Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Variational integrators are numerical integrators for Hamiltonian systems derived from the Euler-Lagrange equations of a discretized Hamilton's principle. Variational integrators are momentum-preserving and symplectic. Consider a mechanical system with a single particle degree of freedom described by the Lagrangian L(t,q,v) = frac{1}{2} m v^2 - V(q), where m is the mass of the particle, and V is a potential. To construct a variational integrator for this system, we begin by forming the discrete Lagrangian. The discrete Lagrangian approximates the action for the system over a short time interval: L_dleft(t_0, t_1, q_0, q_1right) = frac{t_1 - t_0}{2} left[ Lleft(t_0, q_0, frac{q_1-q_0}{t_1-t_0}right) + Lleft(t_1, q_1, frac{q_1-q_0}{t_1-t_0}right) right] approx int_{t_0}^{t_1} dt, L(t, q(t), v(t)) . Here we have chosen to approximate the time integral using the trapezoid method, and we use a linear approximation to the trajectory, q(t) approx frac{q_1 - q_0}{t_1-t_0} left( t - t_0 right) + q_0 between t0 and t1, resulting in a constant velocity v approx left(q_1 - q_0 right)/left(t_1 - t_0 right).