Euler-Lagrange Equation
The fundamental partial differential equation of the calculus of variations that a function must satisfy to make a functional stationary.
Governing formula
(d / dt) (∂L / ∂q̇_i) - (∂L / ∂q_i) = 0
SI unit
Calculus of Variations Equation
In depth
Applying Hamilton's principle δS = 0 yields the Euler-Lagrange equations, automatically incorporating holonomic constraints in generalized coordinates q_i.
Examples in the real world
Deriving equations of motion for a double pendulum or brachistochrone curve.