A Generic Lagrangian-Hamiltonian Framework for Optimal Reference Frames in Unsteady Flow

Xingdi Zhang, Amani Ageeli, Thomas Theußl, Peter Rautek and Markus Hadwiger

IEEE Transactions on Visualization and Computer Graphics, Vol.33, No.1 (Proceedings IEEE VIS 2026), to appear , 2027

Flow visualization inherently depends on a reference frame or observer, relative to which velocities are measured. Recent research has addressed the explicit computation of optimal observers. However, although all techniques share many similarities, the differences can be significant, and no method fulfills all desirable criteria. More accurate methods can be slow, whereas faster approaches can be less accurate, which means that the observers are less optimal. In this paper, we present a novel generic variational framework that combines the best of both worlds, and offers better trade-offs between accuracy and computational efficiency. We build on ideas from Lagrangian and Hamiltonian mechanics to unify previous methods, while at the same time enabling more flexibility and novel algorithmic features. In this way, our framework can also serve as a principled foundation for analysis, comparison, and future advances in this area. We first formulate the observer optimization problem in terms of Lagrangian functions, where an optimal solution must solve the second-order Euler-Lagrange equation. We then expand toward a Hamiltonian perspective, which reduces the second-order equation to first-order. However, we use a hybrid Lagrangian-Hamiltonian approach to combine the advantages of each. By defining Hamiltonian vector fields in an observer phase space, optimal observers simply become integral curves. Our hybrid method computes these vector fields directly from the Lagrangian representing the cost to minimize. Finally, we show how this foundation also enables direct insights into flow field properties, by defining the novel notion of intrinsically steady flows using the Hamiltonian function.

@article{Zhang2026LagrangianHamiltonianObserverFramework,
  title = {A Generic Lagrangian-Hamiltonian Framework for Optimal Reference Frames in Unsteady Flow},
  author = {Zhang, Xingdi and Ageeli, Amani and Theu{\ss}l, Thomas and Rautek, Peter and Hadwiger, Markus},
  journal = {IEEE Transactions on Visualization and Computer Graphics (Proceedings IEEE VIS 2026)},
  year = {2027},
  volume = {33},
  number = {1},
  pages = {to appear}
}