Advantages of Lagrangian Mechanics over Newtonian Mechanics

Here, I'm going to pose a very serious list of doubts I have on Lagrangian Mechanics.

• 
  

  Can we learn Lagrangian Mechanics without studying Newtonian Mechanics?

  
  


• 
  
  

  Does Lagrangian help in solving problems easily, which generally seem to be complicated with Newtonian laws?

  
  


• 
  

  Does Lagrangian make problem solving faster?

  
  

Answer

It is necessary to study Newtonian mechanics to truly understand Lagrangian mechanics since its underlying foundation is Newtonian mechanics. It is essentially a different formulation of the same thing. In a way when doing Lagrangian mechanics you are still doing Newtonian mechanics just in the way of energy. For example, under Lagrangian mechanics, say we have a particle with some kinetic energy, ${T=\frac{1}{2}m\dot{q}^{2}}$, that is in a gravitational field, $V=mgq$. Our Lagrangian is defined as $L=T-V$, so using the Euler-Lagrange equation, ${\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}-\frac{\partial L}{\partial q}=0}$, we would get $m\ddot{q}+mg=0$, which you can see is just Newton's usual sum of forces telling us in this case that the acceleration, $\ddot{q}$, here is just due to gravitational acceleration, $g$.

While this may seem like a convoluted way of getting to the same thing, you can use a different example to solve for a much more complicated system like a double pendulum [pdf link] by both methods to drive the point of why Lagrangian mechanics is the method of choice.

You can see that Lagrange mechanics provides a much more elegant and direct way of solving these complicated systems especially if you start adding in damping or driving mechanisms.

One of the attractive aspects of Lagrangian mechanics is that it can solve systems much easier and quicker than would be by doing the way of Newtonian mechanics. In Newtonian mechanics for example, one must explicitly account for constraints. However, constraints can be bypassed in Lagrangian mechanics. You can also modify the Lagrange equations pretty easily as well if you want to account for something like a driving or dissipation forces.