Fermi (1923) generalized Poincaré's theorem on the non-existence of analytic first integrals by proving that a nearly integrable Hamiltonian system (a 'normal canonical system') with more than two degrees of freedom, under genericity conditions on the unperturbed frequencies and on the Fourier coefficients of the perturbation, admits no analytic single-valued ('uniform') invariant hypersurface other than the constant-energy surfaces. For exactly two degrees of freedom the obstruction fails: on an unperturbed surface where the ratio of the two fundamental frequencies is a fixed irrational number, no integer combination of the frequencies vanishes, and the formal power series defining a candidate invariant hypersurface can be constructed order by order — but its convergence was never established. The question that argument leaves open is whether the formal series converges, i.e. whether nearly integrable systems with two degrees of freedom can possess analytic invariant hypersurfaces that are not energy surfaces. The direct 1982 sequel by Benettin, Ferrari, Galgani and Giorgilli extends the non-existence theorem to intermediate-dimensional invariant manifolds but must still exclude precisely the class of invariant-torus families with mutually proportional frequencies; KAM theory constructs invariant tori only in Cantor-set families and does not decide the question. This question remains open.

Background

A fundamental question of Hamiltonian mechanics is which conserved structures survive when an integrable system is subjected to a small perturbation. In action–angle variables (x,y)=(x1,…,xn;y1,…,yn)(x,y)=(x_1,\dots,x_n;y_1,\dots,y_n), where the angles xix_i are defined modulo 2π2\pi and the actions yiy_i are constants of the unperturbed motion, a nearly integrable Hamiltonian system (called a normal canonical system in the Italian literature of the 1920s) has an energy function of the form

F(x,y,μ)=F0(y)+μF1(x,y)+μ2F2(x,y)+⋯ ,F(x,y,\mu) = F_0(y) + \mu F_1(x,y) + \mu^2 F_2(x,y) + \cdots,

where F0F_0 is the unperturbed, integrable part, μ\mu is a small real parameter, and all functions are analytic and 2π2\pi-periodic in each angle. A first integral is a function constant along every trajectory. Poincaré proved in 1892 that, generically, no analytic first integral independent of the energy exists: the Fourier coefficients of the perturbation enter the defining equations divided by integer combinations m⋅ω(y)m\cdot\omega(y) of the unperturbed frequencies ωi(y)=∂F0/∂yi\omega_i(y)=\partial F_0/\partial y_i, where mm ranges over integer vectors, and these denominators vanish at resonances (the small-divisor problem). A function is called uniform in the terminology of that period if it is single-valued in its variables, so that an analytic uniform function admits a convergent Fourier series in the angles.

Fermi, in the paper referred to throughout, generalized Poincaré's theorem from first integrals to single invariant hypersurfaces. A hypersurface SμS_\mu given by Φ(x,y,μ)=0\Phi(x,y,\mu)=0 is invariant if the Poisson bracket [F,Φ]=∑i(∂F/∂xi ∂Φ/∂yi−∂F/∂yi ∂Φ/∂xi)[F,\Phi]=\sum_i(\partial F/\partial x_i\,\partial\Phi/\partial y_i-\partial F/\partial y_i\,\partial\Phi/\partial x_i) vanishes at every point of SμS_\mu; this is strictly weaker than Φ\Phi being a first integral, which requires the bracket to vanish everywhere. Fermi proved that for more than two degrees of freedom (n>2n>2), under genericity conditions on F0F_0 and F1F_1 (no nontrivial integer combination ∑imiωi\sum_i m_i\omega_i vanishes identically on an unperturbed energy surface, frequency-ratio relations are not algebraically dependent on one another, and the Fourier coefficients BmB_m of F1F_1 do not vanish at resonant combinations), every hypersurface defined by an analytic uniform Φ\Phi and invariant under the flow is a constant-energy surface. This is a cornerstone obstruction result for the ergodic foundation of statistical mechanics: it formalizes the statement that a generic Hamiltonian system has no conserved quantities and no invariant partitions of phase space beyond energy.

