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
.
For ,
From Euler-Lagrange equation
we have
Substituting (2) into (1) gives
Consequently, when (3) is expressed as
it becomes clear that
Therefore,
i.e.,
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
For ,
Therefore,