D4 Symmetric Stratum of Quartic Plane Curves

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The quartic Hamiltonian form determines a stratum of plane curves with
symmetry. The toricity
distinguishes between layers of the stratum. Each planar layer contains a family of Hamiltonian level curves, indexed by energy
. Allowing the coordinate variables to take on complex values associates each plane curve to a Riemann surface
with nontrivial topology. This Demonstration depicts a toric section of
and gives a functional form
for period integrals taken around orthogonal contours (see Details).
Contributed by: Brad Klee (September 2018)
Open content licensed under CC BY-NC-SA
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.
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