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Double Pendulum

The double pendulum consists of two pendula, one attached to the origin at $(x,y) = (0,0)$, and the second attached to the first. Each pendulum consists of a point mass $m_i$ attached to a massless rod of length $l_i$ with $i \in (1,2)$. All motion is assumed to be frictionless.

The dynamics of the system is most naturally described in terms of the angles $\theta_i$ between the rods $l_i$ and the vertical axis $y$. In terms of these angles, the cartesian coordinates are given by

\[\begin{align*} x_1 &= l_1 \sin\theta_1 , \\ x_2 &= l_1 \sin\theta_1 + l_2 \sin\theta_2 , \\ y_1 &= - l_1 \cos\theta_1 , \\ y_2 &= -l_1 \cos\theta_1 - l_2 \cos\theta_2 . \end{align*}\]

In terms of the generalized coordinates $\theta_i$, the Lagrangian reads

\[\begin{align*} L (\theta_1, \theta_2, \dot{\theta}_1, \dot{\theta}_2) = \frac{1}{2} (m_1 + m_2) l_1^2 \dot{\theta}_1^2 &+ \frac{1}{2} m_2 l_2^2 \dot{\theta}_2^2 + m_2 l_1 l_2 \dot{\theta}_1 \dot{\theta}_2 \cos(\theta_1 - \theta_2) \\ &+ g (m_1 + m_2) l_1 \cos\theta_1 + g m_2 l_2 \cos\theta_2 . \end{align*}\]

The canonical conjugate momenta $p_i$ are obtained from the Lagrangian as

\[\begin{align*} p_1 &= \frac{\partial L}{\partial \dot{\theta}_1} = (m_1 + m_2) l_1^2 \dot{\theta}_1 + m_2 l_1 l_2 \dot{\theta}_2 \cos(\theta_1 - \theta_2), \\ p_2 &= \frac{\partial L}{\partial \dot{\theta}_2} = m_2 l_2^2 \dot{\theta}_2 + m_2 l_1 l_2 \dot{\theta}_1 \cos(\theta_1 - \theta_2) . \end{align*}\]

After solving these relations for the generalized velocities $\dot{\theta}_i$,

\[\begin{align*} \dot{\theta}_1 &= \frac{l_2 p_{\theta_1} - l_1 p_{\theta_2} \cos(\theta_1 - \theta_2)}{l_1^2 l_2 \left[ m_1 + m_2 \sin^2(\theta_1 - \theta_2) \right] } \\ \dot{\theta}_2 &= \frac{(m_1 + m_2) l_1 p_{\theta_2} - m_2 l_2 p_{\theta_1} \cos(\theta_1 - \theta_2)}{m_2 l_1 l_2^2 \left[ m_1 + m_2 \sin^2 (\theta_1 - \theta_2) \right] } , \end{align*}\]

the Hamiltonian can be obtained via the Legendre transform,

\[H = \sum_{i=1}^2 \dot{\theta}_i p_i - L ,\]

as

\[\begin{align*} H &= \frac{m_2 l_2^2 p^2_{\theta_1} + (m_1 + m_2) l_1^2 p^2_{\theta_2} - 2 m_2 l_1 l_2 p_{\theta_1} p_{\theta_2} \cos(\theta_1 - \theta_2)}{2 m_2 l_1^2 l_2^2 \left[ m_1 + m_2 \sin^2(\theta_1 - \theta_2) \right] } \\ & \qquad\qquad \vphantom{\frac{l}{l}} - g (m_1 + m_2) l_1 \cos\theta_1 - g m_2 l_2 \cos\theta_2 . \end{align*}\]

Library functions

GeometricProblems.DoublePendulum — Module
DoublePendulum

The DoublePendulum module provides functions hodeproblem and lodeproblem each returning a Hamiltonian or Lagrangian problem, respectively, to be solved in the GeometricIntegrators.jl ecosystem. The actual code is generated with EulerLagrange.jl.

The double pendulum consists of two pendula, one attached to the origin at $(x,y) = (0,0)$, and the second attached to the first. Each pendulum consists of a point mass $m_i$ attached to a massless rod of length $l_i$ with $i \in (1,2)$. The dynamics of the system is described in terms of the angles $\theta_i$ between the rods $l_i$ and the vertical axis $y$. All motion is assumed to be frictionless.

System parameters:

  • l₁: length of rod 1
  • l₂: length of rod 2
  • m₁: mass of pendulum 1
  • m₂: mass of pendulum 2
  • g: gravitational constant
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GeometricProblems.DoublePendulum.hodeproblem — Function
Hamiltonian problem for the double pendulum

Constructor with default arguments:

hodeproblem(
    q₀ = [π/4, π/2],
    p₀ = [3.3321622036187746, 7.0685834705770345];
    tspan = (0.0, 10.0),
    tstep = 0.01,
    params = (l₁ = 2.0, l₂ = 3.0, m₁ = 1.0, m₂ = 2.0, g = 9.80665)
)
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GeometricProblems.DoublePendulum.lodeproblem — Function
Lagrangian problem for the double pendulum

Constructor with default arguments:

lodeproblem(
    q₀ = [π/4, π/2],
    p₀ = [3.3321622036187746, 7.0685834705770345];
    tspan = (0.0, 10.0),
    tstep = 0.01,
    params = (l₁ = 2.0, l₂ = 3.0, m₁ = 1.0, m₂ = 2.0, g = 9.80665)
)
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