Weierstrass Solution of Cubic Anharmonic Oscillation

​
energy surface
position and momentum
time fibers
energy α
time evolution
initial condition
Anharmonic oscillation occurs along the Hamiltonian conserved energy surface
α=2H(q,p)=
2
p
+
2
q
-
3
q
whenever
0<α<4/27
. Parameters
g
2
=1/12,
g
3
=1/216-α/16
redefine the surface as a family of elliptic curves in the Weierstrass normal form. The Weierstrass
℘
function completely determines the time parameterization and solutions to Hamilton's equations of motion[1–4]. The period of oscillation motion is given by
T(α)=2
π
2
F
1
(1/6,5/6;1;27α/4)
. Remarkably, this function also appears in Srinivasa Ramanujan's theory for signature-six elliptic functions[5–9].
The first plot draws the contours
α=2H(q,p)
with height
arctan(α-4/27)
. The time-dependent position and momentum functions
q(t),p(t)
are shown in the second plot. The third diagram shows a collection of time fibers
t(p,q)
, which follow approximately helical trajectories. The "time evolution" and "initial condition" sliders trace out one of these fibers. Note that any two equal-energy time fibers differ, at most, by a parallel translation along the vertical time axis. In this example of integrable anharmonic oscillation, each helical time fiber admits a special set of invariant transformations,
t
0
↦
t
0
'=
t
0
+nT(α),n∈
.

Details

The built-in Mathematica function WeierstrassHalfPeriods does not immediately produce the identity with the Gaussian hypergeometric function. One needs to obtain this result from the integral
2
F
1
[1/6,5/6,1,x]=
1
2π
2π
∫
0
2
F
1
[13,23,12,
6
sin
(ϕ)x]dϕ
,
which derives from the invariant differential along the contours of the Hamiltonian energy surface. Alternative proofs may also be found in the mathematics literature[8].
This family of elliptic curves is another special case in which the Picard–Fuchs equation is simply the hypergeometric differential equation[9]. The complex period is then proportional to a second solution
2
F
1
[1/6,5/6,1,1-x]
.

References

[1] G. Pastras, "Four Lectures on Weierstrass Elliptic Function and Applications in Classical and Quantum Mechanics." arxiv.org/abs/1706.07371.
[2] A. Brizard, "Notes on the Weierstrass Elliptic Function." arxiv.org/abs/1510.07818.
[3] J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., New York: Springer, 2009.
[4] D. Husemöller, Elliptic Curves, 2nd ed., New York: Springer, 2004.
[5] N. J. A. Sloane and Michael Somos. The On-Line Encyclopedia of Integer Sequences. "A113424." oeis.org/A113424.
[6] B. C. Berndt, "Flowers Which We Cannot Yet See Growing in Ramanujan's Garden of Hypergeometric Series, Elliptic Functions, and q's," in Special Functions 2000: Current Perspective and Future Directions, Dordrecht: Springer, 2001 pp. 61–85. doi:10.1007/978-94-010-0818-1_ 3.
[7] S. Ramanujan, "Modular Equations and Approximations to
π
," Quarterly Journal of Mathematics, 45, 1914 pp. 350–372. ramanujan.sirinudi.org/Volumes/published/ram06.pdf.
[8] L. C. Shen, "On Three Differential Equations Associated with Chebyshev Polynomials of Degrees 3, 4 and 6," Acta Mathematica Sinica, 33(1), 2017 pp. 21–36. doi:10.1007/s10114-016-6180-1.
[9] M. Kontsevich and D. Zagier, "Periods," in Mathematics Unlimited—2001 and Beyond (B. Engquist and W. Schmid, eds.), Berlin, Heidelberg: Springer, 2001 pp. 771–808. www.maths.ed.ac.uk/~aar/papers/kontzagi.pdf.

Permanent Citation

Brad Klee
​
​"Weierstrass Solution of Cubic Anharmonic Oscillation"​
​http://demonstrations.wolfram.com/WeierstrassSolutionOfCubicAnharmonicOscillation/​
​Wolfram Demonstrations Project​
​Published: February 19, 2018