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 signaturesix elliptic functions [5–9]. The first plot draws the contours with height . The timedependent 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 equalenergy 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, .
The builtin 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. [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 OnLine 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:10.1007/9789401008181_ 3. [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/s1011401661801.
