# Double Pendulum Cuspiness

Does the curve traced out by the tip of a double pendulum have cusps?

• I don't know, I would need the expression of the curve to tell. But speaking qualitatively from the animation, and complete guesswork, it looks like it is possible for the pendulum to be left shortly in some extreme position, only to be jerked back in a non-smooth manner. – Mankind Feb 26 '16 at 1:40
• The path is smooth in $\mathbb R^4$, simply because of the right-hand sides of the differential equations. But in general it is not smooth in the plane (or would you in fact consider smooth the trajectory of a simple pendulum inside the heteroclinics?). – John B Feb 26 '16 at 1:50

• I guess my main hangup is that in order for the derivative to change sign, it needs to be defined for $t<0$, and I'm not sure how that's true from a physical point of view. – Mark B Feb 26 '16 at 2:27