Instability of closed p-elastic curves in 𝕊2

Abstract

For [Formula: see text], we show that non-circular closed [Formula: see text]-elastic curves in [Formula: see text] exist only when [Formula: see text], in which case they are classical elastic curves, or when [Formula: see text]. In the latter case, we prove that for every pair of relatively prime natural numbers [Formula: see text] and [Formula: see text] satisfying [Formula: see text], there exists a closed spherical [Formula: see text]-elastic curve with non-constant curvature which winds around a pole [Formula: see text] times and closes up in [Formula: see text] periods of its curvature. Further, we show that all closed spherical [Formula: see text]-elastic curves for [Formula: see text] are unstable as critical points of the [Formula: see text]-elastic energy.