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. The system exhibits chaotic behaviour and is sensitive to initial conditions. The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations. 
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 formula_1 and can be stretched to length formula_2. The angle of oscillation of the pendulum is formula_3. 
The Lagrangian formula_4 is: 
where formula_6 is the kinetic energy and formula_7 is the potential energy. 
See. Hooke's law is the potential energy of the spring itself:
where formula_9 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 formula_11 is the gravitational acceleration.
The kinetic energy is given by:
where formula_13 is the velocity of the mass. To relate formula_13 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 formula_2 and formula_3, the equations of motion can be found using two Euler-Lagrange equations:
For formula_2:
formula_24 isolated:
And for formula_3:
formula_28 isolated:
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.