Categories: Mathematics, Physics.

# Beltrami identity

Consider a general functional $J[f]$ of the following form, with $f(x)$ an unknown function:

$\begin{aligned} J[f] = \int_{x_0}^{x_1} L(f, f', x) \dd{x} \end{aligned}$Where $L$ is the Lagrangian. To find the $f$ that maximizes or minimizes $J[f]$, the calculus of variations states that the Euler-Lagrange equation must be solved for $f$:

$\begin{aligned} 0 = \pdv{L}{f} - \dv{}{x} \Big( \pdv{L}{f'} \Big) \end{aligned}$We now want to know exactly how $L$ depends on the free variable $x$.
Of course, $x$ may appear explicitly in $L$,
but usually $L$ also has an *implicit* dependence on $x$ via $f(x)$ and $f'(x)$.
To find a relation between this implicit and explicit dependence,
we start by using the chain rule:

Substituting the Euler-Lagrange equation into the first term gives us:

$\begin{aligned} \dv{L}{x} &= f' \dv{}{x} \Big( \pdv{L}{f'} \Big) + \dv{f'}{x} \pdv{L}{f'} + \pdv{L}{x} \\ &= \dv{}{x} \bigg( f' \pdv{L}{f'} \bigg) + \pdv{L}{x} \end{aligned}$Although we started from the “hard” derivative $\idv{L}{x}$,
we arrive at an expression for the “soft” derivative $\ipdv{L}{x}$
describing only the *explicit* dependence of $L$ on $x$:

What if $L$ does not explicitly depend on $x$ at all, i.e. $\ipdv{L}{x} = 0$?
In that case, the equation can be integrated to give the **Beltrami identity**,
where $C$ is a constant:

This says that the left-hand side is a conserved quantity with respect to $x$, which could be useful to know. Furthermore, for some Lagrangians $L$, the Beltrami identity is easier to solve for $f$ than the full Euler-Lagrange equation. The condition $\ipdv{L}{x} = 0$ is often justified: for example, if $x$ is time, it simply means that the potential is time-independent.

When we add more dimensions, e.g. for $L(f, f_x, f_y, x, y)$,
the above derivation no longer works due to the final integration step,
so the name *Beltrami identity* is only used in 1D.
Nevertheless, a generalization does exist
that can handle more dimensions:
Noether’s theorem.

## References

- O. Bang,
*Nonlinear mathematical physics: lecture notes*, 2020, unpublished.