Physics
Scientific paper
Sep 1973
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1973cemec...8..207g&link_type=abstract
Celestial Mechanics, Volume 8, Issue 2, pp.207-212
Physics
1
Scientific paper
If a dynamical problem ofN degress of freedom is reduced to the Ideal Resonance Problem, the Hamiltonian takes the form 1 begin{array}{*{20}c} {F = B(y) + 2μ ^2 A(y)sin ^2 x_1 ,} & {μ ≪ 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 1 α equiv - B'/left| {4AB''} right|^{1/2} μ . The solution isglobal in the sense that it is valid for all values of α2 in the range 1 0 ≤slant α ^2 ≤slant infty , 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 ɛ ≡ 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 = ɛ1/2. 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 1 begin{array}{*{20}c} {φ equiv tfrac{1}{2}(1 + operatorname{sgn} z)exp ( - z^{ - 3} ),} & {z equiv α ^2 } \ - 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 1 α _ * ^2 equiv {1 {/ {{1 μ }} μ } conventionally separates the deep and the shallow resonance regions. The solution appears in parametric form 1 begin{array}{*{20}c} {x_kappa = x_kappa (u)} \ {y_1 = y_1 (u)} \ {begin{array}{*{20}c} {y_kappa = conts,} & {k > 1,} \ } \ {u = u(t).} \ 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).
No associations
LandOfFree
Global Solution of the Ideal Resonance Problem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Global Solution of the Ideal Resonance Problem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Global Solution of the Ideal Resonance Problem will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-1589540