Question

$$ \theta^{\prime \prime}+\sin \theta=0.2 \sin t, \quad \theta(0)=0, \quad \theta^{\prime}(0)=0 ; \quad 0 \leq t \leq 50 $$ This nonlinear differential equation is used to model the forced motion of a pendulum initially at rest in the vertically downward position. For small angular displacements, the approximation \(\sin \theta \approx \theta\) is often used in the differential equation. Note, however, that the solution of the resulting initial value problem \(\theta^{\prime \prime}+\theta=0.2 \sin t, \theta(0)=0, \theta^{\prime}(0)=0\) is given by \(\theta(t)=-0.1(\sin t-t \cos t)\), leading to pendulum oscillations that continue to grow in amplitude as time increases. Our goal is to determine how the nonlinear \(\sin \theta\) term affects the motion. Plot the numerical solutions for \(\theta(t)\) and \(\theta^{\prime}(t)\). Describe in simple terms what the pendulum is doing on the time interval considered.

Short answer

Question: Based on the generated plots of θ(t) and v(t) of a pendulum's motion on the time interval 0 ≤ t ≤ 50, describe the behavior of the angular displacement (θ) and angular velocity (v) and their implications on the pendulum's motion.

Step-by-step solution

Expressing the Second-Order ODE as a System of First-Order ODEs

To solve the second-order ODE, it is helpful to first rewrite the problem as a system of two first-order ODEs. Let v(t) = θ'(t) be the first derivative. Then, we have the following system of equations: $$ \begin{cases} \theta^{\prime}(t) = v(t) \\ v^{\prime}(t) = 0.2 \sin t - \sin \theta(t) \end{cases} $$ with the initial conditions θ(0) = 0 and v(0) = 0.

Applying Numerical Methods

Now that we have rewritten the problem as a system of first-order ODEs, we can apply a numerical method such as the Euler method or the fourth-order Runge-Kutta method to approximate the solutions for θ(t) and v(t). In this case, we will use the fourth-order Runge-Kutta method, as it provides improved accuracy compared to the Euler method.

Generating θ(t) and v(t) Solutions

Using the fourth-order Runge-Kutta method or another numerical ODE solver, generate the numerical solutions for θ(t) and v(t) on the given time interval \(0 \leq t \leq 50\).

Plotting the Solutions

Plot the numerical solutions for θ(t) and v(t) on the given time interval. Observe the behavior of the pendulum according to the obtained plots.

Analyzing the Pendulum Motion

Based on the generated plots of θ(t) and v(t), analyze the motion of the pendulum. Describe in simple terms how the angular displacement (θ) and angular velocity (v) evolve over time and their implications on the pendulum's motion over the given time interval.

Key concepts

Forced Pendulum Motion

Forced pendulum motion refers to the movement of a pendulum under the influence of an external force. In the provided exercise, the pendulum is subject to periodic forcing as indicated by the term \(0.2 \sin t\) in the differential equation. This term represents a force that varies with time, mimicking scenarios like a swinging pendulum with a varying magnet field beneath it, or a child being pushed on a swing at regular intervals.

This external force can inject energy into the pendulum system, potentially leading to complex motion patterns that differ significantly from the simple oscillations of an unforced pendulum. The force can cause the amplitude of the pendulum's oscillations to vary, potentially leading to chaotic behavior depending on the strength and frequency of the force relative to the pendulum’s natural frequency. As the problem under consideration is nonlinear due to the \(\sin \theta\) term, the resulting motion can be quite intricate and is not easily predicted by simple harmonic principles alone.

Initial Value Problem

An initial value problem (IVP) in the realm of differential equations is a problem where the solution is determined by not only the differential equation itself but also by the values of the solution at a starting point. These starting values are known as the 'initial conditions'.

In the context of the forced pendulum motion exercise, the initial conditions are \(\theta(0) = 0\) and \(\theta'(0) = 0\), signifying the pendulum starts at rest from the vertically downward position. Initial value problems are essential in differential equations as they help in finding specific solutions to the equations, which represent the physical reality of the system being modeled. Solving an IVP helps in predicting how a system will evolve over time from a known starting point.

Numerical Methods for ODEs

Numerical methods for ordinary differential equations (ODEs) are algorithms used to approximate solutions to ODEs that cannot be solved analytically. These methods are crucial in translating mathematical models into meaningful real-world predictions.

Numerical methods include, but are not limited to, Euler’s method, the family of Runge-Kutta methods, and the multistep Adams-Bashforth method. Each method has its advantages and trade-offs between accuracy and computational efficiency. These methods work by discretizing the continuous problem into a finite number of steps and incrementally estimating the solution while controlling errors at each step. Such approaches are invaluable for studying complex systems where analytical solutions are impractical or impossible to determine.

Runge-Kutta Method

The Runge-Kutta method is among the most popular numerical methods for solving ODEs, especially for initial value problems. The fourth-order Runge-Kutta method, frequently referred to simply as 'RK4', is highly regarded for its accuracy and reliability.

The method advances a solution through a series of intermediate steps, using weighted averages of different estimates to increment the solution. This makes it more accurate than simpler methods such as Euler’s method, particularly for problems with rapidly changing solutions or for those requiring higher precision. RK4 is widely used in various fields, such as physics, engineering, and computational biology, to solve problems that cannot be addressed analytically, like the motion of the forced pendulum described in our exercise.