  In the graph node embedding problem, embedding spaces can vary significantly
for different data types, leading to the need for different GNN model types. In
this paper, we model the embedding update of a node feature as a Hamiltonian
orbit over time. Since the Hamiltonian orbits generalize the exponential maps,
this approach allows us to learn the underlying manifold of the graph in
training, in contrast to most of the existing literature that assumes a fixed
graph embedding manifold with a closed exponential map solution. Our proposed
node embedding strategy can automatically learn, without extensive tuning, the
underlying geometry of any given graph dataset even if it has diverse
geometries. We test Hamiltonian functions of different forms and verify the
performance of our approach on two graph node embedding downstream tasks: node
classification and link prediction. Numerical experiments demonstrate that our
approach adapts better to different types of graph datasets than popular
state-of-the-art graph node embedding GNNs. The code is available at
\url{https://github.com/zknus/Hamiltonian-GNN}.
