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When studying advanced classical mechanics, we all study about Lagrangians and the Euler-Lagrange equations and their importance. Of course, the Lagrangian is calculated based on the potential and kinetic energy. But usually, this is calculated using the position function and some constants. For example, potential energy can be defined as:

$$U(x) = - \int F(x)dx$$

and

$$F = ma \, .$$

This means that potential energy can be defined with mass and acceleration, which is the second derivative of the position function. Of course kinetic energy is defined as

$$(1/2) mv^2 \, .$$

Again, kinetic energy is defined with mass and velocity, the derivative of the position function. As you can see, it is kind of repetition to use the position function to derive the Lagrangian to figure out the dynamics of a system because that is what the position function tells you. So, what is the point of the Lagrangian? I heard from @Sofia that it may be used when only the potential and kinetic energy functions are known. But in what practical experiments it that done? It id much easier to find the position function. So what uses of the Lagrangian are there that can't be done with the position function?

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    $\begingroup$ Try solving the equations of motion of a double pendulum using Newtonian mechanics, then try it using Lagrangian mechanics. Which is easier? $\endgroup$ – Kyle Kanos Jan 18 '15 at 0:45
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    $\begingroup$ If you know the position as a function of time, then the problem is already solved! Usually one does not initially know the position at any time. Rather, you have some dynamical data, such as the positions of sources of various kinds of fields which act upon your particle. This allows you to calculate the potential seen by the particle, and thus use the Lagrangian (or Newtonian mechanics) to calculate its position. $\endgroup$ – Mark Mitchison Jan 18 '15 at 0:54
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    $\begingroup$ From David Tong's site at damtp.cam.ac.uk/user/tong/dynamics/two.pdf "... there are two very important reasons for working with Lagrange’s equations rather than Newton’s. The first is that Lagrange’s equations hold in any coordinate system, while Newton’s are restricted to an inertial frame. The second is the ease with which we can deal with constraints in the Lagrangian system." $\endgroup$ – mmesser314 Jan 18 '15 at 2:30
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/8903/2451 and links therein. $\endgroup$ – Qmechanic Jan 18 '15 at 7:43
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It's not necessarily the case that the Lagrangian formalism does things Newton's formulation can't do. It does, however, make solving certain systems a whole lot easier. Let's take the example of a double pendulum, or even a single pendulum. When doing Lagrangian mechanics, instead of flailing around in the dark taking sines and cosines and trying to determine the forces exerted on every object individually, you have a very cut-out process:

1) Choose appropriate coordinates. (You should do this in Newtonian mechanics too!)

2) Calculate $T = \frac 1 2 m (\dot x^2 + \dot y^2 + \dot z^2)$ using a coordinate transformation.

3) Find the total potential energy (this can often be as simple as transforming the vertical coordinate and writing $mgh$ in your coordinates!).

4) Turn the mathematical Euler-Lagrange crank, and there's your equation of motion!

Lagrangian dynamics reduces problems that would be very complicated to, basically, a simple and routine calculation. In addition, it's slightly more general - Newton only works in inertial frames while Lagrangian formulations work in arbitrary frames of reference. This is potentially very useful.

In another vein, Lagrangian dynamics is the easiest formulation to generalize to field theory, which makes up a lot of modern theoretical physics. Seeing Lagrangians first in this concrete sense helps a lot with the transition to abstract field theories that the Lagrangian actually ends up defining. In other words, in many ways Lagrangian mechanics is getting ready for the abstraction of field theory.

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    $\begingroup$ Newton only works in inertial frames while Lagrangian formulations work in arbitrary frames of reference Newtonian mechanics works fine in non-inertial frames if you include fictious forces; if properly generalized, it is actually more general than Lagrangian mechanics as it does not require forces to be derived from a potential $\endgroup$ – Christoph Jan 18 '15 at 12:30

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