A Philosophical Note On The Nature Of Nature
I argue that humans have the ability to see and even traverse the essence of true reality. Objective reality must be one entity, for contradicting doctrines of the universe would not be compatible if they co-existed. Nature always takes the shortest path, for energy consumption purposes, there is a wide selection of physical laws where this concept of “economic” fastest path is observed. In fact, this is a fundamental piece of truth about our universe, from it pop up endless derived laws and formulas that describe the behavior of systems in nature with perfect accuracy; Pierre Louis Maupertuis was the one who put forward the concept of Work and The Principle Of Least Action to physics : $$S = \int p , dq$$ Which means that nature’s laws are the byproduct of the optimal way for things to happen (shortest path, lowest energy consumption, etc). He was a mathematician, whose passion for such an idea stemmed mainly from his obsession with math. In a futile way he tried and tried to make the idea stick in respectable scientific communities, but it almost never did. He tried to enlighten those poor physicists of the 19th century but they rejected and mocked his work. Maupertuis’ insight was a true property of existence where he objectively and correctly observed and concluded that optimization is nature’s meta-law, the apple falls in 1 trajectory and not a million different random ones because that’s the most optimal cost-effective way to fall. He wasn’t properly understood until long after his death when more physics “qualified” mathematicians helped develop his theorem further and polished his gaps. One prominent example was the famous Mathematician Leonard Euler and many of his students. Euler’s mathematical rigor and innovations influenced many physicists who later formalized concepts like work, energy, and mechanics. For example, he refined Maupertuis’ ideas mathematically and applied them to mechanics and variational calculus. He developed differential equations that describe how forces and motion evolve in time based on the principle of work and energy. $$\frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}} \right) - \frac{\partial \mathcal{L}}{\partial q} = 0 $$
Further more. their work laid the foundation for analytical mechanics, particularly in the development of the Lagrangian and Hamiltonian formulations.
Lagrangian and Hamiltonian Mechanics: Euler’s work on variational principles and constraints influenced the development of these advanced formulations of mechanics, which are deeply connected to the concept of work and energy.
1. Lagrangian:
The Lagrangian $\mathcal{L}$ is defined as the difference between the kinetic energy $T$ and the potential energy $V$: $$\mathcal{L} = T - V = \frac{1}{2} m \dot{q}^2 - V(q)$$
2. Hamiltonian:
The Hamiltonian $\mathcal{H}$ is defined as the sum of the kinetic energy $T$ and the potential energy $V$, expressed in terms of generalized coordinates $V(q)$ and their conjugate momenta $p$ :
$$\mathcal{H} = T + V = \frac{p^2}{2m} + V(q)$$
We know now, that these laws still hold, they are immutable as the universe itself is from our perspective. If you use newton’s laws, you get the same equations and predictions as if you start your problem solving quest by the principle of least action and all the mathematics that comes out of it. Some people like to joke about how Mapertuis’ made it easier for mathematicians who suck at physics to grasp physics. But I like the other version of the joke, my version. Which is that the principle of least action was a way to make physicists understand math better.
That’s about it for today, Reality is weird but understandable if you gove yourself time to decipher its hidden layers, you can buddy. That was the moral of all of this jargon.