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 ...

The problem One of the celebrated results of Einstein’s special theory of relativity is that no physical object can travel faster than light, no matter the frame of reference. A popular explanation for this is that as an object approaches the speed of light, its relativistic energy \(E = \gamma mc^2\) 1 tends to infinity. Thus, it would require...

In this post, we will derive the conservation of total mass of a closed system from simpler physical facts, within the framework of classical mechanics. We will first recall the key concepts which will be used to build the argument, namely: The conservation of total linear momentum The principle of relativity Then, we wil...

Let us find the covariant derivative of a rank \((p, q)\) tensor field \(\pmb{\phi}\) by applying the Leibniz law repeatedly: [\nabla_\rho \left( \pmb{\phi} \otimes \pmb{\psi} \right) = \left( \nabla_\rho \pmb{\phi} \right) \otimes \pmb{\psi} + \pmb{\phi} \otimes \left( \nabla_\rho \pmb{\psi} \right)] Note that we will be implicitly using a Le...

Welcome to Part 3 of ‘Algebra Done Tensorially’. It’s been a while since the previous posts, so let us resume our investigation of algebras without further ado. For readers who haven’t read earlier posts in this series yet, I’d recommend you to read Part 1 and Part 2 first :) Parts Topics Part 1 (Bilin...

A Supertask A supertask is a countably infinite sequence of tasks or events, which occurs in a finite amount of time. The word ‘supertask’ was coined by the twentieth-century philosopher James F. Thomson. He went on to provide an example of a supertask that soon became his namesake philosophical puzzle. The problem may be stated as: we are give...