Question

Compare and contrast Newtonian, Lagrangian and Hamiltonian method in deriving the dynamics of a physical system.

Answer 1

Newtonian mechanics takes a more local "cause-and-effect, apply a force, get a reaction" view, while Lagrangian mechanics takes a more global "minimize this quantity" view. More axiomatically, Newtonian mechanics starts with Newton's three laws of motion, while Lagrangian mechanics starts with the Principle of Least Action.

Whereas Newtonian mechanics is based on Cartesian coordinates, Lagrangian mechanics is independent of any particular coordinate system (although the two formulations are mathematically identical).

Recognizing the importance of conserved quantities, Hamiltonian mechanics was derived from the Lagrangian formulation. In Hamiltonian mechanics, the conserved quantities become more apparent. The Hamiltonian function H is calculated from the Lagrangian via a change of variables called the Legendre transforms.