“We call a map F reversible if there exists an involution R with R ∘ F ∘ R = F⁻¹. For each orbit O of F, the restriction F|O is either invariant under R (symmetric orbit) or R maps O onto a distinct orbit O′ (conjugate orbits). In the latter case, R defines a conjugacy between F|O and F|O′, reversing the direction of time along the orbits.”

We say that X1 and X2 are locally analytically conjugate at a singular point p if (2) holds in a neighbourhood of p. ... Of primary importance in the study of such systems are the symmetric periodic orbits. These are closed orbits of X which intersect Fix (R) in exactly two points. ... We have shown that every planar analytic vector field reversible with respect to the involution R(u, v) = (u, −v) is locally analytically conjugate to a Hamiltonian vector field whose Hamiltonian function assumes the classical normal form determined by the type of the equilibrium point.

“For each x ∈ Γ ∩ {H + u = 0} such that the (H + u)-orbit of x is reversible periodic of minimal period 2T, and such that the chord (T, x) is transverse, the following result holds …” “Recall that a closed orbit θ(t) is called reversible periodic if there exists an involution R of the phase space such that R ∘ φt = φ−t ∘ R and R(θ(t)) = θ(−t). Thus the orbit is invariant by the reversor and points θ(t) and R(θ(t)) belong to the same periodic trajectory traversed in opposite time directions.”

“In contrast to invariant subspaces of symmetries without time reversal, this subspace is not dynamically invariant. Instead, a trajectory can cross Fix R, which then implies that the whole trajectory is mapped by R onto a time reversed copy of itself. In this way one can distinguish between intersecting trajectories connecting an attractor-repellor pair related by R, and trajectories intersecting more than once, which induces locally conservative dynamics. This coexistence of conservative and dissipative dynamics in different regions of the phase space is a typical property of systems with time-reversibility… Due to the possibility of locally conservative dynamics, reversible systems can have structurally stable homoclinic orbits and heteroclinic cycles…”

“An observable O of a system is said to be time-reversal invariant if there is an anti-unitary operator T which commutes with O: [T, O] = 0… When time is reversed, the particles of a system occupy the same position as during the real evolution, but have reversed velocities… Thus T maps a trajectory of the system onto another physically admissible trajectory, typically with inverted momenta, and possibly onto a different orbit.”

A smooth vector field X on M is then called R-reversible if TR(X) = −X ∘ R. Of primary importance in the study of such systems are the symmetric periodic orbits. These are closed orbits of X which intersect Fix(R) in exactly two points. ... Let φt be the flow of X. Then reversibility means R ∘ φt = φ−t ∘ R. In particular, if x lies on a periodic orbit of period T, then so does R(x), since R(φt(x)) = φ−t(R(x)).

“Given a regular energy level H−1(E) of H, a closed orbit θ ∈ H−1(E) and ε > 0, there is a C∞ function U : M → R whose Ck norm is at most ε such that the Hamiltonian H + U has a closed orbit in H−1(E) Ck-close to θ with prescribed linearized Poincaré map.” “In particular, for reversible Hamiltonian systems, the linearized return map along a reversible periodic orbit satisfies that the action of the reversor on the normal bundle conjugates the linearized dynamics in opposite time directions. More precisely, if R is the reversor, then TR ∘ Dφt = Dφ−t ∘ TR on the normal bundle along the orbit.”

“The time-reversal operator consists of two combined operations of negating the time t and the complex conjugation… We thereby conclude here that if the Hamiltonian has a complex eigenvalue, the corresponding eigenstate breaks the time-reversal symmetry. The time-reversal symmetry of the original equation of motion reflects in the fact that the symmetry-breaking eigenstates can appear always in a complex-conjugate pair.”

This work considers Hamiltonian systems possessing a time-reversal symmetry R with R^2 = Id and R ∘ Φ_t = Φ_{−t} ∘ R. It explains that R maps any trajectory of the Hamiltonian flow to a time-reversed trajectory: “If x(t) is a solution, then R x(−t) is also a solution.” In general, R does not provide a conjugacy between points x and R(x) along the same orbit; instead it reverses the direction of time along the orbit or, for non-symmetric trajectories, maps an orbit to a distinct, time-reversed orbit.

