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Mathematics for Physics

325 Legendre Transform Derivation

A derivation of the Legendre transform that is used to change the independent variables of a function by subtracting a product of conjugate variables.

definition [d] (Legendre Transform Derivation) From Boas, Chapter 4, Section 11: start from a function \(f(x,y)\) and write its differential

  • \(df = p\, dx + q\, dy\) ,

where

  • \(p = \left(\dfrac{\partial f}{\partial x}\right)_{y}\) , \(\qquad q = \left(\dfrac{\partial f}{\partial y}\right)_{x}\) .

To change the independent variables from \((x,y)\) to \((x,q)\), define a new function by the Legendre transformation

  • \(g = f - q y\) .

Its differential is

  • \(dg = df - q\, dy - y\, dq = (p\, dx + q\, dy) - q\, dy - y\, dq\) ,

which simplifies to

  • \(dg = p\, dx - y\, dq\) .

Thus \(g = g(x,q)\), with

  • \(\left(\dfrac{\partial g}{\partial x}\right)_{q} = p\) , \(\qquad \left(\dfrac{\partial g}{\partial q}\right)_{x} = -y\) .

The same method replaces the \(p\, dx\) term by forming

  • \(h = f - x p\) ,

so that

  • \(dh = q\, dy - x\, dp\)

and \(h = h(p,y)\).

where

  • \(f\) is the original function.
  • \(x\) and \(y\) are the original independent variables.
  • \(df\) is the total differential of \(f\).
  • \(p\) and \(q\) are the partial derivatives of \(f\) that appear as coefficients in \(df\).
  • \(g\) is the Legendre transform of \(f\) that makes \(q\) an independent variable.
  • \(dg\) is the total differential of \(g\).
  • \(h\) is the Legendre transform of \(f\) that makes \(p\) an independent variable.
  • \(dh\) is the total differential of \(h\).
  • \(dx\), \(dy\), \(dp\), and \(dq\) are the differentials of \(x\), \(y\), \(p\), and \(q\).

325.1 Examples

325.1.1 Simple

For \(f(x,y) = \dfrac{1}{2}y^{2} + x\),

\[ q = \left(\dfrac{\partial f}{\partial y}\right)_{x} = y,\qquad g = f - q y = x - \dfrac{1}{2}q^{2} \]

where

  • \(f\) is the original function.
  • \(x\) and \(y\) are the original variables.
  • \(q\) is the conjugate coefficient of \(dy\).
  • \(g\) is the Legendre transform.

325.1.2 General

For \(f(x,y) = \dfrac{1}{2}m y^{2} + U(x)\) with constant \(m > 0\),

\[ q = m y,\qquad y = \dfrac{q}{m},\qquad g(x,q) = U(x) - \dfrac{q^{2}}{2m} \]

where

  • \(f\) is the original function.
  • \(m\) is a positive constant.
  • \(U(x)\) is a function of \(x\) alone.
  • \(q\) is the conjugate coefficient of \(dy\).
  • \(g\) is the Legendre transform.