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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. 17

%I

%S 1,-1,7,1,-1,11,-1,319,-143,715,1,-26,559,-221,4199,-2,139,-323,6137,

%T -2261,52003,1,-10897,135983,-4199,527459,-52003,37145,-1,15409,

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

%C 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 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 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 Wolfdieter Lang, <a href="/A273506/a273506_3.pdf">Remarks on this entry and A273507</a>

%H Wolfdieter Lang, <a href="/A273506/a273506_6.pdf">Expansions for phase space coordinates for the plane pendulum</a>

%e n/m 1 2 3 4

%e ------------------------------

%e 1 | 1

%e 2 | -1, 7

%e 3 | 1, -1, 11

%e 4 | -1, 319, -143, 715

%e ------------------------------

%e 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 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 I2 = (1/(2 Pi)) Int dQ (1/2)R2(Q)^2 = 2 k + (1/4) k^2 + (3/32) k^3 + ...

%e (2/Pi) K(k) ~ (1/2)d/dk(I2) = 1 + (1/4) k + (9/64) k^2 + ...

%e From _Wolfdieter Lang_, Jun 11 2016 (Start):

%e The rational triangle r(n,m) = a(n, m) / A273507(n,m) begins:

%e n\m 1 2 3 4 ...

%e 1: 1/6

%e 2: -1/45 7/72

%e 3: 1/630 -1/30 11/144

%e 4: -1/14175 319/56700 -143/3240 715/10368

%e ... ,

%e row n = 5: 1/467775 -26/42525 559/45360 -221/3888 4199/62208,

%e row 6: -2/42567525 139/2910600 -323/145800 6137/272160 -2261/31104 52003/746496,

%e row 7: 1/1277025750 -10897/3831077250 135983/471517200 -4199/729000 527459/13996800 -52003/559872 37145/497664,

%e row 8:

%e -1/97692469875 15409/114932317500 -317281/10945935000 21586489/20207880000 -52877/4199040 7429/124416 -88711/746496 1964315/23887872.

%e ... (End)

%t R[n_] := Sqrt[4 k] Plus[1, Total[k^# R[#, Q] & /@ Range[n]]]

%t Vq[n_] := Total[(-1)^(# - 1) (r Cos[Q] )^(2 #)/((2 #)!) & /@ Range[2, n]]

%t RRules[n_] := With[{H = ReplaceAll[1/2 r^2 + (Vq[n + 1]), {r -> R[n]}]},

%t Function[{rules}, Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][

%t Flatten[R[#, Q] -> Expand[(-1/4) ReplaceAll[ Coefficient[H, k^(# + 1)], {R[#, Q] -> 0}]] & /@ Range[n]]]]

%t RCoefficients[n_] := With[{Rn = ReplaceAll[R[n], RRules[n]]}, Function[{a},

%t Coefficient[Coefficient[Rn/2/Sqrt[k], k^a],

%t Cos[Q]^(2 (a + #))] & /@ Range[a]] /@ Range[n]]

%t 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 Flatten[Numerator@RCoefficients[10]]

%t R2ToEllK[10]

%Y Denominators: A273507. Time Dependence: A274076, A274078, A274130, A274131. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.

%K sign,tabl,frac

%O 1,3

%A _Bradley Klee_, May 23 2016

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Last modified February 19 19:21 EST 2020. Contains 332047 sequences. (Running on oeis4.)