“We show the existence of formal equivalences between reversible and Hamiltonian vector fields.” “Two vector fields X, Y : Rn → Rn are said to be formally orbitally equivalent if there is a smooth function f : Rn → R with no zeros near 0, so that f·X is formally conjugate to Y. The multiplication by f has the interpretation of a time-reparametrization of the orbits of X.” “Theorem A: Let X ∈ Ω2n0,e,∞(ϕ) ∪ Ω2n0,s,∞(ϕ) and suppose that j3X is a Hamiltonian vector field. Then X is generically formally orbitally equivalent to a Hamiltonian vector field.”

“In a reversible Hamiltonian flow (M, φᵗ, R) with reversor R we have R ∘ φᵗ ∘ R = φ⁻ᵗ. Hence, R maps orbits of φᵗ onto orbits of φᵗ. An orbit O is called R-symmetric if R(O) = O; otherwise, R maps O onto a distinct orbit O′ and R provides a topological conjugacy between the restricted flows φᵗ|O and φᵗ|O′ with reversed time.”

The survey distinguishes between Hamiltonian systems and reversible systems, including those that are both. It defines a reversible system via an involution R with R ∘ Φ_t = Φ_{−t} ∘ R and observes that such an R maps trajectories to their time-reversed counterparts: “If x(t) is a trajectory, so is R x(−t).” Symmetric orbits are characterized by intersections with Fix R. The paper emphasizes orbit pairing and symmetry properties, but it does not state a general theorem that for any point x on a reversible orbit, x and R(x) are conjugate under the dynamics restricted to that orbit; instead R is a global time-reversing symmetry of the flow.

An orbit γ is said symmetric if ϕ(γ) = γ. Hence, every critical point of X in S is a symmetric singularity of X. Some classical properties of reversible systems are: ... Definition 2 Two vector fields X1 and X2 are said Ck-orbitally equivalent, k ≥ 1, if there is a Ck-diffeomorphism h in the phase space that sends trajectories of X1 in trajectories of X2 preserving the orientation, but not necessarily the time. If h preserves the time, i.e., dh ◦ X1 = X2 ◦ h, then X1 and X2 are Ck-conjugated.

“Lemma 2.1. Let H be an affine-integrable Hamiltonian. Then the functions H and X and DX are constant along every orbit. Furthermore, (DH)X = 0 and (DX)X = 0 and (DX)2 = 0.” “Affine-integrable Hamiltonian flows have particularly simple orbit structure: each orbit is contained in an affine subspace and, after a linear symplectic conjugacy, the flow becomes a parallel translation along straight lines.” “This illustrates that for suitable Hamiltonians, there exists a symplectic (canonical) conjugacy that relates the dynamics along different points of the same orbit, by straightening the flow.”

“We define the time-reversed state as one in which the position is the same but the momentum is reversed… A system is said to exhibit symmetry under the time reversal if, at least in principle, its time development may be reversed and all physical processes run backwards, with initial and final states interchanged. Symmetry between the two directions of motion in time implies that to every state ψ there corresponds a time-reversed state Θψ, but in general Θ maps a given orbit into another one, and not necessarily onto itself pointwise.”

Consider a reversible flow (ϕt) with respect to an involution R. An orbit is called symmetric if it is invariant under R, that is, if R(γ) = γ. The latter follows from the well-known fact that an orbit of a reversible flow is a symmetric periodic orbit if and only if it intersects the fixed-point set Fix(R). In that case, the intersection points are related by the reversor as R(x) lies on the same periodic orbit as x, with reversed time parameterisation.

“We are led to the following definition: a closed orbit γ ⊂ Ω is called orbitally stable if given a neighborhood U = U(γ) there exists another neighborhood V = V(γ) such that if z ∈ V then φt(z) ∈ U for each t ≥ 0.” “If γ is a closed orbit (not an equilibrium), ∇H(z) ≠ 0 for each z ∈ γ and so the energy level {H = c} where c = H|γ is a 3d-manifold… We can restrict the flow φt(z) to this submanifold and consider the orbital stability only with respect to orbits lying on the same energy level.” “Finally, if |Δ| = 2 there is a double eigenvalue λ1 = λ2 = 1 or λ1 = λ2 = −1, then either A = ±I (parabolic stable) or A is conjugate to (±1 1; 0 ±1) in GL(R2) (parabolic unstable).”

