Analytical Mechanics (Lagrangian and Hamiltonian) are useful in Physics (e.g. in Quantum Mechanics) but are they also used in application, by engineers? For example, are they used in designing bridges or buildings?
I'm not an engineer myself, but as far as I know, Lagrangians and Hamiltonians have their use in complicated but calculable situations. In order to solve the e.o.m. for a bridge or so, most engineers rely on specific programs for exactly such a purpose that calculate the statics using Newtonian mechanics and a finite element approach.