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On any orbit invariant under R, R conjugates φ_t to φ_{-t}.

The Claim

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.

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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.”

The Receipts

  1. Time-reversal symmetry in dynamical systems: A survey

    PHYSICA D (via UNSW preprint)

  2. Time-Reversible Dynamics

    Emergent Mind

  3. Time-reversal symmetry in dynamical systems: A survey

    Physica D (through ADS entry)

  4. Breakdown of Symmetry in Reversible Systems

    Regular and Chaotic Dynamics (author preprint)

  5. Reversible diffeomorphisms and flows

    Harvard University (John Devaney, "Reversible diffeomorphisms and flows")

  6. Show that the system \ddot x+x\dot x+x=0 is reversible

    Math StackExchange (citing Meiss, Differential Dynamical Systems)

  7. Invariant manifolds in a reversible Hamiltonian system: The tentacle

    ScienceDirect (Nonlinear Analysis: Real World Applications)

  8. Branching of periodic orbits in reversible Hamiltonian systems

    HAL / Université de Bourgogne

  9. Analytic conjugation between planar reversible and Hamiltonian vector fields

    arXiv

  10. Orbital Reversibility of Planar Dynamical Systems

    Chinese Journal of Physics (author preprint)

+ 18 more sources — see the full list on Lenz

Filed Under

Hamiltonian FlowReversible Dynamical SystemReversor R