We cannot imagine the existence of classical and modern physics without Newton's second law in its general form.
Newton's second law is a hypothesis of universal law that does not need mathematical proof.
It is inherent in almost all theories of physics such as Hamiltonian and Lagrangian mechanics, statistical mechanics, thermodynamics, Einstein's relativity and even the QM Schrödinger equation.
The famous Potential plus Kinetic law of conservation of mechanical energy, inherent in most formulas of QM and classical mechanics, is a form of Newton's law.
We would like to see a rigorous proof of the famous E = m c ^ 2 without Newton's second law. [1].
The unanswered question arises:
Can Schrödinger's PDE replace Newton's law of motion?
1-Quora Q/A, Does Newton's law of motion agree with the special theory of relativity?
It doesn't matter. Newtonian mechanics is equivalent to Lagrangian mechanics. For ``simple'' problems it's easier to obtain Newton's equations by applying Newton's laws. For ``more complicated'' problems it's easier to obtain Newton's equations from a variational principle, which is what Lagrangian mechanics amounts to.
In all cases the object of the exercise is to obtain Newton's equations.
I do not know what you mean by "master Newtonian mechanics". From a purely conceptual viewpoint, Newtonian mechanics reduces to three laws.
It is enough to understand the content of those three laws.
Lagrangian mechanics can be obtained from those laws plus the D'alambert principle, see for instance the classic book by Goldstein.
Christian Baumgarten
In a sense, you can forget about Newton's laws because we now know they're wrong. That is, you can just postulate relativistic dynamics and throw Newton's 2nd law in the trash.
But the truth of the matter is that the new relativistic laws were found by requiring they reduce to the Newtonian laws in the correct approximation, so in that sense, you cannot just forget about Newton's laws.
Furthermore, the first law still stands mysteriously. We do not know what or why there exists "absolute space". Einstein thought he could get rid of absolute space, which is why, at the beginning, he erroneously believed his gravitation theory was a "General Relativity" theory. The name, of course, stuck and we just call it General Relativity.
The answer is no, if you want to start from The Principle of Least Action,
The answer is yes, if you want to understand what happened as sequential intellectual processes.
Here is a brief response intended, on the one hand, to thank all my fellow Researchgate contributors for their helpful responses and, on the other hand, to offer them a one-sentence challenge:
How to prove the famous mass energy equivalence relation E=mc^2, without Newton's second law in its general form?
Ismail Abbas,
Your challenge how to prove the famous mass energy equivalence relation E=mc^2, without Newton's second law in its general form reflects the fact that your understanding of physics needs to be adjusted.
"In a sense, you can forget about Newton's laws because we now know they're wrong."
The fact is a new kind of bound state is discovered in the 21st century and explained within Newtonian mechanics, skipping the 20st century as whole.
Orbital motion is a bound state mechanism and the new one is a magnetic bound state. Both of them are interplay of force fields and inertial forces. The new one can be shown using ordinary magnets whilst it displays characteristics of the strong force.
Is it true that Paul Ehrenfest did not understand Newton's second law of motion[1]?
We assume that the giant P. Ehrenfest perfectly understood Newton's three laws of motion.
He knew perfectly well that his "derivation" of Newton's second law of motion is nothing other than showing that Shrödinger's equation conforms to the special case of Newoton's law which is Force = mass x acceleration / which represents a kind of conservation of energy.
There is nothing new here since the law of conservation of energy is inherent in the Schrodiger equation itself.
The problem came from some followers of Ehrenfest who did not see the distinction of Newoton's second law in its general form.
It is worth mentioning that this is a universal law of physics and has no mathematical proof.
However, the old iron guards of SE quantum mechanics believe that SE can replace Newoton's second law in its general form and yet, contrary to this belief,
1-This law applies to the case v
In a classical set-up I like to view Newton's 2nd Law written as F(x(t))/m = x''(t) as an axiom because in a way it defines a dynamical force F [maybe dependent on the position x(t) of the object]. It is successful because it leads for some simple cases to the "right" answer, e.g.
(i) Experiment 1: if the force F=const one obtains as solution x(t) = 0.5*(F/m)* t^2 + v_0*t +x(0) which already Galilei has observed.
(ii) Experiment 2: if you assume the gravitational force to be F(x) = g * M1 *M2 / x^2 you get indeed the observed elliptic orbits of the planets and the Kepler laws
This success was enough at the time of Newton to view it as a basic axiom from which you can derive everything, even the philosophical problem of determinism and the paradox of free will.
The big draw-back of Newton's 2nd Law was that it works well for dynamical forces but when coercive forces are coming into the game it gets rather nasty. The easy way out was indicated by d'Alembert with his "virtual displacements" and formalised by Lagrange who introduced generalised coordinates (which in a natural way include the information about coercive forces) and the variational principle of least action.
If you view the world from Karl Popper's critical rationalism philosophy then Newton's 2nd axiom is useful to us as long as it yields the right results observed in the experiments. However, when relativity or quantum effects are relevant, the Newtonian theory fails and needs to be replaced by something "better".
Still I think there is a way in my opinion classical mechanics (including relativity) to get involved in quantum phenomena. This is because a body with an oscillating magnetic dipole with non-zero time average can be aligned with an external static field both in parallel and antiparallel, mimicking spin up and down, in a sense linking Stern-Gerlach experiment and Kapitza pendulum.
As I understanding the question it is about learning, here about learning mechanics (typically the first topic in a course in Theoretical Physics).
One of the most famous textbooks on the matter - that of Landau and Lifschitz - actually begins with the Lagrangian formulation. This works quite
well, at least for students who are really interested in Theoretical Physics.
Thank you for posing this interesting question. As for the answer from Christian Baumgarten, the concept of force and also actions from surroundings would be lost, if starting with energy principles ?
Feynman's formulation of QM is based on the Lagrangian (inspired by a work of Dirac to which Herbert Jehle draw Feynman's attention). Also for the standard model of elementary particle physics the Lagrangian formulation seems to be the most convenient one. This should be taken into account while reasoning about Christian's proposal.