Dynamics

Mechanics, more I dive into it, more it fascinates me. Particularly classical mechanics and dynamics. It extends from Newtonian mechanics to celestial mechanics to solid mechanics and even further classical statistical mechanics.

And then comes the another beauty. Lagrange's equations, Euler–Lagrange equations, Noether's Principle, Principle of least action, Hamilton's principle, and this list goes on. But fascination doesn't end.

Starting from solution of projectile motion, to modelling of the rattleback motion, from solving the rigid body motion in free space, to model vehicle dynamics, it is everywhere. I never cease to admire the beauty that lies within its simplicity. During my master's thesis in solid mechanics too, this went hand in hand with the tire mechanics.

Later I realized, mechanics and dynamics is everywhere, every process has a governing dynamics, be it prey-predator model in population growth, or convergence of biologically inspired algorithm, there is a dynamics included in it. Once that is understood, the problem seems to break down to bit and pieces.

The Hidden Principle That Governs Everything

I mentioned earlier that I am a mechanics junkie. That is true. But over time, I realized something that shifted my entire understanding. Mechanics is not just about solving problems with moving parts. It is about something much deeper.

Let me explain.

The Standard Way of Seeing Things

In school, you learn Newton's laws. Force equals mass times acceleration. Action and reaction. All of that. This approach works fine for simple problems. A ball rolling down a hill. A planet orbiting the sun. You draw free-body diagrams and solve equations.

But Newton's approach has a limitation. You have to account for every single force explicitly. For a complex system – say a robot arm with multiple joints – this becomes a nightmare. The constraint forces alone will fill several pages. And if you make one sign error, the whole solution collapses.

There has to be a better way.

The Reformulation That Changed Everything

In the late 1700s, Joseph-Louis Lagrange looked at mechanics differently. He asked: what if we do not track forces at all? What if we just track energy?

His approach, now called **Lagrangian Dynamics**, starts with a single quantity. The Lagrangian is kinetic energy minus potential energy. That is it. Then you apply a simple equation to this quantity, and out pop the equations of motion. No force diagrams. No constraint equations. Just energy.

A few decades later, William Rowan Hamilton took this even further. **Hamiltonian Dynamics** reformulates the same physics in terms of position and momentum. The equations become symmetric and elegant. They also reveal something profound: the evolution of a physical system is equivalent to the flow of a fluid in phase space. That is a whole other level of beauty.

 The Strangest Idea in Physics

Here is where it gets really interesting. Both Lagrangian and Hamiltonian mechanics can be derived from a single principle called the **Principle of Least Action**.

The idea is almost absurdly simple. A system moving from point A to point B will follow the path that minimizes a quantity called the action. Action is the integral of the Lagrangian over time.

Think of a ball thrown into the air. Out of all possible paths – infinite of them – the ball takes the one that makes the action as small as possible. It is as if the ball "knows" the optimal path before it moves.

This bothered physicists for centuries. How does the particle know? It cannot look ahead and calculate. But the mathematics works perfectly. And it extends far beyond simple mechanics.

The Higher Meaning

Fermat's principle of least time says light travels between two points along the path that takes the least time. This explains reflection and refraction perfectly

In contact mechanics, the actual pressure distribution between two touching surfaces is the one that minimizes the total complementary energy. I used this during my thesis on tire mechanics without fully appreciating the pattern.

In economics, agents optimize utility. In biology, organisms optimize energy expenditure. In machine learning, we minimize loss functions.

The pattern is everywhere. Optimization is not just a mathematical technique. It might be the fundamental rule by which physical systems operate.

That realization hit me hard. I learned optimization as a tool. But it turns out optimization is the language in which physics is written. Lagrange wrote it. Hamilton refined it. Fermat stumbled onto it with light.

And I, with my small thesis on tire contact mechanics, was unknowingly standing on the same ground.

Coming Back

Do not get me wrong. I still love solving practical mechanics problems. I still enjoy drawing free-body diagrams occasionally. The Newtonian approach has its place.

But understanding Lagrangian and Hamiltonian dynamics changed how I see the world. A swinging pendulum is not just a mass on a string. It is a system following the path of least action. A planetary orbit is not just gravity balancing inertia. It is a continuous optimization problem solved by the universe itself.

That is the higher meaning I was searching for. Not better equations. A different way of seeing.

*[Dynamics: link to Wikipedia page]*