Legendre - Legendre Transformation Shina Tan Legendre...

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Legendre Transformation Shina Tan Legendre transformation is useful in both mechanics and thermodynamics. Given any smooth function f ( x ) defined in the domain x 1 < x < x 2 , if f 00 ( x ) > 0 for all x in this domain, so that f ( x ) is strictly convex 1 , we can define g ( y ) = x df dx - f, (1) where y = dx . (2) The function g ( y ) is called the Legendre transform of f ( x ). For example, the Legendre transform of L ( v ) = 1 2 mv 2 (Lagrangian of a free particle in one dimension) is H ( p ) = p 2 2 m (Hamiltonian of a free particle in one dimension). Because f 00 ( x ) = y 0 ( x ) > 0, y is a monotonically increasing function of x , so the function y ( x ) can be inverted to yield x = x ( y ). The existence of the inverse of y ( x ) is necessary, since we need to first compute xf 0 ( x ) - f ( x ), and then express x in terms of y . The function g ( y ) is defined in the domain y 1 < y < y 2 , where y 1 = lim x x 1 dx , y 2 = lim x x 2 dx . (3) Theorem 1: g ( y ) is also a convex function and, moreover, g 00 ( y ) = 1 /f 00 ( x ) . Proof: dg dx = d dx [ xf 0 ( x ) - f ( x )] = x 0 f 0 ( x )+ xf 00 ( x ) - f 0 ( x ) = f 0 ( x )+ xf 00 ( x ) - f 0 ( x ) = xf 00 ( x ). So dg dy = dg/dx dy/dx = xf 00 ( x ) 0 ( x ) /dx = x . Thus dg dy = x, (4) and g 00 ( y ) = d dy dg dy = d dy x = dx dy . (5) On the other hand, f 00 ( x ) = d dx dx = d dx y = dy dx . (6) Thus g 00 ( y ) is the reciprocal of f 00 ( x ), and is positive. So g ( y ) is also a convex function. Theorem 2: Any smooth and strictly convex function’s Legendre transform’s Legendre transform is that function itself.
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Legendre - Legendre Transformation Shina Tan Legendre...

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