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Legendre transformation[lə′zhän·drə ‚tranz·fər′mā·shən]
a transformation given by
X = y′(x), Y(X) = xy′(x) - y(x) Y′(X) = x
It follows from these formulas that, conversely,
x = y′(x), y(x) = XY′(X) - Y(X), y′(x) = X
Thus, the transformation is self-dual. The Legendre transformation converts the first-order differential equation
(1) F(x, y, y′) = 0
into the equation
(2) F(Y′, XY′ - Y, x) = 0
which can sometimes be integrated more easily than the initial equation. If we know a solution of (2), we can obtain a solution of (1). The Legendre transformation is also used in the study of differential equations of hydrodynamics. It was named after A. Legendre, who first investigated it in 1789.