For a reversible Hamiltonian flow with reversor R, for any point x on a reversible orbit, the points x and R(x) are related by a conjugacy of the dynamics restricted to that orbit.
NOT BSTOTAL BS
HARDLY BS — Verdict: Mostly True
Verified by Lenz ·
The Short Version
The claim is correct if “reversible orbit” means an orbit invariant under the reversor R. In that standard sense, the relation R∘φ_t=φ_{-t}∘R makes R a time-reversing conjugacy on the orbit, carrying x to R(x). The wording is the main weakness: not every orbit in a reversible Hamiltonian system is itself R-symmetric.
Caveats
The statement needs the orbit itself to be R-invariant; reversibility of the system alone is not enough.
The conjugacy is time-reversing: R intertwines φ_t with φ_{-t}, not generally φ_t with itself.
The phrase “reversible orbit” is technical and can be misread; a clearer formulation would explicitly say “R-symmetric orbit.”