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.