Module symplectic 

Symplectic integrators for Hamiltonian systems

This module provides specialized integrators for Hamiltonian systems that preserve important geometric properties such as energy conservation (to within bounded error), phase space volume, and symplectic structure.

Symplectic integrators are particularly useful for:

• Long-time integration of mechanical systems
• Molecular dynamics simulations
• Orbital mechanics
• Any conservative system where energy conservation is critical

Methods

Several symplectic integration methods are provided:

• Symplectic Euler: First-order, simple method (forward and backward variants)
• Leapfrog/Störmer-Verlet: Second-order, widely used for molecular dynamics
• Symplectic Runge-Kutta: Higher-order methods with improved accuracy
• Composition methods: Methods constructed by composing lower-order integrators

Basic Usage

use scirs2_integrate::symplectic::potential::HamiltonianSystem;
use scirs2_integrate::symplectic::leapfrog::StormerVerlet;
use scirs2_integrate::symplectic::SymplecticIntegrator;
use scirs2_core::ndarray::array;

// Define a simple harmonic oscillator: H = p²/2 + q²/2
let system = HamiltonianSystem::new(
    |_t, _q, p| p.clone(),  // dq/dt = p
    |_t, q, _p| -q.clone(), // dp/dt = -q
);

// Initial conditions: (q0, p0) = (1.0, 0.0)
let q0 = array![1.0];
let p0 = array![0.0];
let t = 0.0;
let dt = 0.01;

// Create integrator
let integrator = StormerVerlet::<f64>::new();

// Take one step
let (q1, p1) = integrator.step(&system, t, &q0, &p0, dt).unwrap();

// Energy should be conserved (approximately)
let initial_energy = 0.5_f64 * p0.dot(&p0) + 0.5_f64 * q0.dot(&q0);
let final_energy = 0.5_f64 * p1.dot(&p1) + 0.5_f64 * q1.dot(&q1);
// For a single step with dt=0.1, the energy error is acceptable
assert!((initial_energy - final_energy).abs() < 1e-6);

Re-exports

pub use composition::CompositionMethod;

pub use euler::symplectic_euler;

pub use euler::symplectic_euler_a;

pub use euler::symplectic_euler_b;

pub use leapfrog::position_verlet;

pub use leapfrog::velocity_verlet;

pub use leapfrog::StormerVerlet;

pub use potential::HamiltonianSystem;

pub use potential::SeparableHamiltonian;

pub use runge_kutta::GaussLegendre4;

pub use runge_kutta::GaussLegendre6;

Modules

composition

Composition methods for symplectic integration

euler

Symplectic Euler methods

leapfrog

Leapfrog and Störmer-Verlet integrators

potential

Specialized symplectic integrators for separable Hamiltonian systems

runge_kutta

Symplectic Runge-Kutta methods

Structs

SymplecticResult

Result of symplectic integration containing state history

Traits

HamiltonianFn

Trait for Hamiltonian systems

SymplecticIntegrator

Trait for symplectic integrators

Type Aliases

HamiltonianFnBox

Type alias for Hamiltonian function