Variational discretization of constrained Birkhoffian systems

In this paper, we derive a variational characterization of constrained Birkhoffian dynamics in both continuous and discrete settings. When additional algebraic constraints appear, derivation of the necessary conditions under which the Pfaff action is extremized gives constrained Birkhoffian equations. Inspired by this continuous framework, we directly discretize the constraints as well as the Pfaff action and consequently formulate the discrete constrained Birkhoffian dynamics. Via this discrete variational approach which is parallel with the continuous case, the resulting discrete constrained Birkhoffian equations automatically preserve the intrinsic symplectic structure when identified as numerical algorithms. Considering that the obtained algorithms require not only the specification of an initial configuration but also a second configuration to operate, we present a natural, reasonable, and efficient method of initialization of simulations. While retaining the structure-preserving property, the obtained discrete schemes exhibit excellent numerical behaviors, demonstrated by numerical examples dealing with the mathematical pendulum and the 3D pendulum.