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I'm talking about the mechanics which uses the minimisation of integral of $L=T-V$ to deduce motion paths.

I've read it is a more generalised version of Newton's formulation because it can be used with generalised co-ordinates.

But when we say $\vec{f}=\frac{d\vec{p}}{dt}$, we are simply referring to a law independent of cartesian co-ordinates. If one wants, one can encode $\vec{f}$ and $\vec{p}$ in polar co-ordinates and deduce the laws of motion in those co-ordinates using the same equation,$\vec{f}=\frac{d\vec{p}}{dt}$, right? The point is that $\vec{f}$ is simply a vector quantity obeying the triangle law of addition. We can encode the vector in rectangular co-ordinates and perform the vector addition using $(x,y)+(z,w)=(x+y,z+w)$, or we can encode it in polar co-ordinates and perform the addition $(x,t)+(y,p)$ using law of co-sines. In the end, both additions rules are following the triangle law.

So the only difference between the two mechanics is that the Newtonian one gives us the behavior of a particle in immediate future given initial conditions and instantaneous forces, while the Lagrangian one gives us the long-term behavior given initial and final end points and the potential field (potential seems like the substitute for Force in this mechanics as it encodes everything about the Force in a scalar field). Is this correct?

So what makes the Lagrangian formulation more fundamental? If anything, the Newtonian formulation seems more fundamental because it can work fine with non-conservative forces, as non-conservative forces can't be encoded as a potential field. Also, as local behavior leads to global behavior instead of the other way around, Newton's laws cause the action to be minimised instead of the other way around.

And how do we even directly talk about potential as a fundamental quantity in Lagrangian mechanics? Unlike forces, there's no way to directly compute potential. We can arrive at $F=\frac{GMm}{r^2}$ by simply measuring accelerations of planets. With potential, we can't compute it without referring to force first, as the formula is $\int Fds$

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Technically Lagrangian dynamics is not more fundamental. As you say, it is derived from Newton's laws. The original motivation, the use of generalised coordinates, was important at the time because celestial mechanics (the first major application for Newton's laws) is better treated in polar coordinates than Cartesian coordinates which were implicitly assumed in Newton's laws. The introduction of vectors changed this, because Newton's laws are most naturally treated in a vector formulation, which makes possible better treatments of generalised coordinates.

However, it is also possible to derive Newton's laws from the Lagrangian formulation. This means that from a strictly logical viewpoint, the formulations are equivalent, equally fundamental. Many physicists are attracted to the idea of deriving physics from an action principle. This is a matter of philosophy, or opinion. For myself, I think the laws of physics should be derived from empirics, not from metaphysics.

A more fundamental treatment is found by noting the equivalence of Newton's second and third laws with conservation of energy and momentum. Conservation of energy and momentum is proven for particle interactions in the standard model in the derivation of Feynman rules, and relies only on general principles of measurement on which relativity and quantum mechanics can be founded. From this point of view, I would say that the Newtonian formulation is in fact the more fundamental.

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If you limit yourself to “simple” mechanical problems, there is little difference between Lagrangian or Newtonian mechanics, save that former is based on a single scalar function while the latter is a vector-based formulation. Here, the trade off between the scalar and vector formulations is somewhat compensated by the need to do “more complicated” mathematical operations, such as taking partial derivatives.

In such elementary problems, the advantage of the Lagrangian formulation is the possibility of easily dealing with constraints, eliminating the need for intermediate equations linked to those constraints, and the ability to choose a small number of generalized coordinates which may have little obvious geometric or physical meaning.

The situation changes if one starts with a purely variational approach (disregarding non-conservative problems) rather than some sort of virtual work argument to obtain the equations of motion. It is difficult to imagine obtaining so easily these equations of motions for fields using a Newtonian (vectorial) approach.

Moreover, as discussed in

Hojman, S.A. and Shepley, L.C., 1991. No Lagrangian? No quantization!. Journal of mathematical physics, 32(1), pp.142-146.

it turns out that equations of motions which are not obtained from a Lagrangian cannot be quantized consistently, positioning the Lagrangian as a natural bridge between classical and quantized fields.

Answering your question thus depends a bit on the context and type of problem you are looking at, but it is clear the Lagrangian (and eventually the Hamiltonian) formulation have a broader reach conceptually and computationally then Newton’s.

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