Why do we need the Lagrange equation?

An important property of the Lagrangian formulation is that it can be used to obtain the equations of motion of a system in any set of coordinates, not just the standard Cartesian coordinates, via the Euler-Lagrange equation (see problem set #1).

Why do we use Lagrangian formulation over Newtonian formulation?

Whereas the Newtonian formulation requires explicit rewriting of its laws in order to deal with arbitrary coordinate systems, the Lagrangian formulation (which is, if I recall correctly, slightly weaker than the original Newtonian formulation) in turn, allows us to deal with arbitrary coordinate systems on spaces which …

Why do we need Lagrange formalism?

For vectors, unlike scalars care should be taken about the direction as well. So, if we could use scalar quantities in the equations of motion( as we do in Lagrangian) it would be easier for mathematical calculation. The total knowledge of the forces along with the constraints are to be there.

What is the difference between Newtonian and Lagrangian equation of motion?

The Newtonian force-momentum formulation is vectorial in nature, it has cause and effect embedded in it. The Lagrangian approach is cast in terms of kinetic and potential energies which involve only scalar functions and the equations of motion come from a single scalar function, i.e. Lagrangian.

Why is Lagrangian mechanics important?

Lagrangian Mechanics Has A Systematic Problem Solving Method In terms of practical applications, one of the most useful things about Lagrangian mechanics is that it can be used to solve almost any mechanics problem in a systematic and efficient way, usually with much less work than in Newtonian mechanics.

Why is Lagrangian better than Newtonian mechanics?

In short, the main differences between Lagrangian and Newtonian mechanics are the use of energies and generalized coordinates in Lagrangian mechanics instead of forces and constraints in Newtonian mechanics. Lagrangian mechanics is also more extensible to other physical theories than Newtonian mechanics.

What are the advantages of Euler Lagrange formulation?

Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it.

Why are Lagrangian mechanics much more useful compared to Newtonian mechanics in deriving conservation laws?

Lagrangian mechanics, as compared to Newtonian mechanics, is a formulation built on the principle of least action. This makes the Lagrangian formulation extremely useful in almost all areas of physics, because it turns out that, actually, almost all physical theories are based on an action principle.