We consider time-reversible Hamiltonian vector fields with respect to an involution R. The time-reversibility condition can be written R ∘ ϕt = ϕ−t ∘ R, where (ϕt) denotes the flow. As a consequence, if x belongs to a periodic orbit γ, then R(x) belongs to a (generally distinct) periodic orbit R(γ) obtained from γ by reversing time. When γ is symmetric, i.e. R(γ) = γ, the action of R defines a natural conjugacy between x and its image R(x) along the same orbit with opposite time direction.

“Reversibility is defined by the existence of an involution R such that R ∘ φt = φ−t ∘ R (or R ∘ f ∘ R = f−1 in discrete time). On a reversible orbit, the phase points x and R(x) lie on the same geometric orbit but correspond to opposite time directions.” “Thus R defines a conjugacy between the forward and backward dynamics along the orbit: R ∘ φt = φ−t ∘ R on that orbit. In general, there is no conjugacy which identifies the dynamics at x and at R(x) under the same time orientation; the only natural conjugacy given by R is between φt and φ−t.”

“A system is said to exhibit symmetry under the time reversal if… the time development may be reversed and all physical processes run backwards… Symmetry between the two directions of motion in time implies that to every state ψ there corresponds a time-reversed state ψ_Θ and that the transformation Θ preserves the value of all probabilities… The operation R, called the time-reversal operation, is defined by r → r, p → −p. We define the time-reversed state as one whose position is the same but the momentum is reversed. There is no requirement that r and R(r) lie on the same invariant orbit; Θ may take one orbit into a different orbit.”

The page defines fixed points and periodic orbits for ODE flows and discusses stability: “A fixed point or periodic orbit is said to be stable if solutions starting close to it tends to it under the evolution of the flow.” It treats periodic orbits as invariant sets under the flow and does not link the notion of an involutive symmetry R to a conjugacy along a single orbit. The focus is on the flow φ_t itself, not on additional reversor maps acting as conjugacies on individual orbits.

For this reason, time reversal in classical Hamiltonian mechanics is more correctly identified 'anticanonical' or 'antisymplectic'. Many have suggested that the transformation standardly referred to as ‘time reversal’ in quantum theory is not deserving of the name. I argue on the contrary that the standard definition is perfectly appropriate and is indeed forced by basic considerations about the nature of time in the quantum formalism.

The lecturer defines reversibility: if x(t) is a solution, then R(x(−t)) is also a solution, with R an involution. Around timestamp 807–867: “We learned that linear centers are not structurally stable except in a few cases and one of those cases is when you have a conserved quantity… it only has a reversible symmetry… but it turns out again if you plot this thing you will find what looks like closed orbits… those are closed orbits… the interesting thing about a reversible system is that it seems to behave a lot like a conserved system… reversible systems seem to have something in common with conserved systems and that is linear centers persist.” The lecture emphasizes orbit structures and symmetry of fixed points under R, but does not claim that points x and R(x) are conjugate under the dynamics restricted to a given orbit.

In the standard definition, a reversible flow (φᵗ) on a phase space M with reversor R satisfies R ∘ φᵗ ∘ R = φ⁻ᵗ and R² = Id. For any point x, R maps the orbit O(x) = {φᵗ(x): t ∈ ℝ} onto another orbit O(Rx). If O(Rx) = O(x), the orbit is called symmetric, and R acts as a reversing symmetry of the restricted dynamics. If O(Rx) ≠ O(x), then R gives a conjugacy between φᵗ restricted to O(x) and φᵗ restricted to O(Rx). Thus, only points lying on symmetric orbits are related to their images under R by a conjugacy on the *same* orbit; in general, x and R(x) belong to conjugate but distinct orbits.

The Schrödinger equation tells us time reversal must be anti-unitary because it must flip the sign of the imaginary unit. For half-integer spin, time reversal generates a distinct partner at the same energy and forces Kramers degeneracy.