  Optimization under the symplecticity constraint is an approach for solving
various problems in quantum physics and scientific computing. Building on the
results that this optimization problem can be transformed into an unconstrained
problem on the symplectic Stiefel manifold, we construct geometric ingredients
for Riemannian optimization with a new family of Riemannian metrics called
tractable metrics and develop Riemannian Newton schemes. The newly obtained
ingredients do not only generalize several existing results but also provide us
with freedom to choose a suitable metric for each problem. To the best of our
knowledge, this is the first try to develop the explicit second-order geometry
and Newton's methods on the symplectic Stiefel manifold. For the Riemannian
Newton method, we first consider novel operator-valued formulas for computing
the Riemannian Hessian of a~cost function, which further allows the manifold to
be endowed with a weighted Euclidean metric that can provide a preconditioning
effect. We then solve the resulting Newton equation, as the central step of
Newton's methods, directly via transforming it into a~saddle point problem
followed by vectorization, or iteratively via applying any matrix-free
iterative method either to the operator Newton equation or its saddle point
formulation. Finally, we propose a hybrid Riemannian Newton optimization
algorithm that enjoys both global convergence and quadratic/superlinear local
convergence at the final stage. Various numerical experiments are presented to
validate the proposed methods.
