Global solution of the ideal resonance problem

Abstract

If a dynamical problem ofN degress of freedom is reduced to the Ideal Resonance Problem, the Hamiltonian takes the form

$$\begin{array}{*{20}c} {F = B(y) + 2\mu ^2 A(y)\sin ^2 x_1 ,} & {\mu \ll 1.} \\ \end{array} $$

(1)

Herey is the momentum-vectory _k withk=1,2−N, x ₁ is thecritical argument, andx _k fork>1 are theignorable co-ordinates, which have been eliminated from the Hamiltonian.

The purpose of this Note is to summarize the first-order solution of the problem defined by (1) as described in a sequence of five recent papers by the author. A basic is the resonance parameter α, defined by

$$\alpha \equiv - B'/\left| {4AB''} \right|^{1/2} \mu .$$

(1)

The solution isglobal in the sense that it is valid for all values of α² in the range

$$0 \leqslant \alpha ^2 \leqslant \infty ,$$

(1)

which embrances thelibration and thecirculation regimes of the co-ordinatex ₁, associated with α² < 1 and α² > 1, respectively. The solution includes asymptotically the limit α² → ∞, which corresponds to theclassical solution of the problem, expanded in powers of ε ≡ μ², and carrying α as a divisor. The classical singularity at α=0, corresponding to an exact commensurability of two frequencies of the motion, has been removed from the global solution by means of the Bohlin expansion in powers of μ = ε^1/2. The singularities that commonly arise within the libration region α² < 1 and on the separatrix α² = 1 of the phase-plane have been suppressed by means of aregularizing function

$$\begin{array}{*{20}c} {\phi \equiv \tfrac{1}{2}(1 + \operatorname{sgn} z)\exp ( - z^{ - 3} ),} & {z \equiv \alpha ^2 } \\ \end{array} - 1,$$

(1)

introduced into the new Hamiltonian.

The global solution is subject to thenormality condition, which boundsAB″ away from zero indeep resonance, α² < 1/μ, where the classical solution fails, and which boundsB′ away from zero inshallow resonance, α² > 1/μ, where the classical solution is valid. Thedemarcation point

$$\alpha _ * ^2 \equiv {1 \mathord{\left/ {\vphantom {1 \mu }} \right. \kern-\nulldelimiterspace} \mu }$$

(1)

conventionally separates the deep and the shallow resonance regions.

The solution appears in parametric form

$$\begin{array}{*{20}c} {x_\kappa = x_\kappa (u)} \\ {y_1 = y_1 (u)} \\ {\begin{array}{*{20}c} {y_\kappa = conts,} & {k > 1,} \\ \end{array} } \\ {u = u(t).} \\ \end{array} $$

(1)

It involves the standard elliptic integralsu andE((u) of the first and the second kinds, respectively, the Jacobian elliptic functionssn, cn, dn, am, and the Zeta functionZ (u).