Coupled Harmonic Oscillator

CoupledHarmonicOscillator

The CoupledHarmonicOscillator 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 coupled harmonic oscillator is a collection of two point masses that are connected to a fixed wall with spring constants $k_1$ and $k_2$ and are furthermore coupled nonlinearly resulting in the Hamiltonian:

\[H(q_1, q_2, p_1, p_2) = \frac{q_1^2}{2m_1} + \frac{q_2^2}{2m_2} + k_1\frac{q_1^2}{2} + k_2\frac{q_2^2}{2} + k\sigma(q_1)\frac{(q_2 - q_1)^2}{2},\]

where $\sigma(x) = 1 / (1 + e^{-x})$ is the sigmoid activation function.

System parameters:

• k₁: spring constant of mass 1
• k₂: spring constant of mass 2
• m₁: mass 1
• m₂: mass 2
• k: coupling strength between the two masses.

Hamiltonian problem for coupled oscillator

Constructor with default arguments:

hodeproblem(
    q₀ = [1.0, 0.0],
    p₀ = [2.0, 0.0];
    timespan = (0.0, 100.0),
    timestep = 0.4,
    parameters = (m₁ = 2.0, m₂ = 1.0, k₁ = 1.5, k₂ = 0.3, k = 1.0)
)

Lagrangian problem for the coupled oscillator

Constructor with default arguments:

lodeproblem(
    q₀ = [1.0, 0.0],
    p₀ = [2.0, 0.0];
    timespan = (0.0, 100.0),
    timestep = 0.4,
    parameters = (m₁ = 2.0, m₂ = 1.0, k₁ = 1.5, k₂ = 0.3, k = 1.0)
)