Beltrami identity
/ bel-TRAH-mee /
Solving the Euler-Lagrange equation in full means handling a second-order differential equation, which can be heavy. The Beltrami identity is a labor-saving shortcut available whenever the integrand L does not depend explicitly on the independent variable x — that is, L is a function of y and y' only, with no separate x sitting in the formula. In that lucky-but-common case you get a first integral for free: the order of the equation drops from two to one before you do any real work.
The identity states that the quantity L minus y' times (the partial of L with respect to y') stays constant along the extremal: L - y' (dL/dy') = C, a constant. You can read it as a conservation law. In fact it is exactly the variational ancestor of energy conservation: when x is time and L is a mechanical Lagrangian, this conserved combination is (up to sign) the Hamiltonian, the total energy, which is constant precisely because the laws have no explicit time dependence. The derivation is short — differentiate L along the curve, use the Euler-Lagrange equation, and watch the x-derivative of the Beltrami combination collapse to zero.
Practically, the Beltrami identity is the right first move for the brachistochrone (its slide-time integrand has no explicit x), the catenary, the minimal surface of revolution, and geodesics on surfaces of revolution — all the classic problems whose integrand happens to lack the independent variable. It converts each into a first-order, often separable, equation you can integrate directly. The one honest condition to remember: it applies only when L has no explicit x. If x appears on its own in the integrand, the conserved quantity is no longer constant and you must solve the full Euler-Lagrange equation instead.
For the catenary, L = y sqrt(1 + y'^2) has no explicit x, so Beltrami gives L - y'(dL/dy') = y/sqrt(1 + y'^2) = C. This first-order equation separates and integrates straight to y = a cosh(x/a) — far quicker than the second-order Euler-Lagrange route.
When x is absent from L, Beltrami hands you a first integral and halves the order of the problem.
Beltrami applies only when L has no explicit x. If x appears on its own, L - y'(dL/dy') is not constant; reaching for Beltrami anyway is a common and costly mistake.