# What are the most fundamental laws of Newtonian Physics? [closed]

By fundamental, I mean, those laws which if assumed could be used to prove all other laws and provide the essence of the complete picture.

I am a high school student, while learning physics I came across many laws like,

1) Newtons 3 laws of motion

2) Conservation of energy

3) Conservation of momentum etc.

• Conservation of energy can be proved using 2nd law of motion as in here : https://youtu.be/PplaBASQ_3M

• Conservation of momentum could be proved using 3rd law of motion as in here : https://www.zigya.com/study/book?class=11&board=mbose&subject=Physics&book=Physics+Part+I&chapter=Laws+of+Motion&q_type=&q_topic=Conservation+Of+Momentum&q_category=&question_id=PHEN11037453

• As the first law of motion could be proved using 2nd law. I think that the 2nd and the 3rd laws are the most basic ones.

While solving problems on mechanics, I see people applying these laws. Some laws give answers immidiately, some take more mathematical solving. But which are the only ones which if I have with me could solve all of them neglecting the complexity one would have to face.

## closed as primarily opinion-based by Kyle Kanos, AccidentalFourierTransform, StephenG, sammy gerbil, David Z♦Nov 7 '18 at 0:01

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In my opinion, the most fundamental law of Newtonian physics is the classical limit of the most fundamental law of quantum physics, namely the Feynman sum-over-all-histories principle (also known as the Feynman path integral).

Feynman discovered that the probability amplitude for a quantum system to evolve from state A to state B is proportional to $$\sum_{A \rightarrow B}e^{iS(A\rightarrow B)/\hbar}$$ where the "action" $$S(A\rightarrow B)$$ is a number that depends on how the system evolves between A and B and $$\hbar$$ is Planck's constant divided by $$2\pi$$. (The actual probability is proportional to the square of the magnitude of this sum, and the proportionality constant is determined by making the probabilities sum to 1.) With the proper choice of how to compute $$S$$, this explains quantum systems from the hydrogen atom to the Standard Model of particle physics.

In the limit $$\hbar\rightarrow 0$$, the sum-over-all-histories principle becomes the classical Principle of Stationary Action, $$\delta S = 0$$, which says that the classical path from A to B is the one from which deviations cause no first-order variation in the action. When the action is written as a time-integral of a Lagrangian, this principle is equivalent to the Euler-Lagrange equations.

So a single unifying principle lies behind quantum mechanics, quantum field theory (including the Standard Model), Newtonian mechanics, relativistic classical mechanics, electromagnetism, and General Relativity! I consider it the deepest of all known laws of physics.

This leaves the non-trivial question of how to compute $$S$$. We already know what $$S$$ is for Newtonian mechanics, for relativistic classical mechanics, for non-relativistic quantum mechanics, for electromagnetism, for General Relativity, and for the Standard Model.

The most important thing that stands out about the various expressions for $$S$$ that we've found is that they are invariant under various symmetries. The symmetries of $$S$$ are what give rise to the conservation laws. So we know how to invent new possible actions that might be the right one for, say, quantum gravity. The search for the "ultimate" action that describes a Theory of Everything is underway, under the assumption that it will be even more symmetric than the actions we already know about.

In short: The concept of "action" is what unifies physics. The search for the right action to unify all phenomena is what physics is about.

As a specific example, to learn about Newtonian gravity in this way, start with the action $$S=\int_{t_1}^{t_2}(K-U)dt=\int_{t_1}^{t_2}\left[\frac{1}{2}m\left(\frac{d\vec{r}(t)}{dt}\right)^2+\frac{GMm}{|\vec{r}(t)|}\right]dt$$ for a small test mass $$m$$ moving under gravitational field of a much larger mass $$M$$ (assumed to be stationary at the origin). It happens to be the time integral of the kinetic energy of the system minus its potential energy. Show that $$\delta S=0$$ leads to elliptical orbits, and that energy and angular momentum are conserved.

• Note: I reach basically the same conclusion as Maury Markowitz and Ofek Gillon -- namely that the Lagrangian approach to classical mechanics is the way to go -- but from a more general and unifying starting point. – G. Smith Nov 7 '18 at 0:02

By fundamental, I mean, those laws which if assumed could be used to prove all other laws and provide the essence of the complete picture.

All you need is the Lagrangian and some basic statements about the "universe", whatever that is in the setup.

If you are not familiar with the Lagrangian, it is an interesting beast. Imagine you are throwing a ball from A to B. There are an infinite number of paths that the ball could take where everything is hunky-dory according to Newton. However, the ball always takes one path. This is where the Lagrangian comes in, it says motion is not a couple of different rules but only one - objects follow the path the minimizes the total change in energy.

To construct a Lagrangian you have to add certain constraints, either directly or through some fancy footwork with your choice of coordinate system. Ignoring the later for the moment, the former might be things like "there is a brick wall here and the floor here", or if you want to get all cosmic, things like "the universe is spatially homogeneous, so linear momentum is conserved". Actually doing that can be nasty, which is where Hamilton comes in.

The first law can't be proved using the second law because it kind of defines/postulates the existence of an inertial frame of reference. Read more in the 2nd and 3rd answer here: Why is Newton's first law necessary?

The 2nd law, more accurately, states that $$\frac{dp}{dt} = F$$ (and when the mass is constant, $$p=mv\Rightarrow \frac{dp}{dt} = ma = F$$).

The 3rd law, more accurately, should be the conservation of momentum, which is more universal than the third law (See for example some cases in electromagnetism when 2 charges act on each other with forces that don't cancel out. This is not compatible with Newton's 3rd law but it is with conservation of momentum because some momentum is carried by electromagnetic radiation.

In addition to these 3 laws stating how objects move under the influence of forces, you need the 2 equations which describe every classical interaction: Gravitation and electromagnetism: $$F=-\frac{GMm}{r^2}$$ and $$F = q(E+v\times B)$$

In the end, you also need the 4 equations which describe how electromagnetic fields evolve over time and emerge from currents and charges. These are called Maxwell's equations and are too complicated for highschool level because of the advanced mathematics involved.

I'll conclude with saying you need 7 equations to describe classical physics: one is $$F=\frac{dp}{dt}$$ which describes how objects move due to forces; Two are the descriptions of the 2 basic interactions, and 4 are important for describing electromagnetic fields.

This can be even less. In advanced classical mechanics one learns about the concept of the lagrangian - a function that by plugging it to an equation (Euler Lagrange) you can deduce everything about the system. So you can say there is only one equation: Euler Lagrange.