DiffEqPhysics

DiffEqPhysics.jl provides physics-based problem types for the SciML ecosystem. The main feature is HamiltonianProblem, which allows you to define and solve Hamiltonian systems using automatic differentiation.

For N-body gravitational simulations, please use NBodySimulator.jl.

Installation

using Pkg
Pkg.add("DiffEqPhysics")

Quick Start: Simple Pendulum

Define a Hamiltonian system by specifying the Hamiltonian function H(p, q, params, t):

using DiffEqPhysics, OrdinaryDiffEq

# Hamiltonian for a simple pendulum: H = p²/(2ml²) + mgl(1 - cos(q))
function H(p, q, params, t)
    g, m, l = params
    return p^2 / (2 * m * l^2) + m * g * l * (1 - cos(q))
end

# Parameters: gravitational acceleration, mass, length
params = (9.81, 1.0, 1.0)

# Initial conditions: momentum p₀ and angle q₀
p₀ = 0.0
q₀ = π/4  # 45 degrees

# Create and solve the Hamiltonian problem
prob = HamiltonianProblem(H, p₀, q₀, (0.0, 10.0), params)
sol = solve(prob, SofSpa10(), dt=0.01)

The equations of motion q̇ = ∂H/∂p and ṗ = -∂H/∂q are automatically computed using ForwardDiff.jl.

Features

• Automatic differentiation: Derivatives are computed automatically from the Hamiltonian function
• Multiple input types: Works with scalars, StaticArrays, or regular AbstractArrays
• Symplectic solvers: Use symplectic integrators from OrdinaryDiffEq.jl (like SofSpa10) to preserve the Hamiltonian structure
• Manual derivatives: Optionally provide your own derivative functions (dp, dq) for better performance

Providing Manual Derivatives

For better performance, you can provide the derivative functions directly:

# dp = -∂H/∂q, dq = ∂H/∂p
dp(p, q, params, t) = -params[1] * params[2] * params[3] * sin(q)  # -mgl*sin(q)
dq(p, q, params, t) = p / (params[2] * params[3]^2)                # p/(ml²)

prob = HamiltonianProblem((dp, dq), p₀, q₀, (0.0, 10.0), params)

Examples

See the examples/ folder for more detailed examples:

• pendulum.jl: Simple pendulum comparison with traditional ODE methods
• double_pendulum.jl: Double pendulum with animation