Prolongation-collocation variational integrators (Q2902208)

An approach for solving differential systems that arise from variational Hamiltonian equations is proposed. In such a system, the time evolution of the generalized coordinates \( q = q(t)\) in the interval \([0,T]\) from a starting value \( q(0)=q^0\) to an end value \( q(T)= q^1\) is known to minimize the action integral \( S(q) \equiv \int_0^T L (q(t), \dot{q}(t) ) \; dt\), where \( L=L( \dot{q}, q)\) is the Lagrangian function. NEWLINENEWLINENEWLINENEWLINEFor a given discrete sequence of nodes \(t_k = k h\), \(k=0, \dots ,N\), \(N = T/h\) in the integration interval and a discrete curve \( \{ q_k \}_{k=0}^N\), where \( q_k \simeq q(t_k)\), the exact curve solution \( q = q(t)\) at each \( t \in [t_k, t_{k+1}]\) is approximated by a two-point Hermite interpolant of suitable order that ensures not only the continuity of positions and velocities but also some error control on higher order derivatives at the nodes. NEWLINENEWLINENEWLINENEWLINEIn this setting, the exact action in each interval \([t_k, t_{k+1}] \) is substituted by a discrete action \( L_d (q_k, q_{k+1})\) so that the action \( \int_{t_k}^{t_{k+1} }L(q(t), \dot{q}(t) ) \; dt\) is approximated by using the Euler-Maclaurin quadrature formula that only involves values of the derivatives at the end points of the integration interval. For separable systems, where \( L = (1/2) \dot{q}^T \dot{q} - V(q) \) with some potential \( V = V(q)\), several derivatives of the potential are required depending on the order of the Euler-MacLaurin approximation for the definition of the discrete action together with collocation conditions at both end points. NEWLINENEWLINENEWLINENEWLINEAfter obtaining the discrete action, a set of discrete variational equations is derived so that it defines implicitly the coordinates \( \{ q_k \}_{k=0}^N\) at the given nodes. It is proved that if the Lagrangian \(L\) is time independent then the proposed numerical solvers are self-adjoint and therefore the methods have an even order of accuracy. A study of the order is carried out and finally the results of some numerical examples with elementary test problems are presented to show the accuracy and the preservation of geometrical properties.