Level Curve

The set of points where a Constant of Motion is kept constant. Using Total Energy as the Constant of Motion, we get the example of a level curve as the set

$\{(x,\dot{x})\in {\mathbb{R}}^{2n}\mid E(x,\dot{x})=E(x_0,{\dot{x}}_0)\}$

If a Level Curve contains no Critical Points, then the Implicit Function Theorem proves that we have a Smooth Curve.