Elastic pendulum. In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains  elements of both a simple pendulum and a one-dimensional spring-mass system. For specific energy values, the system demonstrates all the hallmarks of chaotic behavior and is sensitive to initial conditions.At very low and very high energy, there also appears to be regular motion.  The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations.This behavior suggests a complex interplay between energy states and system dynamics.

Analysis and interpretation

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 . 

The Lagrangian  is:

where  is the kinetic energy and  is the potential energy. 

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 gravitational acceleration.

The kinetic energy is given by:

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:

So the Lagrangian becomes:

Equations of motion
With two degrees of freedom, for  and , the equations of motion can be found using two Euler-Lagrange equations:

For :

 isolated:

And for :

 isolated:

These can be further simplified by scaling length  and time . Expressing the system in terms of  and  results in nondimensional equations of motion. The one remaining dimensionless parameter  characterizes the system.

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 in this system for various values of the parameter  and initial conditions  and .

See also
 Double pendulum
 Duffing oscillator
 Pendulum (mathematics)
 Spring-mass system

References

Further reading

External links
 Holovatsky V., Holovatska Y. (2019) "Oscillations of an elastic pendulum" (interactive animation), Wolfram Demonstrations Project, published February 19, 2019.
 Holovatsky V., Holovatskyi I., Holovatska Ya., Struk Ya. Oscillations of the resonant elastic pendulum. Physics and Educational Technology, 2023, 1, 10–17, https://doi.org/10.32782/pet-2023-1-2 http://journals.vnu.volyn.ua/index.php/physics/article/view/1093

Chaotic maps
Dynamical systems
Mathematical physics
Pendulums