Weierstrass Solution of Cubic Anharmonic Oscillation

Anharmonic oscillation occurs along the Hamiltonian conserved energy surface whenever . Parameters 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 . Remarkably, this function also appears in Srinivasa Ramanujan's theory for signature-six elliptic functions [5–9].
The first plot draws the contours with height . The time-dependent position and momentum functions are shown in the second plot. The third diagram shows a collection of time fibers , 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, .



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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
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 .
[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 ," 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.
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