Question

$$ \theta^{\prime \prime}+\sin \theta=0, \quad \theta(0)=0, \quad \theta^{\prime}(0)=2 ; \quad 0 \leq t \leq 20 . $$ This problem models pendulum motion when the pendulum is initially in the vertically downward position with an initial angular velocity of \(2 \mathrm{rad} / \mathrm{s}\). For this conservative system, it was shown in Chapter 6 that \(\left(\theta^{\prime}\right)^{2}-2 \cos \theta=2\). Therefore, the initial conditions have been chosen so that the pendulum will rotate upward in the positive (counterclockwise) direction, slowing down and approaching the vertically upward position as \(t \rightarrow \infty\). The phase-plane solution point is moving on the separatrix; thus, loosely speaking, the exact solution is "moving on a knife's edge." If the initial velocity is slightly less, the pendulum will not reach the upright position but will reach a maximum value less than \(\pi\) and then proceed to swing back and forth. If the initial velocity is slightly greater, the pendulum will pass through the vertically upright position and continue to rotate counterclockwise. What happens if we solve this problem numerically? Plot the numerical solutions for \(\theta(t)\) and \(\theta^{\prime}(t)\). Interpret in simple terms what the numerical solution is saying about the pendulum motion on the time interval considered. Does the numerical solution conserve energy?

Short answer

Answer: The objective is to find and interpret the motion of the pendulum on the given time interval, \(0 \leq t \leq 20\), and to check if the numerical solution conserves energy.

Step-by-step solution

- Convert the second-order ODE to a system of two first-order ODEs

We first convert the given second-order ODE into a system of two first-order ODEs, by introducing: $$\phi = \theta^{\prime}$$ This results in the following first-order ODEs: $$ \frac{d\theta}{dt} = \phi \\ \frac{d\phi}{dt} = -\sin\theta $$

- Apply numerical methods to solve the ODE system

We can use numerical methods for solving the ODE system, such as the Runge-Kutta method or the Euler method. Any coding language or software like Python, Matlab, or Mathematica can be used to implement the chosen numerical method, and to obtain the solutions for \(\theta(t)\) and \(\theta^{\prime}(t)\). Make sure to use the given initial conditions while implementing the method.

- Plot the solutions and analyze the motion

After obtaining the numerical solutions for \(\theta(t)\) and \(\theta^{\prime}(t)\) using the chosen numerical method, plot these solutions over the given time interval, \(0 \leq t \leq 20\). Analyze the motion of the pendulum based on the shapes of these plots.

- Check energy conservation

To check if the numerical solution conserves energy, substitute the numerical values of \(\theta(t)\) and \(\theta^{\prime}(t)\) into the energy conservation equation: $$\left(\theta^{\prime}(t)\right)^{2}-2 \cos \theta(t)=2$$ Plot the left-hand side of this equation over the same time interval \(0 \leq t \leq 20\). If the graph remains constant and equal to 2, it indicates that the numerical solution conserves energy.

Key concepts

Pendulum Motion

A pendulum is a weight suspended from a pivot so that it swings freely under the influence of gravity. In this case, the pendulum motion is described by the differential equation: \( \theta^{\prime \prime}+\sin \theta=0 \), where \(\theta(t)\) represents the angular position of the pendulum over time. This equation illustrates the periodic nature of pendulum motion, characterized by swings that oscillate back and forth.
Initially, the pendulum is positioned straight down with an angular velocity of \(2 \mathrm{rad/s}\). It starts rotating upwards and, depending on the initial conditions, it can reach a maximum angle before swinging back, or continuously revolve.
When solving these equations numerically, we aim to predict the future motion of the pendulum based on given conditions. This relies on converting the second-order differential equation into a system of first-order equations for easier computational handling.

Energy Conservation in Numerical Methods

Energy conservation in pendulum motion refers to the principle that, in the absence of external forces like friction, the total mechanical energy of the system should remain constant. For the numerical solutions of the differential equations, this means checking if \( \left(\theta^{\prime}(t)\right)^{2} - 2 \cos \theta(t) = 2 \) holds true over time.
In theory, these equations imply that potential energy at the maximum swing height converts completely into kinetic energy at the lowest point. But numerically solving these systems can sometimes cause small errors that lead to energy loss or gain over time. This is critical for understanding the limitations of numerical methods in maintaining a system's energy.

• Numerical precision: Rounding errors and numerical approximation can introduce energy variations.
• Algorithm choice: The choice of method (e.g., Euler vs. Runge-Kutta) impacts how well energy conservation is preserved.

Understanding this helps in choosing the right numerical techniques and parameters to minimize energy-related inaccuracies.

Runge-Kutta Method

The Runge-Kutta method is a popular approach for solving ordinary differential equations numerically. It combines simplicity with greater accuracy compared to the Euler method and is widely used due to its balance between computational efficiency and accuracy.
The method proceeds by calculating intermediate values of the solution over a small time step, accumulating these to estimate the function's value at future points.

• Each step utilizes weighted averages of calculated slopes, maintaining accuracy.
• It is particularly effective in preserving the qualitative features of conservative systems, like a pendulum.

For this particular pendulum problem, Runge-Kutta can more accurately predict the angular position \(\theta(t)\) and velocity \(\theta^{\prime}(t)\) over time, maintaining the system's energy balance compared to simpler methods. While numerical methods inherently involve some approximation, Runge-Kutta minimizes the deviation in energy due to its iterative refinement approach.