Variational Principle for Outer Contact Billiards

Construct a variational principle whose critical trajectories are precisely the trajectories of outer contact billiards, preferably in a form analogous to Herglotz’ variational principle for contact vector fields.

Background

Outer contact billiards combine projective and contact geometry and generate contact transformations. The unresolved problem is to identify an action functional or discrete contact variational formulation that characterizes their trajectories.

References

Is there a natural variational principle associated with this geometry? More specifically, can outer contact billiard trajectories be characterized as critical points of a suitable action functional in a similar way that Herglotz' variational principle characterizes trajectories of contact vector fields?

— Outer Contact Billiards  (2608.19393 - Caliz et al., 19 Aug 2026) in Section 6, “Open questions,” item 3