Variational Integrator
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High Quality Content by WIKIPEDIA articles! 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…mehr

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High Quality Content by WIKIPEDIA articles! 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).