Classical Mechanics: A Mathematician’s Perspective

The aim of this blog post is to introduce Classical Mechanics from a mathematician’s standpoint. We begin with what is often referred to as the Newtonian Formalism.

What Is Classical Mechanics?

Naively speaking, Classical Mechanics is the study of particles moving continuously through the Euclidean space. In classical / Newtonian mechanics, we have 3 independent concepts or fundamental units namely, mass, length and time.

Modeling the Fundamental Quantities

Let us understand how we model each one of them. The mass is the easiest to model in Newtonian mechanics. It is an intrinsic assignment of a positive number $m$ to each particle, and it measures inertia of an object.

Next we model length. It is modeled over the $\large$d$$ -dimensional Affine Euclidean space $\large$\mathbb E^d$$ with a translation-invariant metric, for example the Euclidean Metric.

Finally, the time, which is modeled by $\large$\mathbb E^1$$ , the $\large$1$$ -dimensional real vector space (aka. the real line) with a translation-invariant metric and an orientation to distinguish between moving forward and backward in time.

Describing Particle Motion

We can describe a particle by a continuous path $\large$x:T\to\mathbb E^d$$ assigning $\large$t\mapsto x(t)=(x_1(t),\dots,x_d(t))$$ , assigning each time $\large$t$$ , a position $\large$x(t)\in\mathbb E^d$$ .

However, not every continuous path corresponds to a physically realizable motion. We need a law of motion to distinguish real trajectories from arbitrary ones.

Enter: The Potential Function

To impose dynamics, we introduce a Potential Function which is a smooth assignment $\large$V:\mathbb E^d\to\mathbb R$$ . We are assigning to each position $\large$x\in\mathbb E^d$$ a real number $\large$V(x)\in\mathbb R$$ interpreted as the potential energy at that point.

Newton’s Second Law (Equation of Motion)

The fundamental postulate of Newtonian mechanics is encoded in Newton’s Second Law, which, in our formalism, reads:

$\large$m\ddot x(t)=-\nabla V(x(t))$$

This second-order differential equation governs the motion of a particle. It is called the Newton’s 2nd Law and is also recognized as the Euler-Lagrange Equation(s) of Motion for the system.

Show that the trajectory of a particle that is launched at an angle is parabolic?

The Solution Space as a Manifold

Let, $\large$M:=\{x:T\to\mathbb E^d$\space$\vert m\ddot x=-\nabla V(x)\}$$ , denote the set of all solutions to the equations of motion—i.e., all physical paths. Generally $\large$V$$ is taken in such a way that the solution $\large$x:T\to\mathbb E^d$$ is defined for all values of the time-parameter $\large$t\in T$$ .

We know that given initial conditions viz. initial position $\large$x_0$$ and initial velocity $\large$\dot x(t_0)$$ , there exists a unique solution to the equations of motion.

Thus, for each fixed $\large$t_0\in T$$ we can define a map $\large$\Phi:M\to T\mathbb E^d(\cong\mathbb E^d\times\mathbb R^d)$$ sending $\large$x\mapsto(x(t_0),\dot x(t_0))$$ which gives a bijection between $\large$M$$ and the tangent bundle $\large$T\mathbb E^d$$ .
We can make $\large$M$$ into a manifold by transferring the structure on the tangent bundle $\large$T\mathbb E^d$$ to $\large$M$$ through the bijection $\large$\Phi$$ , which makes this bijection a diffeomorphism.

One can easily check that this structure on $\large$M$$ is independent of the choice of $\large$t_o\in T$$ .
So, we can take $\large$t_o=0$$ without loss of generality.

Examples

Let us discuss some examples to understand this better.

Example 1: Free particle i.e. without any external force acting on it, i.e. $\large$\nabla V=0$$ or equivalently, $\large$V\equiv 0\text{\space or\space constant}$$ . Then the solution space is, $\large$M=\{x(t)=p+tv:p\in\mathbb E^d,v\in\mathbb R^d\}$$ . This corresponds to Newton’s First Law — a particle in uniform motion unless acted upon by a force.

Newton's First Law of Motion - GeeksforGeeks

Example 2: The Harmonic Oscillator for $\large$d=1$$ , let $\large$V(x):=\frac{1}{2}kx^2$,\space$k>0$\space constant$ . Then the equation of motion is,
$\large$m\ddot x=-kx$$
for which the solution space is $\large$M=\{x(t)=A\cos\big(\sqrt{\frac{k}{m}}t+\delta\big):A\geq 0,\delta\in[0,2\pi)\}$$ .

An Overview of Harmonic Oscillators

Example 3: Let $\large$d=3$$ and $\large$V=-\frac{k}{\|x\|}$$ ,

This gives the inverse-square law in Gravitation and Electrostatics. The solution to this system are conic sections.

Momentum and Energy

We are yet to talk about Momentum and Energy. We define the momentum of a particle by $\large$p(t):=m\dot x(t)$$ . Under no external force, $\large$\nabla V=0$$ (or equivalently $\large$V\equiv c$$ ). Thus, we have $\large$\dot p(t)=m\ddot x(t)=-\nabla V(x(t))=0$$

which gives us the conservation law of momentum.

Momentum Equations, Definition | Zona Land Education

Kinetic Energy and Total Energy

We can define the Kinetic Energy of the particle to be $\large$K(t):=\frac{1}{2}m\|\dot x\|^2$$ and the total energy of the particle to be $\large$E(t):=K(t)+V(x(t))=\frac{1}{2}m\|\dot x\|^2+V(x(t))$$ .

Then, differentiating with respect to time, we get,

$\large$\dot E(t)=m\ddot x\cdot\dot x+\nabla V(x)\cdot\dot x=\dot x\cdot(-\nabla V(x)+\nabla V(x))=0$$ .
Note here that the potential function $\large$V$$ is only dependent on the position $\large$x$$ of the particle and independent of time $\large$t$$ .

Thus, total energy is conserved – this is the conservation law of energy.

Potential And Kinetic Energy Example ...

Closing Thoughts

This framework sets the stage for more advanced topics, such as Lagrangian and Hamiltonian mechanics, symplectic geometry, and Noether’s theorem, all of which arise naturally starting from this Newtonian formalism.

In future posts, we’ll explore these directions, building bridges between physics and pure mathematics.

Manifolds Unfolded