Question

what is the basic difference between Newtonian and Lagrangian mechanics

Answer 1

Newtonian Mechanics:

F=md2xdt2=−dUdx=−kx
So md2xdt2=−kx, which is an easy differential equation.

Lagrangian Mechanics:

First we know the Euler-Lagrange Equations ∂L∂q=ddt∂L∂q˙, we identify coordinates q=x, and we define our Lagrangian L=T−V (T is kinetic energy and V is potential energy).

T=12mx˙2
V=12kx2
So we plug this all into our little Euler Lagrange Equation, and, solving through, you get (drum roll), md2xdt2=−kx!

Answer 2

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 qi=θi, θ 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=−14FμνFμν+AμJμ) and you can derive extremely important results in general relativity from the Hilbert Action (SH=∫−g−−−√Rdnx) and similar variational principles.

Answer 3

thank you very much .....

Answer 4

no pro..

Answer 5

still m thinking so why r we using newtonian mechanics while we have langrangian mechanics?? is this only for convenience?

Answer 6

"Lagrangian mechanics" is, fundamentally, just another way of looking at Newtonian mechanics

Answer 7

i can tell u the use of lagrangian mechanics

Answer 8

So why use it? Because...

... Newtonian mechanics has a problem: It works very nicely in Cartesian coordinates, but it's difficult to switch to a different coordinate system. Something as simple as changing to polar coordinates is cumbersome; finding the equations of motion of a particle acting under a "central force" in polar coordinates is tedious. The Lagrangian formulation, in contrast, is independent of the coordinates, and the equations of motion for a non-Cartesian coordinate system can typically be found immediately using it. That's (most of) the point in "Lagrangian mechanics".

Answer 9

hmm i got it thanks....

Answer 10

http://www.wisegeek.org/what-is-newtonian-mechanics.htm
maybe this can help.

Answer 11

thanks a lot