This article is missing information about the characteristics of chaotic motion in the system, cf.
Double pendulum#Chaotic motion. Please expand the article to include this information. Further details may exist on the
talk page.(October 2019)
The system is much more complex than a simple pendulum, as the properties of the spring add an extra dimension of freedom to the system. For example, when the spring compresses, the shorter radius causes the spring to move faster due to the conservation of
angular momentum. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum.
Lagrangian
The spring has the rest length and can be stretched by a length . The angle of oscillation of the pendulum is .
Hooke's law is the potential energy of the spring itself:
where is the spring constant.
The potential energy from
gravity, on the other hand, is determined by the height of the mass. For a given angle and displacement, the potential energy is:
where is the
velocity of the mass. To relate to the other variables, the velocity is written as a combination of a movement along and perpendicular to the spring:
The elastic pendulum is now described with two coupled
ordinary differential equations. These can be solved
numerically. Furthermore, one can use analytical methods to study the intriguing phenomenon of order-chaos-order[7] in this system.
Holovatsky V., Holovatska Y. (2019)
"Oscillations of an elastic pendulum" (interactive animation), Wolfram Demonstrations Project, published February 19, 2019.