Pendulum Dynamics and Stability Analysis

This lesson explores the application of nonlinear systems of ordinary differential equations using the pendulum equation. It covers the transformation of the pendulum equation into a nonlinear system, identification of critical points, construction of the Jacobian matrix, and determination of eigenvalues for stability analysis. The video also discusses phase diagrams, equilibrium solutions, and the use of the Hamiltonian to determine trajectories.

What is the primary characteristic of the pendulum equation discussed in the introduction?

• It is a linear equation.

• It is a conservative system.

• It includes friction.

• It is a quadratic equation.

In the nonlinear system, what does Theta Prime represent?

• Gravitational acceleration

• Pendulum length

• Angular velocity

• Angular displacement

What is the significance of the Jacobian matrix in analyzing the system?

• It measures the pendulum's length.

• It helps determine the system's stability.

• It calculates gravitational force.

• It predicts angular displacement.

When are the eigenvalues of the Jacobian matrix real?

• When Theta is zero.

• When cosine Theta is negative.

• When sine Theta is zero.

• When cosine Theta is positive.

What type of critical points are found at even multiples of Pi?

• Unstable nodes

• Stable centers

• Saddle points

• Focus points

In the phase diagram, what does a small angular velocity indicate?

• The pendulum is at a stable equilibrium.

• The pendulum is swinging wildly.

• The pendulum is at an unstable equilibrium.

• The pendulum is moving in large circles.

What happens when the pendulum is given a large push?

• It stops moving.

• It remains at equilibrium.

• It swings back and forth.

• It spins around its axis.

What is the Hamiltonian of the system used to describe?

• The pendulum's length

• The angular displacement

• The gravitational force

• The system's energy

How is the constant C in the trajectory equation determined?

• By using the initial condition Theta Sub Zero

• By calculating the initial angular velocity

• By measuring the pendulum's length

• By observing the pendulum's equilibrium

What does the trajectory equation describe in the context of the pendulum?

• The pendulum's equilibrium

• The pendulum's path over time

• The pendulum's energy

• The pendulum's length