A "bob" or "simple" pendulum is a pendulum consisting of a single spherical (or point) mass attached to a wire of negligible weight. A "physical" pendulum has extended size and is a generalization of the bob pendulum. An example would be a bar rotating around a fixed axle. A simple pendulum can be treated as a special case of a physical pendulum with moment of inertia
where m is the bob mass and l is the wire length.
The equation of motion of a physical pendulum can be found from the torque on it,
where g is the gravitational acceleration, I is the moment of inertia, is the angular acceleration, and is the angle of the wire measured from the downward vertical . Then
Now define the resonant frequency as
so (3) can be written in the simple form
Writing the equation of motion as two coupled first-order ordinary differential equations
gives the phase portrait Eric Weisstein's World of Math illustrated above.
If the pendulum oscillates but does not rotate (i.e., does not have enough energy to trace a complete circle by crossing
the separatrix Eric Weisstein's World of Math at ), then there is an angle
at which
so
Coupled Pendula, Double Pendulum, Melnikov-Arnold Integral