For exactly two degrees of freedom the obstruction collapses. Fermi's separate treatment of this case, as reconstructed in the secondary literature, runs as follows: one may choose the unperturbed surface S0S_0 on which the frequency ratio ω2/ω1\omega_2/\omega_1 equals a fixed irrational number α\alpha; then m⋅ω≠0m\cdot\omega\neq 0 for every nonzero integer pair mm, no division by zero ever occurs in the recursion for the Fourier coefficients of the series Φ=∑rμrΦr\Phi=\sum_r \mu^r \Phi_r, and the formal series for an invariant hypersurface exists order by order. The convergence of this series was not established, so the existence of analytic uniform invariant hypersurfaces other than energy surfaces is not excluded when n=2n=2.

The direct sequel by Benettin, Ferrari, Galgani and Giorgilli (1982) extended the non-existence theorem to invariant manifolds of intermediate dimension (between the invariant tori of dimension nn and the energy hypersurfaces of dimension 2n−12n-1), but its conclusion excludes precisely two classes: manifolds of resonant type and (n+1)(n+1)-dimensional families of nn-tori with mutually proportional frequencies. For n=2n=2 the second class is exactly Fermi's candidate construction. In parallel, Kolmogorov–Arnold–Moser (KAM) theory proves that individual invariant tori whose frequency vectors are Diophantine (badly approximable by rationals, i.e. ∣m⋅ω∣≥c/∣m∣τ|m\cdot\omega|\ge c/|m|^{\tau} for some constants c,τ>0c,\tau>0) survive small perturbations; as refined by Moser (1967) and Pöschel (1982), the surviving tori form families parametrized over Cantor sets with Whitney-smooth dependence (smooth in the sense of extending from the Cantor parameter set to open sets). These are not analytic hypersurfaces, and they neither prove convergence of Fermi's two-degree-of-freedom series nor refute the existence of the hypersurfaces. The boundary between the Poincaré–Fermi obstruction and the KAM persistence phenomenon therefore remains unmapped at two degrees of freedom — the case of greatest physical importance, since it governs the local dynamics of the restricted three-body problem (the motion of a massless body under the gravitational attraction of two massive bodies orbiting each other) and of area-preserving maps obtained as surfaces of section (two-dimensional cross-sections of the flow on which the return map preserves area).

Problem Statement

Consider an analytic nearly integrable Hamiltonian system with two degrees of freedom,

F(x,y,μ)=F0(y)+μF1(x,y)+μ2F2(x,y)+⋯ ,F(x,y,\mu) = F_0(y) + \mu F_1(x,y) + \mu^2 F_2(x,y) + \cdots,

where x∈T2x\in\mathbb{T}^2 (the two angles, each defined modulo 2π2\pi), the actions yy range over an open subset of R2\mathbb{R}^2, FF is analytic and 2π2\pi-periodic in each angle, the unperturbed frequency map ω(y)=∇yF0(y)\omega(y)=\nabla_y F_0(y) has rank 2, and F1F_1 is a perturbation satisfying the Poincaré–Fermi genericity condition that its Fourier support is not contained in any single line through the origin of Z2\mathbb{Z}^2 (equivalently, F1F_1 is not a function of any single angle combination); this excludes the degenerate perturbations that leave a nonzero integer combination of the actions conserved. Write [F,Φ]=∑i=12(∂F∂xi∂Φ∂yi−∂F∂yi∂Φ∂xi)[F,\Phi]=\sum_{i=1}^{2}\left(\frac{\partial F}{\partial x_i}\frac{\partial\Phi}{\partial y_i}-\frac{\partial F}{\partial y_i}\frac{\partial\Phi}{\partial x_i}\right) for the Poisson bracket. Call a hypersurface Sμ={Φ=0}S_\mu=\{\Phi=0\} invariant if [F,Φ][F,\Phi] vanishes at every point of SμS_\mu, and call Φ\Phi uniform if it is single-valued in its variables. Following the question left open by Fermi's 1923 analysis, determine the following.

