Consider a general functional of the following form, with an unknown function:
Where is the Lagrangian. To find the that maximizes or minimizes , the calculus of variations states that the Euler-Lagrange equation must be solved for :
We now want to know exactly how depends on the free variable . Of course, may appear explicitly in , but usually also has an implicit dependence on via and . 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:
Although we started from the “hard” derivative , we arrive at an expression for the “soft” derivative describing only the explicit dependence of on :
What if does not explicitly depend on at all, i.e. ? In that case, the equation can be integrated to give the Beltrami identity, where is a constant:
This says that the left-hand side is a conserved quantity with respect to , which could be useful to know. Furthermore, for some Lagrangians , the Beltrami identity is easier to solve for than the full Euler-Lagrange equation. The condition is often justified: for example, if is time, it simply means that the potential is time-independent.
When we add more dimensions, e.g. for , 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.
- O. Bang, Nonlinear mathematical physics: lecture notes, 2020, unpublished.