The behavior of the classical pendulum is governed by Newton's law
Up: CDS 280 - Info
Next: damped pendulum with torque
The following references were used for this example
H.W. Broer & F. Verhulst, Dynamische Systemen en Chaos,
Epsilon uitgaven 14, Utrecht 1990.
V.I. Arnold, Mathematical Methods of Classical Mechanics,
second edition, Springer-Verlag, 1989.
The pendulum is the classical example of a deterministic system. Its behavior is goverened by Newton's law F = m a, which is a second-order differential equation, since the acceleration a is the second derivative of the displacement. The pendulum consists of a point-mass m on a stiff massless rod of length l. We assume that the pendulum oscillates in a plane, and that its pivot is such that the pendulum can make full rotations.
The behavior of the classical pendulum is governed by Newton's law
Let 
denote the angle between the
rod and the vertical axis. We assume that only gravitation influences the
pendulum, such that the force F in
Newton's law is given by
|
F = -m g sin
. |
The point-mass is at arclength distance
l
from the vertical. Hence, the
acceleration a is given by the
second derivative of l
with respect to time
t. The behavior of the pendulum is
therefore given by the differential equation
m a = m l
= -m g sin
, |
or equivalently
| (1) |
= -
 2
sin , |
where =
(g/l).
From now on we assume g = 9.8
m/s2 and l = 1.
Note that the mass m no longer
appears in the equations. This means that mass does not influence the
oscillations of the pendulum. This fact has been known
experimentally since Galilei.
Equation (1) is an example of a system with one
degree of freedom ( is
one-dimensional). If we assume that
m = 1 and
l = 1, then the total
energy, H = H(
,
), is
defined as the sum of the kinetic energy,
2,
|
and the potential energy,
 |
The law of conservation of energy says that the total
energy of points moving according to equation
(1) is conserved: the energy function
H(
(t),
(t)) is independent of
t. This means that on each solution
of (1) the value of the total energy is
constant. Therefore, the energy level sets H(
,
) =
constant, correspond to solutions of equation
(1).
| 10 |  |
||
|
|||
| -10 | |||
-2 |
|
+2 |
|
The level sets of the energy function give the solutions to
equation (1).
We will now consider the case where apart from gravitation other forces influence the behavior of the pendulum.
Written by Hinke Osinga
Last modified: Sat Feb 27 18:55:15 1999