Without external drive a pendulum has constant energy, as determined by the initial conditions. At rest (ωA = 0) and at height yA above the lowest point y0 = 0 the initial energy is 100% potential energy:
Epot_max = mg ( yA - y0 )
In movement potential and kinetic energy are exchanged continuously with constant sum. For a single pendulum all energy is kinetic at the lowest point y = 0.
Ekin_max = m v2/2 = m (rω)2/2 = mg (yA - y0) = Epot_max
For the double pendulum the initial potential energy is distributed between kinetic and potential energy of both bobs, while the secondary one oscillates around the primary one.
The simple pendulum is described by one nonlinear ordinary second order differential equation with two initial condition (angle and angular momentum):
It is nonlinear because of the nonlinear trigonometric function that connects oscillation angle with angular velocity. The second order differential equation is equivalent to description by two first order differential equations, the first of which (1) is linear, the second one (2) nonlinear. The nonlinearity lies in the simple relation between angle and angular velocity, described by f. With α angle of oscillation, ω angular velocity, g gravitiy acceleration, L pendulum length:
dα2/dt2 = -g/L sin a
➡
(1) dα/dt = ω;
(2) dω/dt = f (α)
f (α) = - g/L sin α
The double pendulum is described by two nonlinear, coupled, ordinary differential equations of second order with four initial conditions (angle and angular momentum of both pendulums). This is equivalent to four equations of first order, all of which are nonlinear because of trigonometric functions and the quadratic coupling terms shown in red:
(1) dα1/dt = ω1;
(2) dω1/dt = f 1(α1 , α2 , ω1 , ω2)
f1 () = (-g/L1*((m1+m2)sina1- m2sina2cos(a1-a2))
- m2sin(a1-a2)(L2/L1ω22- ω12cos(a1-a2))) / (m1+m2sin2(a1-a2))
(3) dα2/dt = ω2;
(4) dω2/dt = f 2(α1 , α2 , ω1 , ω2)
f2 () = -g/L2*sina2+L1/L2ω12sin(a1-a2)
-L1/L2cos(a1-a2)*(-g/L1*((m1+m2)sina1- m2sina2cos(a1-a2))
- m2sin(a1-a2)(L2/L1ω22- ω12cos(a1-a2))) /(m1+m2sin2(a1-a2))
The equations for the second derivatives f are far more complicated than for the single pendulum. They depend on the relations of the pendulum lengths and masses, and on the angle and angular velocity of both pendulums. The fact that there are two coupled nonlinear differential equations is the deeper cause of deterministic chaotic behavior. When restarting the movement with exactly the same initial conditions (as the computer does with Reset), the same trace is reproduced. Yet when trying to create the same initial condition by moving the pendulum with the mouse, the path will develop differently, as it depends critically on the exact value of the initial conditions. This is best observed when superimposing traces of two oscillations (choose pause but not clear, draw bob and start).
The chaotic behavior remains when some of the nonlinear terms are removed or changed, as long as the nonlinear character of the equations is preserved (the equations then no longer model the double pendulum). The decisive origin of chaotic behaviour is the existence of more than one nonlinear differential equation.
With m2 = 0 (0r L2 = 0) the first equation is reduced to that of the single pendulum, the second equation becomes identical to zero. The simulation then shows the periodic, generally nonlinear, oscillation of a single pendulum.
With drive the differential equation for the primary pendulum is
f1_drive = f1 + A cos(delta*t)
where A is the amplitude of the drive, delta its frequency in relation to that of the free pendulum at small amplitudes.