Related papers: Huygens' cycloidal pendulum: an elementary derivat…
A heuristic but pedagogical derivation is given of an explicit formula which accurately reproduces the period of a simple pendulum even for large amplitudes. The formula is compared with others in the literature.
This work shows how two simple derivations of the centripetal acceleration are fundamentally related to seminal works of Huygens and Newton. A different, more physically motivated derivation is then given.
The simple pendulum is one of the first experiments that students of higher physics do. There are certain precautions which the students are asked to take while performing the experiment. In this note we will try to explain as to why these…
Simple Hamiltonian systems, such as mathematical pendulum or Euler equations for rigid body, are solved without computation. It is nothing but a joke but maybe you will find it nice.
The period of oscillation of a simple pendulum ($T = 2\pi\sqrt{l/g}$) is a familiar formula to the average first-year physics student. However, deriving this expression from first principles involves solving a non-linear differential…
We extend the usual definition of the derivative in a way that Calculus I students can easily comprehend and which allows calculations at branch points.
The Doppler effect has many applications in science and engineering fields. Although the format of the classical Doppler effect equation is simple, the derivation for the equation in physics textbooks is not intuitive to many students. This…
A discussion of Lagrangian and Hamiltonian dynamics is presented at a level which should be suitable for advanced high school students. This is intended for those who wish to explore a version of mechanics beyond the usual Newtonian…
This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra…
In this paper we discuss how teaching of mathematics for middle school and high school students can be improved dramatically when motivation of concepts and ideas is done through the classical problems and the history of mathematics. This…
Constructive-deductive method for plane Euclidean geometry is proposed and formalized within Coq Proof Assistant. This method includes both postulates that describe elementary constructions by idealized geometric tools (pencil, straightedge…
This paper outlines a deceptively complex problem in classical mechanics which the paper names the "Falling Astronaut Problem," and it explores a method for teachers to implement this problem in an undergraduate classroom. The paper…
Perfect fractals are mathematical objects that, because they are generated by recursive processes, have self-similarity and infinite complexity. In particular, they also have a fractional dimension. Although several proposals for the study…
We present a pedagogical treatment of the electroweak Higgs mechanism based solely on Feynman diagrams and S-matrix elements, without recourse to (gauge) symmetry arguments. Throughout, the emphasis is on Feynman rules and the…
We derive Huygens' principle for electrodynamics in terms of 4-vector potentials defined as distributions supported on a surface surrounding the charge-current density. By combining the Pauli algebra with distribution theory, a compact and…
This work originates from part of a final year undergraduate research project on the Eisenhart lift for Hamiltonian systems. The Eisenhart lift is a procedure to describe trajectories of a classical natural Hamiltonian system as geodesics…
We discuss the geometry behind classical Heisenberg model at the level suitable for third or fourth year students who did not have the opportunity to take a course on differential geometry. The arguments presented here rely solely on…
An elementary derivation of the Newton "inverse square law" from the three Kepler laws is proposed. Our proof, thought essentially for first-year undergraduates, basically rests on Euclidean geometry. It could then be offered even to…
There have been several modifications of how basic calculus has been taught, but very few of these modifications have considered the computational tools available at our disposal. Here, we present a few tools that are easy to develop and…
A simple and transparent derivation of the formally exact probability distribution for classical non-equilibrium systems is given. The corresponding stochastic, dissipative equations of motion are also derived.