Cartesian equation: , or , with . Special case (blue curve) k = 1: double sinusoid: .

Since the total energy of the simple pendulum of mass m and length l forming an angle  with the vertical is equal to , the equation of the isoenergy curve is . They are the curves above, with .
The two blue sinusoids above are the limit between the oscillating pendulum (closed internal curves) and the turning pendulum (external curves).

See on this page a superb animation tracing the isoenergy curves, including the case of the motion with friction.
 
 

With , we get a cylindrical winding of this curves: they are bicylindrics with equation: .

 
 

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© Robert FERRÉOL  2017