Adiabatic invariance V: Multiple periods

✍ Scribed by J.E Littlewood

• Book ID: 102986653
• Publisher: Elsevier Science
• Year: 1964
• Tongue: English
• Weight: 615 KB
• Volume: 30
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1016/0003-4916(64)90307-0

✦ Synopsis

We consider a system of n linear differential equations

where the & , Bx,, are slowly varying functions of t in -* < t < m . Suppose that if the coefficients are frozen at their values for any value 7 of t the modified equations have distinct real periods 27r/wk(7). We now ask whether, subject to appropriate conditions about the slow variation, in terms of a small E, and in analogy with the results for the equation 2 + 02x = 0 ("Lorentz's pendulum problem"), "invariants" JA(t) will exist, of the usual lp dq form, in the sense that Jh(t) = Jx(O) + O(E) for all t. The answer is no. But we find that functions FA(t) exist, depending only on the coefficients, such that Jx*(t) = Fh(t),Jh(t)