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In physics, the conjugate relationship between ##\textbf{r}## and ##\textbf{p}## is considered equivalent to the conjugate relationship ##\textbf{x}## and ##\textbf{x}^*## such that for the Legendre transform between the Lagrangian and the Hamiltonian ##\textbf{r}## and ##\textbf{p}## are used in place of ##\textbf{x}## and ##\textbf{x}^*##. To me at least, this equivalence is not at all obvious.

What is the mathematical basis for this substitution?

In other words, are both the following true for the same variables, and if so, under what conditions?

##\mathscr{F}[g(\textbf{r})]=G(\textbf{p})##

And

##f^*(\textbf{r})=\sup[\langle \textbf{r}, \textbf{p}\rangle - f(\textbf{p}) ]##