Struct VelocityVerlet 

pub struct VelocityVerlet { /* private fields */ }

Explicit, second-order, fixed-step method for dynamics.¹

\ddot{x} = a(t, x, \dot{x})

v_{n+1/2} = v_n + \tfrac{h}{2}\,a_n

x_{n+1} = x_n + h\,v_{n+1/2}

a_{n+1} = a(t_{n+1}, x_{n+1}, v_{n+1/2})

v_{n+1} = v_{n+1/2} + \tfrac{h}{2}\,a_{n+1}

It requires one evaluation of the acceleration per step, and is stable for $h \le 2/\omega_{max}$. It is symplectic, and conserves a nearby energy, for conservative accelerations and a fixed time step. Accelerations that depend on velocity are evaluated at the half-step velocity, which is no longer symplectic.

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1. Also known as the leapfrog or kick-drift-kick method, and equivalent to the central difference method for dynamics. ↩

Implementations

impl VelocityVerlet

pub fn new<T>(dt: Quantity<T>) -> Self

Constructs the integrator with a fixed time step.

Trait Implementations

impl Debug for VelocityVerlet

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Default for VelocityVerlet

fn default() -> VelocityVerlet

Returns the “default value” for a type.

impl<X, UX, UV, UA, T> ExplicitDynamics<X, UX, UV, UA, T> for VelocityVerlet

where X: Differentiable<T> + Tensor, Derivative<X, T>: Differentiable<T> + Tensor, for<'a> &'a Derivative<X, T>: Mul<Quantity<T>, Output = X>, for<'a> &'a Derivative<Derivative<X, T>, T>: Mul<Quantity<T>, Output = Derivative<X, T>>, UX: TensorVec<Item = X>, UV: TensorVec<Item = Derivative<X, T>>, UA: TensorVec<Item = Derivative<Derivative<X, T>, T>>,

fn integrate( &self, function: impl FnMut(Quantity<T>, &X, &Derivative<X, T>) -> Result<Derivative<Derivative<X, T>, T>, String>, time: &[Quantity<T>], initial_position: X, initial_velocity: Derivative<X, T>, ) -> Result<(Times<T>, UX, UV, UA), IntegrationError>

Solves an initial value problem for dynamics, given the acceleration as a function of time, position, and velocity.

fn integrate_bounded( &self, function: impl FnMut(Quantity<T>, &X, &Derivative<X, T>) -> Result<Derivative<Derivative<X, T>, T>, String>, bound: impl FnMut(Quantity<T>, &X, &Derivative<X, T>) -> Result<Quantity<T>, String>, safety: Scalar, time: &[Quantity<T>], initial_position: X, initial_velocity: Derivative<X, T>, ) -> Result<(Times<T>, UX, UV, UA), IntegrationError>

Solves an initial value problem like integrate, subject to a stability bound.

impl<T> FixedStep<T> for VelocityVerlet

fn dt(&self) -> Quantity<T>

Returns the time step.