Note on bicharacteristics

Let \(H \colon T^*M \to \mathbb{R}\) be a Hamiltonian function and consider the differential equation

\begin{align} H(d\psi) & = 1. \end{align}

The equation \(H = 1\) defines a hypersurface \(\Sigma\) and the graph of the solution \(\psi\) defines a Lagrangian submanifold \(L \subset T^*M\). The differential equation is equivalent to \(L \subset \Sigma\).

Now consider the Hamiltonian vector field \(X\) defined using the canonical symplectic form of the cotangent bundle. The hypersurface \(\Sigma\) is foliated by the integral curves, which are called bicharacteristics. When restricting \(X\) to \(L\), the bicharacteristics also foliate \(L\): note that for any tangent vector \(\xi \in TL\), we have \(\omega(X, \xi) = dH(\xi) = 0\), since \(H\) is constant on \(\Sigma\); by definition of Lagrangian submanifolds, this implies that \(X \in TL\).

Now let \((x^\lambda, p_i)\) be an integral curve of \(X\) contained in \(L\), which means that \(p(t) = d\psi(x(t))\) for all \(t\). We have:

\begin{align} \frac{d}{dt}\psi(x(t)) = \langle p(t), \dot{x}(t)\rangle. \end{align}

Thus we may integrate the right hand side to obtain \(\psi\) on \(x(t)\).

For a concrete example using Maxwell's equations see Guillemin and Sternberg, Geometric Asymptotics, Chapter III.