Is Lagrangian mechanics **circular** at its heart (in that it *assumes* Newton's laws)?

Question

Sorry for just copy-pasting this question from SE here. But I don't know why the user QMechanic there closed my question since "This question does not appear to be about physics within the scope defined in the help center."

The following is what I posted.

This is a question on Lagrangian formulation of mechanics and not Newton's formulation. So, we don't a priori take Newton's laws to be true.

This SE post has answers which brilliantly define mass explicitly using Newton's laws.

Now all the resources which I've come across, which "claim" to formulate Lagrangian mechanics (including Susskind himself) begin by postulating Lagrangian as $L(q, \dot q):={1\over 2}m\dot q^2-V(q)$, and just state $m$ to be the mass of the object.

That's either just brushing details under the rug or total ignorance! We can't just take $m$ to be granted from Newton's laws to formulate the supposedly "independent" (but equivalent to Newtonian mechanics) Lagrangian mechanics! Otherwise it'll be circular -- we'd have used Newton's laws (in the form that there exists a quantity called mass for each object; see the second top-voted answer in the linked post, which takes it to be the second law) in formulating Lagrangian mechanics! We need to *define* mass independently here (and later need to show that this definition is equivalent to that in Newtonian formalism).

Question: So what is a mathematically precise (at least at precise as the answers in the linked post are!) definition of mass in Lagrangian mechanics?

I'll also greatly appreciate if you can show that Lagrangian and Newtonian definitions of mass are equivalent.

Comments

Mass is always defined as that number that makes momentum conservation come out correctly (in collisions, for example). This is the only definition that is possible. There are not two different definitions of mass that need to be shown to be equivalent.

Answer 1

Tha "Lagrangian Mechanics" (LM) is not "circular", but "secondary" with respect to the Newton Mechanics (NM). Maximum that LM can do is to reproduce the NM equations of motion, and it uses the Newtonian initial conditions. In Physics no information about the system at $t=t_2$ is known (but searched), and in LM the initial velocities are hence arbirtary; thus the trajectories are arbitrary too. There is no unique trajectory in such a problem setup. Although mathematically the final contitions at $t=t_2$ make sense, it is unprobable situation in Physics. So the Newtonian aproach prevails.

As to the masses, in the non relativistic case they are additive - when you calculate the center of inertia of a compound system. In the relativistic formulation they are not completely additive, but they are still constants by definition (conserved quantities), as any other constant coefficients in the equations ;-).

Comments

Concerning the first paragraph in you answer, things are the other way round:

The Lagrangian framework is very general, Newton s law can be derived as the corresponding EOMs when applying it to classical mechanics. But the Lagrangian framework can be applied beyond classical mechanics ...
 

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@Dilaton: You have not got my point at all.

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@Dilaton: Also, in order to say the LM is more general, i.e., it describes "something else", you have first to have this "something else" to compare with LM. It means that "something else" constructed by physicists, had been constructed without LM. And normally, "something else" constructions contain inequalities outlining the "something else" notions applicability region. This is absent in LM.

As my Physics professor said (while I was a student), the main physical equations are guessed (or "obtained"), rather than "derived".