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In the spatially periodic case, the Camassa-Holm equation can be given the following geometric interpretation. The group $$\mathrm{Diff}(S^1)$$ of diffeomorphisms of the unit circle $$S^1$$ is an infinite-dimensional Lie group whose Lie algebra $$\mathrm{Vect}(S^1)$$ consists of smooth vector fields on $$S^1$$. The $$H^1$$ inner product on $$\mathrm{Vect}(S^1)$$,

\left\langle u \frac{\partial}{\partial x}, v\frac{\partial}{\partial x} \right\rangle_{H^1} = \int_{S^1}(uv+u_xv_x)dx, $$ induces a right-invariant Riemannian metric on $$\mathrm{Diff}(S^1)$$. Here $$x$$ is the standard coordinate on $$S^1$$ and the subscripts denote differentiation. Let
 * $$U(x,t)=u(x,t)\frac{\partial}{\partial x}$$

be a time-dependent vector field on $$S^1$$, and let $$\{\varphi_t\}$$ be the flow of $$U$$, i.e. the solution to
 * $$\frac{d}{dt}\varphi_t(x)=u(\varphi(x,t),t).

$$ Then $$u$$ is a solution to the Camassa-Holm equation with $$\kappa=0$$, if and only if the path $$t\mapsto\varphi_t\in\mathrm{Diff}(S^1)$$ is a geodesic on $$\mathrm{Diff}(S^1)$$ with respect to the right-invariant $$H^1$$ metric.

For general $$\kappa$$, the Camassa-Holm equation corresponds to the geodesic equation of a similar right-invariant metric on the universal central extension of $$\mathrm{Diff}(S^1)$$, the Virasoro group.