Lagrangian mechanics

Newtonian mechanics asks you to identify every force, including the ones that exist only to hold the system together — the normal force from a track, the tension in a rigid rod, the reaction at a hinge. These constraint forces are usually unknown at the start, do no work, and are of no interest in the answer, yet you have to solve for them anyway. Lagrangian mechanics is a reformulation that lets you skip them entirely.

The move is to describe the system with as many coordinates as it has genuine freedoms, and no more: one angle for a pendulum, one distance along the track for a bead. Then form the Lagrangian, L = T − V, kinetic energy minus potential energy. The minus sign matters and is the most common slip — L is not the total energy. Feeding L through the Euler–Lagrange equation produces one equation of motion per coordinate, with the constraint forces already gone.

This is not a rival theory. For any problem both methods can handle, they give identical motion, because the Lagrangian formulation is derivable from Newton's laws and vice versa. What changes is the labour and the reach: the same recipe works in whatever coordinates suit the problem, extends cleanly to fields and to quantum mechanics, and makes conservation laws fall out of symmetries — if L does not depend on a coordinate, the quantity paired with it is conserved, with no separate argument required.