Here we implement a Runge-Kutta method for solving a two-variable second-order differential equation and use it to demonstrate chaotic motion in the plane:
If we choose the initial angles to be vertical to working precision, we get a very nice demonstration of the inexactness of floating point representation.
For the modeling, we simply write down the kinematic equations and balance forces. To avoid some of the hairy algebra, we use SymPy to solve for the second derivatives in terms of the zeroth and first dervivatives. We then have a system of first order differential equations in four variables which is ripe for presentation to our Runge-Kutta solver.
⎛ 2 ⎞
⎜ ⎛d ⎞ ⎟
⎜ l₁⋅m₂⋅sin(2⋅θ₁(t) - 2⋅θ₂(t))⋅⎜──(θ₁(t))⎟ 2⎟
⎜ g⋅m₂⋅sin(θ₁(t) - 2⋅θ₂(t)) g⋅m₂⋅sin(θ₁(t)) ⎝dt ⎠ ⎛d ⎞ ⎟
-⎜g⋅m₁⋅sin(θ₁(t)) + ───────────────────────── + ─────────────── + ───────────────────────────────────────── + l₂⋅m₂⋅sin(θ₁(t) - θ₂(t))⋅⎜──(θ₂(t))⎟ ⎟
⎝ 2 2 2 ⎝dt ⎠ ⎠
─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
⎛ 2 ⎞
l₁⋅⎝m₁ - m₂⋅cos (θ₁(t) - θ₂(t)) + m₂⎠
⎛ 2⎞ ⎛ 2⎞
⎜ ⎛d ⎞ ⎟ ⎜ ⎛d ⎞ ⎟
- (m₁ + m₂)⋅⎜g⋅sin(θ₂(t)) - l₁⋅sin(θ₁(t) - θ₂(t))⋅⎜──(θ₁(t))⎟ ⎟ + ⎜g⋅m₁⋅sin(θ₁(t)) + g⋅m₂⋅sin(θ₁(t)) + l₂⋅m₂⋅sin(θ₁(t) - θ₂(t))⋅⎜──(θ₂(t))⎟ ⎟⋅cos(θ₁(t) - θ₂(t))
⎝ ⎝dt ⎠ ⎠ ⎝ ⎝dt ⎠ ⎠
────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
⎛ 2 ⎞
l₂⋅⎝m₁ - m₂⋅cos (θ₁(t) - θ₂(t)) + m₂⎠