Math & Kinematics

Euler-Lagrange Equation

Euler-Lagrange Equation is the differential equation of motion obtained from a system's Lagrangian, the difference between kinetic and potential energy, via the principle of stationary action. Applied to a manipulator with joint coordinates, it yields the standard robot dynamics form involving the mass matrix, Coriolis and centrifugal terms, and gravity vector. This formulation is the basis for computed-torque control, inverse dynamics, and the dynamics engines inside physics simulators.

Why it matters for physical AI

The manipulator equation derived this way is what simulators integrate and model-based controllers invert, so its structure quietly defines both the training environments and the control layers beneath learned policies.