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A275790
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Triangle T(n, m) appearing in the expansion of the scaled phase space coordinate qhat of the plane pendulum in terms of the Jacobi nome q and sin(v) multiplying even powers of 2*cos(v), with v = u/((2/Pi)*K(k)).
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1
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1, 8, 1, -32, 11, 3, -736, -92, 9, 15, 2816, -593, -249, -65, 35, 48976, 6122, 1581, -970, -1295, 315, -951424, 61252, 67791, 46030, 18515, -21735, 3465, -1045952, -130744, -92082, -30445, 14455, 53928, -25179, 3003, 26933248, 1069361, -1666047, -634255, -1167740, -1258236, 1562253, -471471, 45045, 634836808, 79354601, 24881793, 17914550, 30289840, 12635028, -71064609, 42480438, -9594585, 765765
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OFFSET
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0,2
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COMMENTS
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The dimensionless scaled phase space coordinates of the plane pendulum are (qtilde(tau, k), ptilde(tau, k)) with tau = omega_0*t, omega^2 = g/L (L is the length of the pendulum, g the acceleration), and the energy variable E = 2*k^2 = 2*sin^2(Theta_0/2), with the maximal deflection angle Theta_0 (from [0, Pi/2]). qtilde = Theta/(2*k)) with the deflection angle Theta. Similarly ptilde = (d(Theta)/d(tau))/(2*k).
The exact solution is qtilde(tau, k) = (1/k)*arcsin(k*sn(tau, k)) with Jacobi's elliptic sn function, and ptilde(tau,k) = cn(tau, k) with the elliptic cn function.
Here the expansion in new variables v and q is used where v = tau/((2/Pi)*K(k)) and q = exp(-Pi*K'(k)/K(k)) with the real and imaginary quarter periods K and K'. This leads to qhat(v, q) = qtilde(tau(v, q), k(q)) with tau(v, q) = theta_3^2(0, q)*v. (For theta_3^2(0, q) see A004018.) Because k is actually a function of k^2 one uses the q expansion of (k/4)^2 given in A005798.
Using the result for the sn expansion in q and v from A274662 one obtains qhat(v, q) = sin(v)*Sum_{n >= 0} q^n/L(n)*Sum_{m=0..n} T(n, m)*(2*cos(v))^(2*m) with L(n) = A025547(n+1) = lcm{1, 3, ..., (2*n+1)}.
This entry is inspired by a paper of Bradley Klee giving an approximation to the phase space solution of the plane pendulum (see A273506). Thanks for discussions via e-mail go to him.
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LINKS
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FORMULA
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T(n, m)*(2*cos(v))^(2*m)), n >= 0, m = 0, 1, ..., n, gives the contribution to q^n/L(n) (L(n) = A025547(n+1)) in the rescaled phase space coordinate qhat(v, q) expansion of the plane pendulum. See a comment above for details.
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EXAMPLE
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The triangle T(n, m) begins:
n\m 0 1 2 3 4 5 ...
0: 1
1: 8 1
2: -32 11 3
3: -736 -92 9 15
4: 2816 -593 -249 -65 35
5: 48976 6122 1581 -970 -1295 315
...
row n=6: -951424 61252 67791 46030 18515 -21735 3465,
row n=7: -1045952 -130744 -92082 -30445 14455 53928 -25179 3003,
row n=8: 26933248 1069361 -1666047 -634255 -1167740 -1258236 1562253 -471471 45045,
row n=9: 634836808 79354601 24881793 17914550 30289840 12635028 -71064609 42480438 -9594585 765765.
...
The corresponding L(n) = A025547(n+1) numbers are 1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535,...
n=4: the contribution to qhat(v, q) of order q^4 is (q^4/315)*(2816 - 593*(2*cos(v))^2 - 249*(2*cos(v))^4 - 65*(2*cos(v))^6 + 35*(2*cos(v))^8).
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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