A276814 Irregular triangle read by rows T(n,m), coefficients in power/Fourier series expansion of an arbitrary anharmonic oscillator's exact phase space angular velocity.

-3, -4, 6, -5, 22, -30, -6, 36, 16, -168, 192, -7, 54, 46, -294, -266, 1428, -1386, -8, 76, 64, -480, 30, -832, 2560, -128, 3520, -12800, 10752, -9, 102, 86, -738, 78, -1260, 4356, -594, -558, 11484, -23166, 3564, -42900, 118404, -87516, -10, 132, 112, -1080, 100, -1840, 7040, 48, -1680, -800, 18240, -40320, -760, 8640

Offset: 1

Author

Bradley Klee, Sep 18 2016

Keywords

• sign
• tabf

Comments

Irregular triangle read by rows ( see examples ). The phase space trajectory of A276738 has one time dependent variable, the phase space angle "x" defined as Tan[x]=p/q. Then dx/dt = cos[x]^2* d/dt(p/q), which can be written as a function of Q=cos[x] by application of the classical equations of motion d/dt(p,q) = ( -d/dq H, d/dp H ), with H the anharmonic oscillator Hamiltonian. Substituting the result of A276738 and expanding in powers of b, we obtain dx/dt = -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. Observe the limit where Q --> 0, dx/dt --> -1, the harmonic oscillator value. Similarly if v_i --> 0 then dx/dt --> -1.

Examples

			n/m  1    2     3     4     5     6      7
---------------------------------------------
1  | -3
2  | -4,  6
3  | -5,  22,  -30
4  | -6,  36,   16,  -168   192
5  | -7,  54,   46,  -294  -266   1428  -1386
---------------------------------------------
		

Links

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

Crossrefs

Arbitrary Oscillator: A276738, A276815,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}]
basis[n_] :=  Times[Times @@ (v /@ #), Q^Total[#],2] & /@ (IntegerPartitions[n] /. x_Integer :> x + 2)
TriangleRow[n_, fun_] := Coefficient[fun, b^n #] & /@ basis[n]
With[{xd = xDot[10]},TriangleRow[#, xd] /. v[_] -> 0 & /@ Range[10]]