Lagrange's Equations from Newton's Law and Elementary Calculus

Newton’s equations for a constrained system contain both the motion and the constraint reactions, although the motion is often the quantity sought. Starting with the planar pendulum, we compare two positions satisfying the constraint at the same time, take the scalar product of Newton’s equation with their difference, divide by the coordinate increment, and pass to the limit. The unknown tension then disappears. Repeating the calculation for the two angles of a spherical pendulum gives two independent equations of motion. We next express their inertial terms as conditions on the velocity and coordinate partial derivatives of a common function, and construct that function by successive integration. Kinetic energy, potential energy, and the Lagrangian emerge from this calculation. The same procedure extends to particle systems with any finite number of independent generalized coordinates. The derivation uses Newton’s second law, scalar products, ordinary derivatives, partial derivatives, and integration. It takes no virtual displacement, virtual time, virtual work, d’Alembert principle, or total-differential formula as a prerequisite. Complete calculations, geometric illustrations, and Lean checks of selected identities are provided. AI assistance and verification scope ChatGPT and Codex were used to assist with English translation and Lean code. Formal verification is limited to the selected identities and explicit assumptions described in the manuscript. The author takes responsibility for the content of this preprint.