Lagrangian is used in path integral calculations in quantum field theory.

In non-equilibrium statistics, when using the Fokker-Planck dynamics, it is possible to associate a Lagrangian to this equation, then the solution of the Fokker-Planck (propagator) can also be obtained using an extremal criterium like in clasical mechanics.

There may be be also advantages from the numerical point of view : see Lagrangian numerical solvers, Eulerian solvers, Lagrangian remapping, ...

It is also easier to handle multiphysics coupling in the Lagrangian frame, or get mathematical properties.

I think it is important to mention variational calculus. The Lagrangian method comes from the so-called "principle of least action".

@ David, Some examples in engineering field where Lagrangian concept is used are:

Civil Engineering: Brachistochrone problem, Given two points 1 and 2, with 1 higher above the ground, in what shape should we build a frictionless roller coaster track so that a car released from point 1 will reach point 2 in the shortest possible time?  The concept is used in the designing of ski-jump platforms.

Another example is the design of a bridge of "catenary" in shape.

Automobile and Rocket Engineering: There are various moving parts in cars/bikes/rockets. Lagrangian mechanics is used to achieve optimal control status of these dissipative systems.

Hope that helps.

while I am not  a physict I like of the hamiltonian not a physicist but I view it as the sum of kinetic and potential energy, however those needn't be numbers and are contingent for potential and kinetic energy.

@David

L - deals with q, so the solution of differential equation becomes easier..

B.Rath