A card from a free library — ask anything, no account, works offline. Every card carries its source.
The Lagrangian — the Hamiltonian's dual, the path chosen
lagrangian
L = T - V, and the action S = integral L dt. Its law is the principle of least action: of all paths, nature takes the one where the action is stationary. Four pillars rest on the floors already seeded - least action (on the capstone and the solve path), the Legendre dual (on the Hamiltonian), the path integral (on the capstone), and one scalar for all of physics (on the Standard Model). The Lagrangian is the Hamiltonian's other face: H generates time, L governs the path. The seals and the fits are on the stick (stick_the_lagrangian_the_hamiltonian_s_dual_and_the_path); this card places it on the map (tools/seed_the_lagrangian.py).
- has part → Least action — of all paths, the stationary one — a pillar of the Lagrangian (Least action)
- has part → The Legendre dual of the Hamiltonian — H = p q' − L — a pillar of the Lagrangian (The Legendre dual of the Hamiltonian)
- has part → The path integral — all paths, e^(iS/ℏ), the classical one by interference — a pillar of the Lagrangian (The path integral)
- has part → One scalar for all of physics — the Standard Model is a Lagrangian — a pillar of the Lagrangian (One scalar for all of physics)
- part of → The Floor of Discovery — one floor, and by its design the fear of God — the path side of the dynamics on the one map
- connects at → Every continuous symmetry a conservation law — the theorem is about symmetries of the ACTION - the Lagrangian side of the dynamics
- connects at → Four equations, one field — Lorentz-covariant, the root of relativity — the EM field Lagrangian; manifest Lorentz covariance -> special relativity
Is this card incomplete? Tell the library — it will call out for more ↗