Definition A.4.16 Conjugate Variable.
The conjugate variable of \(f(\mathbf{x})\) is the defined as
\begin{align*}
\mathbf{p} \amp \triangleq \nabla f(\mathbf{x}) \, .
\end{align*}
Subsection A.4.3 The Legendre Transform and Dual Coordinates
Consider a strictly convex, differentiable function function \(f: U \subseteq \mathbb{R}^n \to \mathbb{R}\) .
Definition A.4.17 Legendre Transform.
The Legendre transform of \(f\) is the function \(f^*: U^* \to \mathbb{R}\) defined by:
\begin{gather*}
f^*(\mathbf{p}) \triangleq \sup_{\mathbf{x} \in U} \left\{ \langle \mathbf{p}, \mathbf{x} \rangle - f(\mathbf{x}) \right\} \, .
\end{gather*}
Remark A.4.18.
For an invertible mapping \(\mathbf{x} \mapsto \mathbf{p}(\mathbf{x})\text{,}\) the transformed function may be written as:
\begin{gather*}
f^*(\mathbf{p}) = \langle \mathbf{p}, \mathbf{x}(\mathbf{p}) \rangle - f\big(\mathbf{x}(\mathbf{p})\big) \, .
\end{gather*}
Context A.4.3.1 Application to Variational Mechanics
In variational mechanics, the Legendre transform converts the Lagrangian formulation with coordinates \((q, \dot{q}, t)\) into the Hamiltonian formulation with canonical phase-space coordinates \((q, p, t)\text{:}\)
\begin{gather*}
p_i \triangleq \frac{\partial L}{\partial \dot{q}_i}\\
H(q, p, t) \triangleq \sum_i{p_i \dot{q}_i} - L(q, \dot{q}, t) \, .
\end{gather*}
Remark A.4.19.
Here, a second-order system of \(n\) Euler-Lagrange differential equations are mapped to a first-order system of \(2n\) Hamilton’s canonical equations.
