information:extraction
Pendula

A classic, extensively studied nonlinear problem is the dynamics of a frictionless pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation

where gravity points "downwards" and θ {\displaystyle \theta } is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use d θ / d t {\displaystyle d\theta /dt} as an integrating factor, which would eventually yield

which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless C 0 = 2 {\displaystyle C_{0}=2}).

Another way to approach the problem is to linearize any nonlinearity (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at θ = 0 {\displaystyle \theta =0}, called the small angle approximation, is

since sin (θ) ≈ θ {\displaystyle \sin(\theta)\approx \theta } for θ ≈ 0 {\displaystyle \theta \approx 0}. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at θ = π {\displaystyle \theta =\pi }, corresponding to the pendulum being straight up:

since sin (θ) ≈ π − θ {\displaystyle \sin(\theta)\approx \pi -\theta } for θ ≈ π {\displaystyle \theta \approx \pi }. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that | θ | {\displaystyle |\theta |} will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.

One more interesting linearization is possible around θ = π / 2 {\displaystyle \theta =\pi /2}, around which sin (θ) ≈ 1 {\displaystyle \sin(\theta)\approx 1}:

This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.