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A273506 T(n,m), numerators of coefficients in a power/Fourier series expansion of the plane pendulum's exact phase space trajectory.

%I A273506 #67 May 13 2017 23:45:06
%S A273506 1,-1,7,1,-1,11,-1,319,-143,715,1,-26,559,-221,4199,-2,139,-323,6137,
%T A273506 -2261,52003,1,-10897,135983,-4199,527459,-52003,37145,-1,15409,
%U A273506 -317281,21586489,-52877,7429,-88711,1964315,1,-76,269123,-100901,274873,-8671,227447,-227447,39803225,-2,466003,-213739,522629,-59074189,226061641,-10690009,25701511,-42077695,547010035
%N A273506 T(n,m), numerators of coefficients in a power/Fourier series expansion of the plane pendulum's exact phase space trajectory.
%C A273506 Triangle read by rows ( see examples ). The phase space trajectory of a simple pendulum can be written as (q,p) = (R(Q)cos(Q),R(Q)sin(Q)), with scaled, canonical coordinates q and p. The present triangle and A273507 determine a power / Fourier series of R(Q): R(Q) = sqrt(4 *k) * (1 + sum k^n * (A273506(n,m)/A273507(n,m)) * cos(Q)^(2(n+m)) ); where the sum runs over n = 1,2,3 ... and m = 1,2,3...n. The period of an oscillator can be computed by T(k) = dA/dE, where A is the phase area enclosed by the phase space trajectory of conserved, total energy E. As we choose expansion parameter "k" proportional to E, the series expansion of the complete elliptic integral of the first kind follows from T(k) with very little technical difficulty ( see examples and Mathematica function R2ToEllK ). For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).
%C A273506 For some remarks on this pendulum problem and an alternative way to compute a(n,m) / A273507(n,m) using Lagrange inversion see the two W. Lang links. - _Wolfdieter Lang_, Jun 11 2016
%H A273506 Bradley Klee, <a href="http://arxiv.org/abs/1605.09102">Plane Pendulum and Beyond by Phase Space Geometry</a>, arXiv:1605.09102 [physics.class-ph],  2016.
%H A273506 Wolfdieter Lang, <a href="/A273506/a273506_3.pdf">Remarks on this entry and A273507</a>
%H A273506 Wolfdieter Lang, <a href="/A273506/a273506_6.pdf">Expansions for phase space coordinates for the plane pendulum</a>
%e A273506 n/m  1    2     3     4
%e A273506 ------------------------------
%e A273506 1  |  1
%e A273506 2  | -1,  7
%e A273506 3  |  1, -1,    11
%e A273506 4  | -1,  319, -143, 715
%e A273506 ------------------------------
%e A273506 R2(Q) = sqrt(4 k) (1 + (1/6) cos(Q)^4 k +  (-(1/45) cos(Q)^6 + (7/72) cos(Q)^8) k^2)
%e A273506 R2(Q)^2 = 4 k + (4/3) cos(Q)^4 k^2 + ( -(8/45) cos(Q)^6 + (8/9) cos(Q)^8)k^3 + ...
%e A273506 I2 = (1/(2 Pi)) Int dQ (1/2)R2(Q)^2 = 2 k + (1/4) k^2 + (3/32) k^3 + ...
%e A273506 (2/Pi) K(k) ~ (1/2)d/dk(I2) = 1 + (1/4) k + (9/64) k^2 + ...
%e A273506 From _Wolfdieter Lang_, Jun 11 2016 (Start):
%e A273506 The rational triangle r(n,m) = a(n, m) / A273507(n,m) begins:
%e A273506 n\m   1          2          3         4   ...
%e A273506 1:   1/6
%e A273506 2: -1/45        7/72
%e A273506 3:  1/630      -1/30      11/144
%e A273506 4: -1/14175   319/56700 -143/3240  715/10368
%e A273506 ... ,
%e A273506 row n = 5: 1/467775 -26/42525 559/45360 -221/3888 4199/62208,
%e A273506 row 6: -2/42567525 139/2910600 -323/145800 6137/272160 -2261/31104 52003/746496,
%e A273506 row 7: 1/1277025750 -10897/3831077250 135983/471517200 -4199/729000 527459/13996800 -52003/559872 37145/497664,
%e A273506 row 8:
%e A273506 -1/97692469875 15409/114932317500 -317281/10945935000 21586489/20207880000 -52877/4199040 7429/124416 -88711/746496 1964315/23887872.
%e A273506 ... (End)
%t A273506 R[n_] := Sqrt[4 k] Plus[1, Total[k^# R[#, Q] & /@ Range[n]]]
%t A273506 Vq[n_] :=  Total[(-1)^(# - 1) (r Cos[Q] )^(2 #)/((2 #)!) & /@ Range[2, n]]
%t A273506 RRules[n_] :=  With[{H = ReplaceAll[1/2 r^2 + (Vq[n + 1]), {r -> R[n]}]},
%t A273506 Function[{rules}, Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
%t A273506    Flatten[R[#, Q] ->  Expand[(-1/4) ReplaceAll[ Coefficient[H, k^(# + 1)], {R[#, Q] -> 0}]] & /@ Range[n]]]]
%t A273506 RCoefficients[n_] :=  With[{Rn = ReplaceAll[R[n], RRules[n]]}, Function[{a},
%t A273506     Coefficient[Coefficient[Rn/2/Sqrt[k], k^a],
%t A273506        Cos[Q]^(2 (a + #))] & /@ Range[a]] /@ Range[n]]
%t A273506 R2ToEllK[NMax_] := D[Expand[(2)^(-2) ReplaceAll[R[NMax], RRules[NMax]]^2] /. {Cos[Q]^n_ :> Divide[Binomial[n, n/2], (2^(n))], k^n_ /; n > NMax -> 0},k]
%t A273506 Flatten[Numerator@RCoefficients[10]]
%t A273506 R2ToEllK[10]
%Y A273506 Denominators: A273507. Time Dependence: A274076, A274078, A274130, A274131. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.
%K A273506 sign,tabl,frac
%O A273506 1,3
%A A273506 _Bradley Klee_, May 23 2016