Convergence at two degrees of freedom. Fix an irrational number α\alpha and consider the three-dimensional unperturbed set S0={(x,y):ω2(y)/ω1(y)=α}S_0=\{(x,y):\omega_2(y)/\omega_1(y)=\alpha\}, which is foliated by the unperturbed two-tori whose frequency vectors are all proportional to (1,α)(1,\alpha). On these tori the small divisors m⋅ω=ω1 (m1+m2α)m\cdot\omega=\omega_1\,(m_1+m_2\alpha) are nonzero for every nonzero integer pair m=(m1,m2)m=(m_1,m_2), because α\alpha is irrational, so the formal series Φ=∑r≥0μrΦr\Phi=\sum_{r\ge 0}\mu^r\Phi_r for the defining function of the candidate hypersurface, whose zeroth-order term Φ0\Phi_0 vanishes exactly on S0S_0, exists order by order without ever dividing by zero. Does this series converge for all sufficiently small ∣μ∣|\mu| — for some Diophantine irrational α\alpha (satisfying ∣qα−p∣≥c/qτ|q\alpha-p|\ge c/q^{\tau} for all integers pp and all positive integers qq, with constants c>0c>0 and τ>0\tau>0) and under a sufficiently rapid decay of the Fourier coefficients of F1F_1 — so that a nearly integrable system with two degrees of freedom can possess an analytic uniform invariant hypersurface that is not an energy surface?

This is exactly the exceptional class left open by Fermi's theorem and by its 1982 extension to invariant manifolds of intermediate dimension, namely (n+1)(n+1)-dimensional families of nn-tori with mutually proportional frequencies, which for n=2n=2 is the one-parameter family filling S0S_0.

References

  1. E. Fermi, 'Generalizzazione del teorema di Poincaré sopra la non esistenza di integrali uniformi di un sistema di equazioni canoniche normali', Il Nuovo Cimento 26, 105–114 (1923). DOI: 10.1007/BF02959350, https://doi.org/10.1007/BF02959350
  2. G. Benettin, G. Ferrari, L. Galgani, A. Giorgilli, 'An extension of the Poincaré–Fermi theorem on the nonexistence of invariant manifolds in nearly integrable Hamiltonian systems', Il Nuovo Cimento B 72, 137–148 (1982). DOI: 10.1007/BF02829400, https://doi.org/10.1007/BF02829400
  3. E. Fermi, 'Dimostrazione che in generale un sistema meccanico normale è quasi-ergodico', Il Nuovo Cimento 25, 267–269 (1923). DOI: 10.1007/BF02959600, https://doi.org/10.1007/BF02959600
  4. J. Moser, 'Convergent series expansions for quasi-periodic motions', Mathematische Annalen 169, 136–176 (1967). DOI: 10.1007/BF01399536, https://doi.org/10.1007/BF01399536
  5. J. Pöschel, 'Integrability of Hamiltonian systems on Cantor sets', Communications on Pure and Applied Mathematics 35, 653–696 (1982). DOI: 10.1002/cpa.3160350504, https://doi.org/10.1002/cpa.3160350504
  6. R. S. MacKay, 'A variational principle for invariant odd-dimensional submanifolds of an energy surface for Hamiltonian systems', Nonlinearity 4, 155–157 (1991). DOI: 10.1088/0951-7715/4/1/010, https://doi.org/10.1088/0951-7715/4/1/010
  7. I. Mezić, 'Break-up of invariant surfaces in action–angle–angle maps and flows', Physica D: Nonlinear Phenomena 154, 51–67 (2001). DOI: 10.1016/S0167-2789(01)00226-3, https://doi.org/10.1016/S0167-2789(01)00226-3