What is the difference between Newtonian and Lagrangian mechanics in a nutshell? What is Lagrangian mechanics, and what's the difference compared to Newtonian mechanics? I'm a mathematician/computer scientist, not a physicist, so I'm kind of looking for something like the explanation of the Lagrangian formulation of mechanics you'd give to someone who just finished a semester of college physics.
Things I'm hoping to have explained to me:


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*What's the overall difference in layman's terms? From what I've read so far, it sounds like Newtonian mechanics takes a more local "cause-and-effect"/"apply a force, get a reaction" view, while Lagrangian mechanics takes a more global "minimize this quantity" view. Or, to put it more axiomatically, Newtonian mechanics starts with Newton's three laws of motion, while Lagrangian mechanics starts with the Principle of Least Action.

*How do the approaches differ mathematically/when you're trying to solve a problem? Kind of similar to above, I'm guessing that Newtonian solutions start with drawing a bunch of force vectors, while Lagrangian solutions start with defining some function (calculating the Lagrangian...?) you want to minimize, but I really have no idea.

*What are the pros/cons of each approach? What questions are more naturally solved in each? For example, I believe Fermat's Principle of Least Time is something that's very naturally explained in Lagrangian mechanics ("minimize the time it takes to get between these two points"), but more difficult to explain in Newtonian mechanics since it requires knowing your endpoint.

 A: The Lagrangian formulation assumes that in a system, the forces of constraints don't do any work, they only reduce the number degrees of freedom of the system. So one need not know the form of force the constraint forces have unlike Newtonian mechanics!!!
A: In Newtonian mechanics you have to use mainly rectangular co ordinate system and consider all the constraint forces. Lagrange's scheme avoids the considerations of the constraint forces deftly and you can use any set of "generalized coordinates" like angle, radial distance etc. consistent with the constraint relations. The number of those generalized coordinates are the same with the number of degrees of freedom of the system.
In all dynamical systems we arbitrarily choose some generalized co ordinates consistent with the constraints of the system. In Newtonian mechanics, the difference between the kinetic and potential energy of the system gives you the so called Lagrangian. Then we have n number of differential equations. $n$ is the number of degrees of freedom of the system.
The main advantage of Lagrangian mechanics is that we don't have to consider the forces of constraints and given the total kinetic and potential energies of the system we can choose some generalized coordinates and blindly calculate the equation of motions totally analytically unlike Newtonian case where one has to consider the constraints and the geometrical nature of the system.
A: main advantage of lagrangian and hamiltonian mechanics over Newtonian mechanics we can deal with scalar quantities , energy, whereas in the later we have to deal with vector quantities.
Besides this we can approach easily to any system ( e.g. mechanical, electrical, optical etc.) with the lagrangian and Hamiltonian mechanics. But this easy acess can not be achieved with  NEWTONIAN MECHANICS. 
A: To answer the second part of your question, I will give a classic example of harmonic motion. The potential energy of a spring is $U=\frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is the displacement.
Newtonian Mechanics:
$F=m\frac{d^2x}{dt^2}=\frac{-dU}{dx}=-kx$
So $m\frac{d^2x}{dt^2}=-kx$, which is an easy differential equation.
Lagrangian Mechanics:
First we know the Euler-Lagrange Equations $\frac{\partial L}{\partial q} = \frac{d}{dt}\frac{\partial L}{\partial \dot{q}}$, we identify coordinates $q=x$, and we define our Lagrangian $L=T-V$ ($T$ is kinetic energy and $V$ is potential energy).
$T=\frac{1}{2}m\dot{x}^2$
$V=\frac{1}{2}kx^2$
So we plug this all into our little Euler Lagrange Equation, and, solving through, you get (drum roll), $m\frac{d^2x}{dt^2}=-kx$!
Conclusion
So after all of this we get the same equation as with Newtonian mechanics and with a lot more work right? In this example probably, and most other simple systems. However, Lagrangian Mechanics has some very powerful applications. 
Consider the following system: You have multiple pendula connected by springs, and each pendulum begins with some initial position and velocity. How do you go about solving this system? In Newtonian Mechanics it will get extremely complex to work out all the forces involved. However, taking a Lagrangian perspective, much of the hard work is taken care of as you can easily define $q_i=\theta_i$, $\theta$ being the angular displacement of each the pendulum. And instead of having to deal with the various forces you just deal with the potential and kinetic energy. 
An even more, exponentially more, important application is in classical field theory (I know it has some important connections to with QFT, but I am not in any position to knowledgeably comment on that). Electromagnetism and General Relativity are two excellent examples. You can derive Maxwell's equations entirely from the electromagnetic Lagrangian ( $L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+A_{\mu}J^\mu$) and you can derive extremely important results in general relativity from the Hilbert Action ($S_H=\int \sqrt{-g}Rd^nx$) and similar variational principles.
A: *

*Lagrangian mechanics can be derived from Principle of Least Action, or from Newtonian mechanics. This is not a fundamental difference.

*Pretty much. In Newtonian mechanics you start with drawing a bunch of vectors and then list your equations. In Lagrangian mechanics you first identify all constraints, choose generalized coordinates, then write down Lagrangian, and plug it into the Lagrangian equation $\frac{d}{dt}\frac{\partial L}{\partial \dot{q_i}}-\frac{dL}{dq_i}=0$

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*Lagrangian mechanics are better when there are lots of constraints. The more the constraints, the simpler the Lagrangian equations, but the more complex the Newtonian become. Lagrangian mechanics is not very suited for non-ideal or non-holonomic systems, such as systems with friction.

*Lagrangian mechanics is also much more extensible. It can remain almost the same form in hydrodynamics, electrodynamics, electric circuits, special and general relativity, etc.

*Principle of Least Time is closely related to Principle of Least Action, but they are in fact very different. I don't see how you can derive the former from the latter.


A: It is all about Frame of reference. In Newtonian physics you stand at a point and watch something move in relation to the stationary observation point based on forces applied. In lagrangian physics you are the something moving that experiences the forces.  Reynolds transport theorem relates the two frames of reference. 
A: According to Newton: 
$r(t+dt)=r(t)+v(t)dt$    
and 
$\dot{r}(t+dt)=\dot{r}(t)+\frac{F(t)}{m}dt$. 
As you can see, there is no freedom to choose the trajectory - it is determined with the instant values of force and velocity. "Future" is determined with "present".
A particle never "chooses" the optimal trajectory to go from a known position in the past $r(t_1)$ to a known position in the future $r(t_2)$. The future data are not involved in the dynamics. But the least action "principle", apart from good equations, is based on the future data $r(t_2)$ which is mathematically possible but physically meaningless. 
There is no a "least action principle" proceeding only from the initial data. Instead, the Newton equations with the initial data suffice to solve physical problems ;-)
