Hi everyone, hope you’re all doing well :) In this short post, I’ll be sharing the slides from my recent presentation at SASMS (Short Attention Span Math Seminars), hosted by the Pure Math Club at the University of Waterloo. The presentation was on an informal construction of the ideas developed in some posts on this blog: Scalar Field Lagr...

In A Brief Geometric Analysis of Harmonic Oscillators: Part 2 (Tensor Algebra), we generalized the methods used in A Brief Geometric Analysis of Harmonic Oscillators to analyze the behaviour of harmonic oscillators in phase space. We began with Hooke’s law, \(\omega^{-1} \ddot{x} + \omega x = 0\) and explored the Hamiltonian flow described by i...

Statement Consider the 2 well-known properties of a linear operator \(T : \mathbb{R}^m \to \mathbb{R}^n : m, n \in \mathbb{N}\), [\begin{align} T \left( \sum_a \pmb{u}_a \right) & = \sum_a T \left( \pmb{u}_a \right) & \forall : \pmb{u}_a \in \mathbb{R}^m && \left( 1 \right) T \left( c \pmb{u} \right) & = c T \left( \pmb{u}...

In A Brief Geometric Analysis of Harmonic Oscillators, we examined how a convenient choice of phase space for the motion of a harmonic oscillator reveals its periodicity, without explicitly solving the equation of motion (Hooke’s law). Here, we will apply the same intuition but in the language of tensor algebra, introducing greater rigour as wel...

Some thoughts Hi everyone! It’s been a few months since the last post here. A lot (of good) has happened since then. Before coming to that, I would like to thank all my old readers for sticking by, and new readers for visiting this blog :) I hope you all are doing great and had a wonderful year … Happy New Year! In this period of absence, I’ve...

Related concepts from older posts In Scalar Field Lagrangian From Symmetry Considerations, we derived the Lagrangian for the Klein-Gordon theory of a scalar field \(\phi\) evolving in a potential \(V\), [\mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - V \left( \phi \right)] We did so, by deriving the field-theoretic energy-mom...

Much of analytical mechanics dedicates itself to studying the trajectories of dynamical systems. Newtonian mechanics describes trajectories in physical space. On the other hand, Lagrangian mechanics deals with trajectories in configuration space, which is the space of all generalized position and velocity coordinates; while Hamiltonian mechanics...

In Combining Valid Solutions Into New Ones in Classical Field Theory, we showed that any linear combination of valid solutions to classical field equations, is in turn a solution. We then took the idea of solution fields behaving like coordinates further, in Gauge Invariance in Classical Field Theory. Here, we found that it is possible for cert...

In Combining Valid Solutions Into New Ones in Classical Field Theory, we have seen how an arbitrary field \(\phi\) constructed from a class of solutions \(\left\{ \phi_{\left( i \right)} \right\}\) for some equations of motion, must obey: [\frac{\partial \mathcal{L}}{\partial \phi} - \nabla_\mu \frac{\partial \mathcal{L}}{\partial \left( \parti...

A popular theme in some field theories is linearity — wherein valid solutions of the field equations in question can be added and scaled to generate new valid solutions. For example, Maxwell’s equations are linear in vacuum and homogeneous media. A different kind of field equation that is linear, is the time-independent Schrödinger equation for ...