General formulas for adiabatic invariants in nearly periodic Hamiltonian systems

Abstract

While it is well known that every nearly periodic Hamiltonian system possesses an adiabatic invariant, extant methods for computing terms in the adiabatic invariant series are inefficient. The most popular method involves the heavy intermediate calculation of a non-unique near-identity coordinate transformation, even though the adiabatic invariant itself is a uniquely defined scalar. A less well-known method, developed by S. Omohundro, avoids calculating intermediate sequences of coordinate transformations but is also inefficient as it involves its own sequence of complex intermediate calculations. In order to improve the efficiency of future calculations of adiabatic invariants, we derive generally applicable, readily computable formulas for the first several terms in the adiabatic invariant series. To demonstrate the utility of these formulas, we apply them to charged-particle dynamics in a strong magnetic field and magnetic field-line dynamics when the field lines are nearly closed.

Normal stability of slow manifolds in nearly-periodic Hamiltonian systems

“…Given that µ 0 = 0, Eq. (3.15) in [15] then provides µ 1 = ι R 0 ⟨ϑ 1 ⟩, where we have ϑ 1 = mv ⋅ dx. Using again the fact that Φ θ leaves x fixed, the average is simple to compute, giving ⟨ϑ 1 ⟩ = mv ⋅ bb ⋅ dx.…”

“…According to Eq. (3.14) in [15], µ 0 = ι R 0 ⟨ϑ 0 ⟩, where we have R 0 = v × b ⋅ ∂ v and ϑ 0 = qA ⋅ dx, and the angle brackets denote averaging over the U(1)-action Φ θ that is generated by R 0 . Since Φ θ in (38) leaves the x-position fixed and ϑ 0 depends only on x, we have that ⟨ϑ 0 ⟩ = ϑ 0 , and since R 0 has only velocity components, we conclude µ 0 = 0.…”

“…According to Eq. (3.14) in [15], µ 0 = ι R 0 ⟨ϑ 0 ⟩, where R 0 is defined in Lemma 8, ϑ 0 = ζ A 0 , and the angle brackets denote averaging over the U(1)-action Φ θ generated by R 0 , i.e. that given in (65).…”

“…According to Eq. (3.15) in [15], µ 1 = ι R 0 ⟨ϑ 1 ⟩, where ϑ 1 = ζ A 1 + ⟨V, dR⟩. Again using the fact that Φ θ leaves R fixed, the average is simple to compute, giving ⟨ϑ 1 ⟩ = ζ A 1 + ⟨V ∥ , dR⟩.…”

“…By working at this level of generality, our abstract results exhibit an interesting competition between stabilizing and destabilizing influences on the slow manifolds that we construct. Namely, stronger degeneracy of Ω ǫ as ǫ → 0 appears to destabilize the slow manifolds, while vanishing of early terms in the adiabatic invariant series (see [15] for explicit formulas for the first few terms) has a stabilizing effect. From this perspective, the guiding center embeddings we study are remarkable because the stabilizing and destabilizing influences balance, leading to normal stability results that would be expected for ǫ-independent symplectic manifolds.…”