We study geodesics of the $H^1$ Riemannian metric$$ « u,v » = ∫_0^1 ‹ u(s), v(s)› + α^2 ‹ u'(s), v'(s)› ds$$on the space of inextensible curves $\gamma\colon [0,1]\to\mathbb{R}^2$ with $| γ'|≡ 1$.This metric is a regularization of the usual $L^2$ metric on curves, for which the submanifold geometry and geodesicequations have been analyzed already. The $H^1$ geodesic equation represents a limiting case of the Pochhammer-Chree equation from elasticity theory. We show the geodesic equation is $C^{\infty}$ in the Banach topology $C^1([0,1], \mathbb{R}^2)$, and thus there is a smooth Riemannian exponential map. Furthermore, if we hold one endpoint of the curves fixed, we have global-in-time solutions. We conclude with some surprising features in the periodic case, along with an analogy to the equations of incompressible fluid mechanics.