Related papers: Some Novel Aspects of the Plane Pendulum in Classi…
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…
The solutions that describe the motion of the classical simple pendulum have been known for very long time and are given in terms of elliptic functions, which are doubly periodic functions in the complex plane. The independent variable of…
About twenty years ago, Rabinowitz showed firstly that there exist heteroclinic orbits of autonomous Hamiltonian system joining two equilibria. A special case of autonomous Hamiltonian system is the classical pendulum equation. The phase…
The Jacobi and Weierstrass elliptic functions used to be part of the standard mathematical arsenal of physics students. They appear as solutions of many important problems in classical mechanics: the motion of a planar pendulum (Jacobi),…
Revisiting canonical integration of the classical pendulum around its unstable equilibrium, normal hyperbolic canonical coordinates are constructed
The mathematical model representing the equation of motion of a pendulum is nonlinear. Solutions that satisfy the equation cannot be represented by elementary functions, such as trigonometric functions. To solve such problems, it is common…
In this work we solve the nonlinear second order differential equation of the simple pendulum with a general initial angular displacement ($\theta(0)=\theta_0$) and velocity ($\dot{\theta}(0)=\phi_0$), obtaining a closed-form solution in…
The author considers the planar rotational motion of the mathematical pendulum with its pivot oscillating both vertically and horizontally, so the trajectory of the pivot is an ellipse close to a circle. The analysis is based on the exact…
We discuss a new class of coordinate systems for a plane, which provide an analytical representation of arbitrary straightline, and then define the form of potential on the plane, under which the equations of motion of a mass point are…
Galileo, in the XVII century, observed that the small oscillations of a pendulum seem to have constant period. In fact, the Taylor expansion of the period map of the pendulum is constant up to second order in the initial angular velocity…
We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schr\"odinger equations admit separation of variables in polar coordinates and are exactly…
A simple derivation of the classical solutions of a nonlinear model describing a harmonic oscillator on the sphere and the hyperbolic plane is presented in polar coordinates. These solutions are then related to those in cartesian…
The spatial Kepler problem with a perturbation satisfying the rotational symmetry w.r.t. the $z$-axis and the reflection symmetry w.r.t. the $(x, y)$-plane, can be reduced to an Hamiltonian system with 2 degrees of freedom after fixing the…
We introduce a continuous one-parameter family of elliptic sine-Gordon equations (SGE) characterized by the modulus $0 \le m \le 1$ of Jacobi elliptic functions and analyze some of its properties and obtain its kink solution for various…
In this work, we have obtained exact solutions of Einstein equations for static and axially symmetric magnetized matter, specifically in plane-symmetric and almost-plane symmetric cases. Although these solutions impose constraints on the…
We study harmonic maps from a subset of the complex plane to a subset of the hyperbolic plane. In \cite{FotDask}, harmonic maps are related to the sinh-Gordon equation and a B{\"a}cklund transformation is introduced, which connects…
A new class of static plane symmetric solution of Einstein field equation generated by a perfect fluid source is put forward. A special family of this new solution is investigated in detail. The constraints on the parameters by different…
A geometric interpretation is given for certain elliptic-hyperbolic systems in the plane. Among several examples, one which reduces in the elliptic region to the equations for harmonic 1-forms on the projective disc is studied in detail. A…
We derive the exact analytical solutions to the symmetron field theory equations in the presence of a one or two mirror system. The one dimensional equations of motion are integrated exactly for both systems and their solutions can be…
We study the elliptic sinh-Gordon and sine-Gordon equations on the real plane and we introduce new families of solutions. We use a Backlund transformation that connects the elliptic versions of sinh-Gordon and sine-Gordon equations. As an…