Lagrangian Mechanics: Physics Through Optimization

Lagrangian mechanics reformulates physics by focusing on energy, not forces.
The Lagrangian is defined as kinetic energy minus potential energy.
Instead of tracking forces, it finds the path that minimizes “action.”
The principle of least action says nature always takes the most efficient route.
Lagrange’s equations replace Newton’s laws for complex systems.
This approach works especially well with constraints, like pendulums or orbits.
It generalizes easily to fields, forming the basis of modern physics theories.
Electromagnetism and quantum mechanics can both be written in Lagrangian form.
Symmetries in the Lagrangian lead directly to conservation laws via Noether’s theorem.
At its heart, Lagrangian mechanics reveals the universe as an optimizer of motion.