Related papers: Weierstrass elliptic functions for the pendulum
In this paper, we expand the theory of Weierstrassian elliptic functions by introducing auxiliary zeta functions $\zeta_\lambda$, zeta differences of first kind $\Delta_\lambda$ and second kind $\Delta_{\lambda,\mu}$ where…
Einstein's perihelion advance formula can be given a geometric interpretation in terms of the curvature of the ellipse. The formula can be obtained by splitting the constant term of an auxiliary polar equation for an elliptical orbit into…
Lame equation arises from deriving Laplace equation in ellipsoidal coordinates; in other words, it's called ellipsoidal harmonic equation. Lame function is applicable to diverse areas such as boundary value problems in ellipsoidal geometry,…
Applied problems of oil and gas recovery are studied numerically using the mathematical models of multiphase fluid flows in porous media. The basic model includes the continuity equations and the Darcy laws for each phase, as well as the…
In this paper we relate some classical normal forms for complex elliptic curves in terms of 4-point sets in the Riemann sphere. Our main result is an alternative proof that every elliptic curve is isomorphic as a Riemann surface to one in…
An analytical solution to the nonlinear differential equation describing the equation of motion of a particle moving in an unforced physical system with linear damping, governed by a cubic potential well, is presented in terms of the Jacobi…
We study moduli of planar ring domains whose complements are linear segments and establish formulas for their moduli in terms of the Weierstrass elliptic functions. Numerical tests are carried out to illuminate our results.
In a previous paper, a point of order 8 on an elliptic curve was calculated. Exploiting the well-known correspondence of the points on an elliptic curve with the points of a respective period parallelogram, we proceed to calculating all…
Analysis of the generalized Weierstrass-Enneper system includes the estimation of the degree of indeterminancy of the general analytic solution and the discussion of the boundary value problem. Several different procedures for constructing…
A covariant functor on the elliptic curves with complex multiplication is constructed. The functor takes values in the noncommutative tori with real multiplication. A conjecture on the rank of an elliptic curve is formulated.
We study relations of the Weierstrass's hyperelliptic al-functions over a non-degenerated hyperelliptic curve $y^2 = f(x)$ of arbitrary genus $g$ as solutions of sine-Gordon equation using Weierstrass's local parameters, which are…
We first normalize the derivative Weierstrass $\wp'$-function appearing in Weierstrass equations which give rise to analytic parametrizations of elliptic curves by the Dedekind $\eta$-function. And, by making use of this normalization of…
We consider the problem of obtaining higher order in regularization parameter $\epsilon$ analytical results for master integrals with elliptics. The two commonly employed methods are provided by the use of differential equations and direct…
A simple expression for the zeros of Weierstrass' function is given which follows from a formula for relativistic orbits.
We investigate oscillating solutions of the equation of motion for the Higgs potential. The solutions are described by Jacobian elliptic functions. Classifying the classical solutions, we evaluate a possible parameter-space for the initial…
For billiards in an ellipse with an ellipse as caustic, there exist canonical coordinates such that the billiard transformation from vertex to vertex is equivalent to a shift of coordinates. A kinematic analysis of billiard motions paves…
It is proved that the rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. As an illustration, we consider a family of elliptic curves with complex multiplication.
We show how Jacobian elliptic functions (JEF) can be used to solve ordinary differential equations (ODE) describing nonlinear dynamics of microtubules (MT). We demonstrate that only one of JEFs can be used while the remaining two do not…
The small angle approximation often fails to explain experimental data, does not even predict if a plane pendulum's period increases or decreases with increasing amplitude. We make a perturbation ansatz for the Conserved Energy Surfaces of…
A minimal Lorentz surface in $\mathbb R^4_2$ is said to be of general type if its corresponding null curves are non-degenerate. These surfaces admit canonical isothermal and canonical isotropic coordinates. It is known that the Gauss…