Question

In the pendulum problem, $\theta =x\sin \sqrt{g/l}t$ is onc solution when the amplitude $x$ is small enough for the motion to be considered simple harmonic. Show that the corresponding exact solution when $\alpha$ is not small is $$\sin \frac{\theta }{2}=\sin \frac{x}{2}\mathrm{sn}\sqrt{\frac{g}{l}}t$$ where $k=\sin (\alpha /2)$ is the modulus of the elliptic function. Show that this reduces to the simple harmonic motion solution for small amplitude $\alpha$.

Short answer

$\sin \frac{\theta }{2}=\sin \frac{x}{2}\mathrm{sn}(\sqrt{\frac{g}{l}}t,k)$ reduces to $\theta =x\sin \sqrt{g/l}t$ for small $\alpha$.

Step-by-step solution

Understand the Given Simple Harmonic Motion

The given simple harmonic motion solution for the pendulum is $\theta =x\sin \sqrt{g/l}t$ where $x$ is the amplitude. This solution is valid under the assumption that $x$ is small and the motion can be considered simple harmonic.

Introduce the Exact Solution

The problem states the exact solution when $\alpha$ is not small: $$\sin \frac{\theta }{2}=\sin \frac{x}{2}\mathrm{sn}(\sqrt{\frac{g}{l}}t,k)$$ where $k=\sin \left(\frac{\alpha }{2}\right)$ is the modulus of the elliptic function.

Understand Elliptic Functions

Elliptic sine functions $\mathrm{sn}(u,k)$ generalize the sine function for larger angles $\alpha$ by accounting for the nonlinearity in the pendulum's motion. $k$ is the modulus parameter.

Relate the Exact Solution to Simple Harmonic Motion

To see how the exact solution reduces to the simple harmonic motion for small $\alpha$, use the fact that elliptic functions approximate trigonometric functions for small arguments of $k$. As $\alpha$ approaches 0, $k=\sin (\alpha /2)$ also approaches 0.

Simplification for Small $\alpha$

When $k$ is small, $\mathrm{sn}(u,k)\approx \sin (u)$. Therefore, for small $\alpha$, the exact solution $\sin \frac{\theta }{2}\approx \sin \frac{x}{2}\sin \sqrt{\frac{g}{l}}t$ simplifies to $\theta \approx x\sin \sqrt{\frac{g}{l}}t$, recovering the simple harmonic motion.

Key concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) describes the motion of a pendulum under certain simplifying assumptions. In SHM, the restoring force is directly proportional to the displacement, leading to a sinusoidal motion. The key equation provided is: $\theta =x\sin (\sqrt{\frac{g}{l}}t)$. Here, $x$ denotes the amplitude, or maximum displacement from the equilibrium position. For SHM to hold true, the angle $\theta$ must be small. This relationship arises because the small angle approximation (which we will discuss later) makes the restoring force linearly proportional to the angle.

Elliptic Functions

Elliptic functions extend the concept of trigonometric functions to account for more complex types of motion, especially nonlinear ones. They are particularly useful in problems like the pendulum, where large angles invalidate simple harmonic assumptions. The specific form given in this problem involves the Jacobi elliptic sine function, $\mathrm{sn}(u,k)$. The equation presented is: $$\sin \frac{\theta }{2}=\sin \frac{x}{2}\mathrm{sn}(\sqrt{\frac{g}{l}}t,k)$$ The modulus parameter $k$ in $\mathrm{sn}(u,k)$ modifies the function based on the initial conditions, such as amplitude. For small angles, these functions can approximate simpler trigonometric forms but provide more accuracy for larger displacements.

Nonlinear Motion

Nonlinear motion arises when the relationship between restoring force and displacement is not simply proportional. For a pendulum, this nonlinearity becomes significant at larger angles. In such cases, the small-angle approximation fails. The exact solution involving elliptic functions: $$\theta =x\sin (\sqrt{\frac{g}{l}}t)$$ morphs into a more complex form to account for nonlinearities. When angles increase, the exact solution better describes the system behavior: $$\sin \frac{\theta }{2}=\sin \frac{x}{2}\mathrm{sn}(\sqrt{\frac{g}{l}}t,k)$$.

Small Angle Approximation

The small angle approximation simplifies the analysis of pendulum motion by approximating $\sin (\theta )$ as $\theta$ (when \ \theta \ is in radians and small). This assumption leads to linear relationships and is the basis for the simple harmonic motion equation given earlier. When angles are small, $\mathrm{sn}(u,k)$ approximates $\sin (u)$. Therefore, the exact solution $$\sin \frac{\theta }{2}=\sin \frac{x}{2}\mathrm{sn}(\sqrt{\frac{g}{l}}t,k)$$ simplifies to $$\theta =x\sin (\sqrt{\frac{g}{l}}t)$$. In this context, $k$, which is the modulus, approaches zero as the angle gets smaller, making $\mathrm{sn}(u,k)$ close to $\sin (u)$. This reduction bridges the exact nonlinear model to the simpler linear model when angles are minimal.