Recast all of mechanics as a single function L = T − V and let one equation fall out of it. The Euler–Lagrange equation d/dt(∂L/∂q̇) − ∂L/∂q = 0 reproduces Newton's F = ma, works in any coordinate system, and turns every symmetry into a conservation law. Rendered, not quoted.
SOURCE Lagrange, Mécanique analytique (1788) — the origin of the Euler–Lagrange equation. archive.org scan — 1788 treatise, no canonical DOI: AMBER.
State a system by two coordinates: position q and velocity q̇. Write one scalar, the Lagrangian:
The true motion is the one that makes the action S = ∫ L dt stationary. Setting δS = 0 gives the Euler–Lagrange equation for every coordinate:
For L = ½mv² − V(x) this is exactly m ẍ = −dV/dx — Newton, recovered from a scalar.
Mechanics from one function — Lagrange, 1788. The action the-principle-of-least-action is what L extremizes; the Legendre transform carries L into its twin, the-hamiltonian (H = T + V, phase space). Noether reads conservation laws off L's symmetries. Same physics, three coordinate-free faces.
Re-runs the full selfcheck() live against the engine's pure functions — Newton recovery, cyclic momentum, the pendulum EOM, coordinate independence, and symplectic energy conservation. Confirms green; flips red the instant window 6 corrupts the sign of V.
Natural units: m = 1, k = 1, g = 1, l = 1. A potential V(q), a coordinate q, and a Lagrangian L = T − V. Nothing else — no forces are supplied; they are derived.
Live engine. Forces come from numeric derivatives of L; the pendulum is advanced by a symplectic (leapfrog) integrator that conserves energy.
Phase portrait θ vs θ̇ — a closed orbit means energy is conserved.
Proven, from the pure functions:
“A scalar cannot contain a vector law. T − V is bookkeeping; the physics is still F = ma, so the Lagrangian adds nothing.”
Rebuttal: the sign and structure are forced. Only T − V makes δS = 0 reproduce Newton; T + V gives an anti-restoring force (see window 6). And the payoff is real — constraints, curved coordinates, and fields all follow the same one equation with no free-body diagrams.
“L is total energy, T + V.” No — L = T − V. Energy T + V is the Hamiltonian.
“Plain Euler integration conserves energy.” No — explicit Euler pumps energy and spirals out; only a symplectic scheme keeps the orbit closed.
“The equations only hold in Cartesian coordinates.” No — Euler–Lagrange is coordinate-free; polar gives the same physics.
The disclosed planted void. Flip the Lagrangian to L = T + V (plus, not minus). The Euler–Lagrange force becomes m a = +dV/dx — anti-restoring, physically wrong. The Witness (7) catches it live.