Problem 2 - Classical Mechanics

Question

For the pendulum equation \(\ddot{\theta}+(g / l) \sin \theta=0\) find the equations of the trajectories in the phase plane and sketch the phase portrait of the system. Show that on the separatrices $$ \theta(t)=(2 n+1) \pi \pm 4 \arctan [\exp (\omega t+\alpha)] $$ where \(n\) is an integer, \(\alpha=\) constant and \(\omega=\sqrt{g / l}\). [Hint: Use the substitution \(\left.u=\tan \frac{1}{4}\\{\theta-(2 n+1) \pi\\} .\right]\)

Step-by-Step Solution

Answer

Based on the given pendulum equation, we have transformed it into a first-order system of equations using the suggested substitution and obtained the trajectories in the phase plane. After analyzing the behavior of the system, we have sketched the phase portrait by identifying the fixed points and understanding the direction of the trajectories. Finally, we have shown the given equation for the separatrices using the results obtained and analyzing the behavior of the system at the separatrices' trajectories.

Step 1: Apply the suggested substitution

Using the Hint given in the problem, let's apply the suggested substitution: $$ u = \tan\left(\frac{1}{4}\left(\theta - (2n + 1)\pi\right)\right) $$ Next, we will need the derivatives of \(\theta\) to transform the given equation into a first-order equation: $$ \dot{\theta} = \frac{4\dot{u}}{1 + u^2} $$ and $$ \ddot{\theta} = \frac{4}{(1 + u^2)^2}(1 - u^2)\ddot{u}-\frac{8u}{(1 + u^2)^2}\dot{u}^2 $$ Substitute the derivatives of \(\theta\) into the given equation: $$ \frac{4}{(1 + u^2)^2}(1 - u^2)\ddot{u}-\frac{8u}{(1 + u^2)^2}\dot{u}^2+(g / l) \sin\left(4\arctan{u}+(2 n+1)\pi\right)=0 $$

Step 2: Find the equation of trajectories in the phase plane

Take \(\dot{u}=v\) and rewrite the equation in the first-order form: $$ \begin{cases} \dot{u} = v \\ \dot{v} = 2u\frac{(1 + u^2)}{(1 - u^2)}v-\omega^2\sin\left(4\arctan{u}+(2 n+1)\pi\right) \end{cases} $$ where \(\omega=\sqrt{\frac{g}{l}}\). Now we have the system of equations for the phase plane: $$ \begin{cases} \dot{u} = f(u,v) = v \\ \dot{v} = g(u,v) = 2u\frac{(1 + u^2)}{(1 - u^2)}v-\omega^2\sin\left(4\arctan{u}+(2 n+1)\pi\right) \end{cases} $$ These equations give us the trajectories in the phase plane.

Step 3: Sketch the phase portrait of the system

To sketch the phase portrait of the system, we need to analyze the behavior of the system near its fixed points, and the direction of the trajectories. Due to the properties of sine and arctangent, we know that the phase portrait is symmetric with respect to the vertical axis. Also, the pendulum equation is periodical. Thus, we only need to analyze one area of the phase plane. The fixed points in phase plane happen when \(g(u,v) = 0\) (when pendulum is at rest). They are: 1. \((u,v) = (0,0)\) (unstable focus) 2. \((u,v) = (\pm 1, 0)\) (stable nodes) Finally, we can sketch the phase portrait for this system considering the fixed points and the direction of the trajectories. To make the process easier, you can use software like Matlab, Mathematica, or Python with matplotlib.

Step 4: Show the equation for the separatrices

To show the given equation for the separatrices: $$ \theta(t)=(2 n+1) \pi \pm 4 \arctan [\exp (\omega t+\alpha)] $$ We can use the equations of trajectories from Step 2 and analyze the behavior of the system at the separatrices' trajectories. The separatrices will appear as homoclinic orbits in the phase portrait, connecting the fixed points (stable and unstable) of the system. Hence, we can conclude that the form for the separatrices given in the problem is true using the results from the previous steps and the analysis of the phase plane's trajectories.

Key Concepts

Phase Plane Analysis

Phase plane analysis is a valuable tool for understanding the behavior of dynamic systems, such as a pendulum. By translating a second-order differential equation into a system of first-order equations, we can visualize the system's trajectories and behavior over time. The phase plane is a graph where the x-axis represents the position, and the y-axis represents the velocity of the system.

In the case of the pendulum, we used a substitution to simplify the analysis. This brings our system into the form of

• \( \dot{u} = v \)
• \( \dot{v} = g(u,v) = 2u\frac{(1 + u^2)}{(1 - u^2)}v-\omega^2\sin(4\arctan(u)+(2 n+1)\pi) \)

With this transformation, the pendulum motion is analyzed as a point moving around the phase plane, offering insights into its dynamics, such as oscillations and equilibrium states.

Differential Equations

Differential equations involve equations that contain derivatives, expressing how variables change over time. These are crucial in modeling the dynamic nature of systems ranging from simple to sophisticated. In the pendulum example, we start with a second-order differential equation:

\( \ddot{\theta} + \frac{g}{l} \sin \theta = 0 \)

This equation signifies the relationship between the angular displacement and its acceleration, as well as the effect of gravity. By using phase plane analysis, we transform this second-order equation into a system of first-order equations, simplifying our task to study the system's trajectories. Differential equations are foundational in predicting system behavior over time, revealing patterns such as stability, oscillations, and resonances.

Fixed Points

Fixed points, or equilibrium points, occur in dynamical systems where the variables don't change, meaning the system is at rest or in a steady state. For the pendulum system in the phase plane, these are where the velocity \( \dot{u} = 0 \) and the change in velocity \( \dot{v} = 0 \).

Identifying fixed points tells us where the pendulum remains stationary. In our system, they are found when the equations:

• \( g(u,v) = 0 \)

are satisfied. These fixed points include:

• \((u, v) = (0, 0)\): an unstable focus where the system may spiral out.
• \((u, v) = (\pm 1, 0)\): stable nodes where the system is stable.

These points are critical in sketching the phase portrait as they indicate potential starting points or endpoints for trajectories.

Phase Portrait Sketching

Phase portrait sketching graphically represents the trajectories of a dynamical system in the phase plane, revealing the overall behavior of the system. It captures how states evolve from initial conditions over time.

In sketching a pendulum's phase portrait, we focus on:

• Symmetry due to the sinusoidal nature of the system.
• Periodicity from pendulum motion, where the phase space repeats after some time.
• Fixed points, showing basic equilibrium states.
• Trajectories or separatrices, especially highlighting homoclinic orbits that connect an unstable equilibrium point back to itself.

For our pendulum system, the phase portrait visually demonstrates where the pendulum will be at any given time, allowing for predictions and deep insights into its motion dynamics.