Integration Methods =================== The following integration methods are included in ode: * Euler's method * Backward Euler method * Verlet method The integration methods operate on systems of either first or second order differential equations. By convention :math:`X` is the vector containing the state variables of the system, :math:`f(t,X)` is a function returning either the first or second derivative of the system, and :math:`h` is the timestep. The current state and derivative of the system are represented as lists. Euler's method -------------- Euler's method is an explicit method for solving a system of first order differential equations. .. math:: f(t,X) = \dot{X} .. math:: X_{n+1} = X_n + h \cdot f(t_n, X_n) Backward Euler method --------------------- Euler's method is an implicit method for solving a system of first order differential equations. .. math:: f(t,X) = \dot{X} .. math:: X_{n+1} = X_n + h \cdot f(t_{n+1}, X_{n+1}) Verlet method ------------- The Verlet method, also called Störmer–Verlet method, is an explicit method for solving a system of second order differential equations. An initial velocity vector :math:`V_0` is needed as well as the initial condition :math:`X_0`. .. math:: f(t,X) = \ddot{X} The first step is calculated with: .. math:: X_1 = X_0 + V_0 h + \frac{1}{2} f(X_0) h^2. Subsequent steps are calculated with: .. math:: X_{n+1} = 2 X_{n} - X_{n-1} + f(t, X_n) h^2. If subsequent velocities are needed, they can be calculated with: .. math:: V_n = \frac{X_{n+1} - X_{n-1}}{2 h}.