant. Our spring is of course a harmonic oscillator with frequency
and energy E = (1/2)KA2 So we have
(2)
or
So from (2), the new amplitude is
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