By: Tao Steven Zheng (鄭濤)

【Problem】

The Beltrami identity, named after the Italian mathematician Eugenio Beltrami (1835 - 1900), is a simplified and less general version of the Euler–Lagrange equation in the calculus of variations.

Show that the Euler-Lagrange equation

can be written as

Then show that if does not explicitly depend on , then

where is constant.

Hint：Use the shorthand and .

【Solution】

Note that the total derivative

can be expressed as

Also,

Substituting the above two expressions into

gives

Simplify this expression and factor out :

Divide away :

Therefore,

is equivalent to the Euler-Lagrange equation.

If does not explicitly depend on , then and

becomes

So by integration

where is constant.