The Beltrami’s identity
where is a constant, is a reduced form of Euler-Lagrange equation for the special case when does not dependent explicitly on .
From Euler-Lagrange equation
Substituting (2) into (1) gives
Consequently, when (3) is expressed as
it becomes clear that
In “A Relentless Pursuit”, we derived differential equation
from Euler-Lagrange equation without taking advantage of the fact that has no explicit dependency on . The derivation was mostly done by a CAS.
Let’s derive (4) from Beltrami’s Identity. This time, we will not use CAS.
For , the Beltrami’s Identity is
It simplifies to
Further simplification yields