How To Set Up The Second Order Differently

In Week 4 We Showed How To Set Up The Second Order Difierential Equat

In Week 4 We Showed How To Set Up The Second Order Difierential Equat

In Week 4, we demonstrated the method for establishing the second-order differential equation that models the motion of a pendulum. The original nonlinear differential equation, expressed as m d²θ/dt² + c dθ/dt + g/l sin(θ) = 0, describes the dynamics of the pendulum, where m is mass, c is damping coefficient, g is gravitational acceleration, and l is the pendulum length. For our analysis, we assign specific values: m = t, L = 1, c = 0.5, and g = 9.8.

The goal involves transforming this second-order nonlinear differential equation into a system of first-order equations via change of variables, linearizing set points, and analyzing stability. We will walk through each step meticulously to understand the pendulum's behavior under different conditions.

Paper For Above instruction

Part a): Conversion to First-Order System

Starting with the nonlinear second-order differential equation, we introduce a change of variables to convert it into a system of first-order equations. Define:

  • θ(t) as the angular displacement
  • ω(t) = dθ/dt as the angular velocity

Rewriting the original equation, we have:

m d²θ/dt² + c dθ/dt + (g/l) sin(θ) = 0.

Dividing through by m yields:

d²θ/dt² + (c/m) dθ/dt + (g/(l m)) sin(θ) = 0.

Expressed as a system, the derivatives become:

  • dθ/dt = ω
  • dω/dt = - (c/m) ω - (g/(l m)) sin(θ)

Substituting the specific parameters (m = t, l= 1, c= 0.5, g= 9.8), the system simplifies to:

dθ/dt = ω

dω/dt = - (0.5 / t) ω - (9.8 / t) sin(θ)

This system provides a first-order representation capturing the pendulum dynamics.

Part b): Linearization Around the Critical Point (0, 0)

To analyze the stability near the equilibrium point (θ, ω) = (0, 0), we linearize the system about this point. Taking the Jacobian matrix involves calculating the partial derivatives of the right-hand side functions with respect to θ and ω at (0, 0):

F(θ, ω) = (ω, - (c/m) ω - (g/l m) sin(θ))

Partial derivatives:

  • ∂F₁ / ∂θ = 0
  • ∂F₁ / ∂ω = 1
  • ∂F₂ / ∂θ = - (g / (l m)) cos(θ). At (0, 0), cos(0) = 1, so ∂F₂ / ∂θ = - (g / (l m)) = - 9.8 / t
  • ∂F₂ / ∂ω = - (c / m) = - 0.5 / t

Thus, the linearized system matrix at (0, 0) with m = t is:

A(t) =   | 0   1 |


| - (9.8 / t)   - (0.5 / t) |

Since this matrix explicitly depends on time, its eigenvalues are time-dependent. Examining these eigenvalues at specific times allows us to evaluate the local stability.

The eigenvalues λ satisfy:

λ² + (0.5 / t) λ + (9.8 / t) = 0

Discriminant D is:

D = (0.5 / t)² - 4 * (9.8 / t) = (0.25 / t²) - (39.2 / t)

Because D is negative for small t, eigenvalues are complex conjugates with negative real parts for sufficiently large t, indicating oscillatory decay and local stability. For smaller t, the system's stability depends on the eigenvalues' real parts, which are influenced by the damping term and pendulum parameters.

Part c): Linearization About the Critical Point (2π, 0)

Next, considering the equilibrium at (θ, ω) = (2π, 0), which is physically equivalent to the pendulum returning to the initial position after a full rotation, linearization around this point involves similar derivatives.

The Jacobian matrix at (2π, 0) is analogous since sin(2π) = 0 and cos(2π) = 1:

A(t) =   | 0   1 |


| - (9.8 / t)   - (0.5 / t) |

Therefore, the stability analysis mirrors that of (0, 0). The eigenvalues follow the same quadratic, and their real parts are identical at each time t, suggesting similar local stability characteristics. Because the equilibrium corresponds to a full rotation, analyzing stability here helps understand whether small perturbations cause the pendulum to continue oscillating or diverge.

Part d): Critical Points and Stability of the System

To verify all equilibria, substitute into the original first-order system:

  • At (θ, ω) = (0, 0):
  • dθ/dt = 0
  • dω/dt = - (g/(l m)) sin(0) = 0
  • Similarly, at (2π, 0):
  • dθ/dt = 0
  • dω/dt = - (g/(l m)) sin(2π) = 0

Thus, both (0, 0) and (2π, 0) are equilibrium points. Linearizing around (0, 0) produces the same coefficient matrix as previously derived, confirming their nature as critical points.

Eigenvalues of the linearized system at (0, 0) indicate stability if their real parts are negative, which is confirmed when the discriminant is negative, leading to oscillatory decay. At (2π, 0), the stability depends on the same factors. Typically, (0, 0) is stable (a center or focus), whereas (2π, 0) may be unstable, representing a saddle or unstable focus, due to the nature of the nonlinear potential energy landscape of the pendulum.

Part e): Numerical Solutions and Motion Interpretation

Using numerical methods such as those implemented in NumSysDE, the differential equations can be solved for specified initial conditions, providing graphs of θ(t) and ω(t). For initial conditions θ(0) = 0, ω(0) = 0, and small initial velocities, the graphs typically reveal oscillatory motion with decreasing amplitude if damping is present, illustrating energy loss over time and movement toward equilibrium.

When graphing solutions with initial velocities away from zero, the pendulum exhibits more complex behavior, including potentially full rotations or oscillations about the vertical. The periodicity observed in the graphs, often with e^{2π i} or similar terms, indicates the system's tendency to repeat its motion after integer multiples of period 2π, consistent with the pendulum’s physical periodicity.

At t = 0, the pendulum is at the lowest point (assuming initial θ = 0). As time progresses, if initial velocities are small, the pendulum oscillates back and forth. For larger initial velocities, the motion can include complete rotations if the energy exceeds the potential barrier at θ = π.

The sign of ω(t) indicates the direction of motion: positive for counterclockwise and negative for clockwise. The duration of these directions and the transition points can be inferred from the graphs, illustrating whether the pendulum continues rotating, swings back, or settles into a stable oscillation around equilibrium.

References

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