Editing Legendre transformation (section)

=== Formal definition in physics context ===
In analytical mechanics and thermodynamics, Legendre transformation is usually defined as follows: suppose <math>f</math> is a function of <math>x</math>; then we have 

::<math>\mathrm{d} f = \frac{\mathrm{d} f}{\mathrm{d} x} \mathrm{d} x.</math>

Performing the Legendre transformation on this function means that we take <math>p = \frac{\mathrm{d} f}{\mathrm{d} x}</math> as the independent variable, so that the above expression can be written as

::<math>\mathrm{d} f = p \mathrm{d} x,</math>

and according to Leibniz's rule <math>\mathrm{d} (uv) = u\mathrm{d} v + v\mathrm{d} u,</math> we then have 

::<math>\mathrm{d} \left(x p - f \right) = x \mathrm{d} p + p \mathrm{d} x - \mathrm{d} f = x\mathrm{d} p,</math>

and taking <math>f^* = xp-f,</math> we have <math>\mathrm d f^* = x \mathrm{d} p,</math> which means 

::<math>\frac{\mathrm{d} f^*}{\mathrm{d} p} = x.</math>

When <math>f</math> is a function of <math>n</math> variables <math>x_1, x_2, \cdots, x_n</math>, then we can perform the Legendre transformation on each one or several variables: we have 

::<math>\mathrm{d} f = p_1\mathrm{d} x_1 + p_2 \mathrm{d} x_2 + \cdots + p_n \mathrm{d} x_n,</math>

where <math>p_i = \frac{\partial f}{\partial x_i}.</math> Then if we want to perform the Legendre transformation on, e.g. <math>x_1</math>, then we take <math>p_1</math> together with <math>x_2, \cdots, x_n</math> as independent variables, and with Leibniz's rule we have 

::<math>\mathrm{d} (f - x_1 p_1) = -x_1 \mathrm{d} p_1 + p_2 \mathrm{d} x_2 + \cdots + p_n \mathrm{d} x_n.</math>

So for the function <math>\varphi(p_1, x_2, \cdots, x_n) = f(x_1, x_2, \cdots, x_n) - x_1 p_1,</math> we have

::<math>\frac{\partial \varphi}{\partial p_1} = -x_1,\quad \frac{\partial \varphi}{\partial x_2} = p_2,\quad \cdots, 
\quad \frac{\partial \varphi}{\partial x_n} = p_n.</math>

We can also do this transformation for variables <math>x_2, \cdots, x_n</math>. If we do it to all the variables, then we have 

::<math>\mathrm{d} \varphi = -x_1 \mathrm d p_1 - x_2 \mathrm{d} p_2 - \cdots - x_n \mathrm{d} p_n </math> where <math>\varphi = f-x_1 p_1 - x_2 p_2 - \cdots - x_n p_n. </math>

In analytical mechanics, people perform this transformation on variables <math>\dot q_1, \dot q_2, \cdots, \dot q_n </math> of the Lagrangian <math>L(q_1, \cdots, q_n, \dot{q}_1, \cdots, \dot{q}_n) </math> to get the Hamiltonian:

<math>H(q_1, \cdots, q_n, p_1, \cdots, p_n) = \sum_{i=1}^n p_i \dot{q}_i -
L(q_1, \cdots, q_n, \dot{q}_1 \cdots, \dot{q}_n). </math>

In thermodynamics, people perform this transformation on variables according to the type of thermodynamic system they want; for example, starting from the cardinal function of state, the internal energy <math>U(S,V)</math>, we have

::<math>\mathrm{d}U = T \mathrm{d} S - p \mathrm{d} V, </math>

so we can perform the Legendre transformation on either or both of <math>S, V </math> to yield

::<math>\mathrm{d} H = \mathrm{d} (U + pV) \ \ \ \ \ \ \ \ \ \ = \ \ \ \ T\mathrm{d} S + V \mathrm{d} p</math> 

::<math>\mathrm{d} F = \mathrm{d} (U - TS) \ \ \ \ \ \ \ \ \ \ = -S\mathrm{d} T - p \mathrm{d} V</math>

::<math>\mathrm{d} G = \mathrm{d} (U - TS + pV) = -S\mathrm{d} T + V \mathrm{d} p,</math>

and each of these three expressions has a physical meaning.

This definition of the Legendre transformation is the one originally introduced by Legendre in his work in 1787,<ref name=":0" /> and is still applied by physicists nowadays. Indeed, this definition can be mathematically rigorous if we treat all the variables and functions defined above: for example, <math>f,x_1,\cdots,x_n,p_1,\cdots,p_n, </math> as differentiable functions defined on an open set of <math>\R^n </math> or on a differentiable manifold, and <math>\mathrm{d} f, \mathrm{d} x_i, \mathrm{d} p_i </math> their differentials (which are treated as cotangent vector field in the context of differentiable manifold). This definition is equivalent to the modern mathematicians' definition as long as <math>f </math> is differentiable and convex for the variables <math>x_1, x_2, \cdots, x_n. </math>