A Legendre transform replaces dependence on a variable by dependence on its conjugate slope. For a differentiable convex function $f(x)$, define $$p=f'(x).$$ When this relation can be inverted to give $x(p)$, the Legendre transform is $$f^*(p)=px-f(x),$$ with $x=x(p)$.

The geometric idea is that $f^(p)$ describes the intercept information of the tangent line whose slope is $p$. Differentiating gives $$\frac{df^}{dp}=x,$$ so the transformation exchanges the roles of $x$ and the slope $p$.

For $$f(x)=\frac12ax^2,\qquad a>0,$$ we have $p=ax$, hence $x=p/a$. Therefore $$f^*(p)=p\frac{p}{a}-\frac12a\left(\frac{p}{a}\right)^2 =\frac{p^2}{2a}.$$ The quadratic transforms into another quadratic expressed in the conjugate variable.

Legendre transforms appear when a problem is easier in terms of a response variable than the original variable. In mechanics they replace velocities by canonical momenta to construct the Hamiltonian. In thermodynamics they replace inconvenient natural variables to produce potentials such as enthalpy, Helmholtz free energy and Gibbs free energy.

The transform is not merely a change of notation. It reorganizes a function so that derivatives with respect to a different set of independent variables carry the physical information.