Details

In this analysis, coordinate variables take on complex values from (or ). The Hamiltonian function has four lines of reflection symmetry in a plane spanned by the real coordinates and . Each line and intersects one circular point and two hyperbolic points. Reflection symmetry allows us to construct real-valued contours,

of the Riemann surface,

.

The surface is a two-dimensional subset of a four-dimensional space and thus resists geometric intuition. Nevertheless, we color the cross sections and arrange them along to suggest the underlying topology. The depiction of this Demonstration raises to above and to above . Curve meets each curve and in four real points of intersection. The transformation model preserves dihedral symmetry up to a permutation of colors (dimensions).

To get a general idea of time dynamics along , we calculate period integrals along the contour curves. Transforming to action-angle variables,

,

reduces the Hamiltonian from quartic to quadratic degree,

.

Solving for the root leads directly to the period-energy function,

,

with the loop integral taken around the curve . Regardless of toricity, the period function obviously converges on a region . Adapting an algorithm from [1], we routinely produce a Picard–Fuchs type differential equation. Application of the left operator,

,

to integrand makes for a sum over derivatives,

.

Such a sum necessarily leaves a remainder on the right-hand side, . The certificate function

,

contributes to an exact differential , which integrates to zero on a complete cycle around , that is, . The Picard–Fuchs differential equation follows immediately.

Definition of the complex period function around either curve or follows from a beautiful symmetry. Inversion of parameters,

,

acts on annihilation operator as a scale transformation,

.

However, annihilation relations are scale free, so the transformation property implies an identity between real and complex periods,

,

with real period defined as . Both functions and have the same series expansion coefficients, quickly generated by a -recurrence (the Frobenius solution of the Picard–Fuchs equation with two initial values). Our plot of these functions uses a dynamic sum over 500 powers of to achieve adequate convergence.

The graph of period functions and obviously shows energy inversion symmetry at the elliptic fixed point . This special configuration corresponds to Edwards's family of elliptic curves [2], with genus and period function

.

The differential equation takes a hypergeometric form. In straightforward analysis of and , we find that boundary values also allow solutions in terms of Gauss's hypergeometric function (see [3–5]). However, the stratum limits do not lead to families of elliptic curves. For nonsingular curves , toricity parameter determines the genus,

.

To prove the validity of function , observe the self-intersection topology of curve in the limit (see [6]).

The fixed point is also an expectation value. We give a dataset for the librational motion of a plane pendulum over a wide range of energies [7]. The data allows construction of a likelihood function with variation of the toricity . It is then possible to fit function to period versus energy data at precision.

References

[1] B. Klee, "Approximating Pi with Trigonometric-Polynomial Integrals" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/ApproximatingPiWithTrigonometricPolynomialIntegrals.

[2] H. Edwards, "A Normal Form for Elliptic Curves," Bulletin of the American Mathematical Society, 44, 2007 pp. 393–422. doi:10.1090/S0273-0979-07-01153-6. ams.org

[3] N. J. A. Sloane. "The On-Line Encyclopedia of Integer Sequences." (Sep 14, 2018) oeis.org/A002894.

[4] N. J. A. Sloane. "The On-Line Encyclopedia of Integer Sequences." (Sep 14, 2018) oeis.org/A113424.

[5] N. J. A. Sloane. "The On-Line Encyclopedia of Integer Sequences." (Sep 14, 2018) oeis.org/A318417.

[6] S. S. Abhyankar and C. L. Bajaj, "Computations with Algebraic Curves", Symbolic and Algebraic Computation: ISSAC 1988 (P. Gianni, ed.), Berlin, Heidelberg: Springer, 1989. doi:10.1007/3-540-51084-2_26.

[7] B. Klee, "Fidget Spinner Libration Data," (Sep 25, 2018) wolframcloud.com.