A276815 Irregular triangle read by rows T(n,m), coefficients in power/Fourier series expansion of an arbitrary anharmonic oscillator's exact differential time dependence.

3, 4, -24, 5, -70, 210, 6, -96, -48, 960, -1920, 7, -126, -126, 1386, 1386, -12012, 18018, 8, -160, -160, 1920, -80, 3840, -17920, 640, -26880, 143360, -172032, 9, -198, -198, 2574, -198, 5148, -25740, 2574, 2574, -77220, 218790, -25740, 437580, -1662804, 1662804, 10, -240, -240, 3360, -240, 6720, -35840, -120, 6720, 3360

Offset: 1

Author

Bradley Klee, Sep 18 2016

Keywords

• sign
• tabf

Comments

The phase space trajectory A276738 has phase space angular velocity A276814, which allows expansion of dt = dx /(dx/dt) = dx(-1 + sum b^n*T(n,m)*f(n,m)); where the sum runs over n = 1, 2, 3 ... and m = 1, 2, 3, ... A000041(n). The basis functions f(n,m) are the same as in A276738. To obtain period K, we integrate the function of Q=cos[x] over a range of [2*pi,0]. All odd powers of Q integrate to zero, so the period is an expansion in E=(1/2)*b^2 (Cf. A276816). This sequence transforms into A274076/A274078 by setting v_i=0 for odd i, v_i=(-1)^(i/2-1)/2/(i!) otherwise, and (1/2)*b^2 = 2*k. For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

Examples

			n/m  1    2     3     4      5      6      7
------------------------------------------------
1  | 3
2  | 4   -24
3  | 5   -70    210
4  | 6   -96   -48   960   -1920
5  | 7   -126  -126  1386   1386  -12012  18018
------------------------------------------------
		

Links

• Bradley Klee, Plane Pendulum and Beyond by Phase Space Geometry, arXiv:1605.09102 [physics.class-ph], 2016.

Crossrefs

Arbitrary Oscillator: A276738, A276814, A276816, A276817.

Pendulum: A273506, A273507, A274076, A274078, A274130, A274131, A038534, A056982, A000984, A001790, A038533, A046161, A273496.

Programs

Mathematica

R[n_] := b Plus[1, Total[b^# R[#, q] & /@ Range[n]]]
Vp[n_] := Total[2 v[# + 2] q^(# + 2) & /@ Range[n]]
H[n_] := Expand[1/2*r^2 + Vp[n]]
RRules[n_] :=  With[{H = Series[ReplaceAll[H[n], {q -> R[n] Q, r -> R[n]}], {b, 0, n + 2}]},  Function[{rules},
    Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
   Flatten[R[#, q] -> Expand[-ReplaceAll[ Coefficient[H, b^(# + 2)], {R[#, q] -> 0}]] & /@ Range[n]]]]
xDot[n_] := Expand[Normal@Series[ReplaceAll[ Q^2 D[D[q[t], t]/q[t], t], {D[q[t], t] -> R[n] P, q[t] -> R[n] Q, r -> R[n], D[q[t], {t, 2}]
->  ReplaceAll[D[-(q^2/2 + Vp[n]), q], q -> R[n] Q]} ], {b, 0, n}] /. RRules[n] /. {P^2 -> 1 - Q^2}]
dt[n_] := Expand[Normal@Series[1/xDot[n], {b, 0, n}]]
basis[n_] :=  Times[Times @@ (v /@ #), Q^Total[#],2] & /@ (IntegerPartitions[n] /. x_Integer :> x + 2)
TriangleRow[n_, fun_] := Coefficient[fun, b^n #] & /@ basis[n]
With[{dt10 = dt[10]}, TriangleRow[#, dt10] /. v[_] -> 0 & /@ Range[10]]