Phase space Consider a one-dimensional harmonic oscillator evolving according to the equation of motion: [m \ddot{x} + kx = 0] It is possible to investigate the essential properties of \(x \left( t \right)\) using a geometric intuition solely by playing with the equation of motion, rather than directly finding the general solution analytically. Firstly, for future ease, we define a paramete...

# Subtraction Is More Fundamental Than Addition

When we first learn mathematics in school, we are taught the four fundamental operations of arithmetic: addition, subtraction, multiplication and division. It is a common practice to teach addition first. After all, we’re adding things all the time in real life: apples to apples, oranges to oranges and so on. Only then are we taught that the reverse operation is necessary to model the removal o...

# Scalar Field Lagrangian From Symmetry Considerations

Welcome to another post on, well, you guessed it: scalar field theory. Today, we will be deriving a Lagrangian density for scalar fields that appears almost everywhere in physics. Common arguments for assuming the form of the Lagrangian in question either take motivation from too specific systems such as chains (which are promoted to scalar fields using a procedure that does not work for highe...

# Factorials as Invariant Points

In ‘Deriving the Gamma Function from Scratch’, we investigated the analytic origin of the extended factorial. Namely, it comes from the complex solution for the functional equation of the discrete factorial function. Today, we will take a more general route and interpret factorials as invariant points under a certain transformation, in order to solve for those points and obtain an explicit exp...

# Harmonic Oscillators in Scalar Field Theory

Once again, let us see how classical scalar field theory naturally gives rise to common natural phenomena. Today, we will derive the evolution of harmonic oscillators. Scalar form Scalar fields Given a scalar field (in flat spacetime) \(\phi \left( x^\rho \right)\) with the following Lagrangian density, [\mathcal{L} = \frac{1}{2} \partial_\alpha \phi : \eta^{\alpha \beta} : \partial_\beta ...

# The Discrete Antiderivative Operator

Discrete derivative operator Let a function \(f: \mathbb{R} \mapsto \mathbb{R}\). On discretizing the domain of \(f\) into quanta \(h\) centred at \(a_0\), \(f: \mathbb{A} \mapsto \mathbb{A}\) where \(\mathbb{A} = \left\{ kh+a_0 : k \in \mathbb{Z}, a_0 \in \mathbb{R} \right\}\), the derivative operator is replaced by the discrete derivative operator \(\mathcal{D}: \mathbb{A}^{\mathbb{A}} \maps...

# The Impossible Cut

A sweet problem The classic statement Imagine you have a cake. How can you slice it into \(8\) pieces in exactly \(3\) steps? Well, you divide the cake into two, three times, so that the number of pieces compounds to \(2^3 = 8\). This can be done by cutting the cake along different planes, in the following manner: How mathematicians probably cut cakes Notice how the planes of the cuts in t...

# All the World's Not a Stage

A story without a film Consider a gravitational system comprising the earth and an apple. The apple is released from a certain height and it plummets to the ground. How would the evolution of this system proceed, if, instead, time ran backwards? Our intuition tells us that if time runs backwards, the apple should, as if by definition, rise up from the ground and shoot to the sky in eternally ...

# Demystifying the Definition Of a Covector Basis

A covector basis \(\left\{ \pmb{\theta}^i \right\}\) is defined to act on the corresponding vector basis \(\left\{ \pmb{e}_j \right\}\) in the manner, [\pmb{\theta}^i \left( \pmb{e}j \right) = \delta^i{\phantom{i} j}] Where \(\delta^i_{\phantom{i} j}\) represents the Kronecker delta. But where does the above definition even come from? Well, turns out it’s not so mysterious after all. Say a c...

# Algebra Done Tensorially: Part 2 (Algebras Over Fields)

Welcome to Part 2 of ‘Algebra done Tensorially’. If you haven’t already done so, make sure to check out Part 1 (Bilinear Products) before reading this post :) I will start right from where we stopped in Part 1. Recap: bilinear products as linear maps We had learnt in the previous post that passing only one argument to a bilinear product makes it a linear map on the vector space characterized ...