Related papers: Huygens' cycloidal pendulum: an elementary derivat…
The pedal of a curve in the Euclidean plane is a classical subject which has a singular point at the inflection point of the original curve or the pedal point. The primitive of a curve is a curve given by the inverse construction for making…
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…
Integration of polynomials over the classical groups of unitary, orthogonal and symplectic matrices can be reduced to basic building blocks known as Weingarten functions. We present an elementary derivation of these functions.
An elementary pedagogical derivation of the Lense-Thirring precession is given based on the use of Hamilton vector. The Hamilton vector is an extra constant of motion of the Kepler/Coulomb problem related simply to the more popular…
In classical geometry, there is no such well-known and much-studied topic as the construction of conic sections (or briefly conics) from its five points. Its importance in many applications of mechanical engineering, civil engineering and…
We do not present any original or new material. This is a tutorial addressed to students who need to study the microscopic derivation of the quantum-mechanical master equation encountered in many practical physical situations.
The differential cross-section for the reflection of light beams off rigid bodies obtained by the rotation of a generic derivable convex function is calculated. The calculation is developed using elementary notions of calculus and is…
The harmonic oscillator propagator is found straightforwardly from the free particle propagator, within the imaginary-time Feynman path integral formalism. The derivation presented here is extremely simple, requiring only elementary…
Dynamical two-particle susceptibilites are important for a wide range of different experiments in condensed-matter physics and beyond. Nevertheless, most textbooks avoid describing how to derive such response functions, perhaps because they…
It is proved that a triangulation of a polyhedron can always be transformed into any other triangulation of the polyhedron by using only elementary moves. One consequence is that an additive function (valuation) defined only on simplices…
In this article using elementary school level Geometry we observe an alternative proof of Pythagorean Theorem from Heron's Formula.
Using a theorem of partial differential equations, we present a general way of deriving the conserved quantities associated with a given classical point mechanical system, denoted by its Hamiltonian. Some simple examples are given to…
We discuss how a class of difficult kinematic problems can play an important role in an introductory course in stimulating students' reasoning on more complex physical situations. The problems presented here have an elementary analysis once…
A self-contained introduction is presented of the notion of the (abstract) differentiable manifold and its tangent vector fields. The way in which elementary topological ideas stimulated the passage from Euclidean (vector) spaces and linear…
We quantitatively analyze a familiar classroom demonstration, Van Waltenhofen's eddy current pendulum, to predict the damping effect for a variety of plate geometries from first principles. Results from conformal mapping, finite element…
We provide a simple and efficient algorithm for computing the Euclidean projection of a point onto the capped simplex---a simplex with an additional uniform bound on each coordinate---together with an elementary proof. Both the MATLAB and…
We formulate a problem on diamagnetic levitation, which may be suitable for specialized high-school or first-year students in physical sciences. We guide the students, step-by-step, through the physics of diamagnetic levitation. The…
We introduce new algebraic structures associated with heptagon relations -- higher analogue of the well-known pentagon. The main points we deal with are: (i) polygon relations as algebraic imitations of Pachner moves, on the example of…
How can we convince students, who have mainly learned to follow given mathematical rules, that mathematics can also be fascinating, creative, and beautiful? In this paper I discuss different ways of introducing non-Euclidean geometry to…
We consider gradient Ricci solitons conformal to a $n$-dimensional pseudo-Euclidean space and we completely describe the most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary…