PoincarÉ Theorems

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Encyclopedia of Nonlinear Science

POINCARÉ THEOREMS

One of the greatest of all French mathematicians, Jules Henri Poincaré (1854–1912) graduated from the École Polytechnique in Paris and later studied at the École des Mines. In 1879, he became a docteur es sciences at the University of Paris, where he was appointed as a professor in 1881. Poincaré became a member of Académic des Sciences in 1887.

A mathematical genius of rare power, Poincaré’s approach to science was to solve concrete problems arising from mathematics, mechanics, and physics, rather than to present his results in a “pure axiomatic form.” However, a complete understanding of Poincaré’s scientific works has yet to be achieved. Outside mathematics, Poincaré is also known for his works in theoretical physics including his seminal contribution in the special theory of relativity (1904–1905) and for his works on the philosophy of science.

Many of Poincaré’s papers gave birth to whole new branches of mathematics, a prime example being algebraic topology, but the matter that occupied his life throughout was the geometrical approach to nonlinear differential equations—in particular, the long-time behavior of orbits of the Newtonian N-body problem in celestial mechanics.

The qualitative approach to nonlinear dynamics, introduced by Poincaré in his seminal papers “Mémoire sur les courbes définies par une équation différentielle” (1881–1886), which focuses on orbits rather than formulas was of a geometric and global nature. This is how the qualitative theory of ordinary differential equations was born.

Studying smooth vector fields on the plane, he classified their simplest equilibria (i.e., points where the given vector field vanishes): foci, nodes, saddle points, and centers. The typical smooth planar vector field has only the first three types of equilibria, but those of a more complicated nature are not excluded.

Poincaré outlined the proof that if a half trajectory γ of the planar vector field υ is confined in a compact domain K in which υ is free of equilibrium points, but the whole trajectory γ is not confined in K, then K contains a closed orbit of υ to which γ is asymptotically attracted (the Poincaré-Bendixson theorem in its simplest form). This type of closed orbit is called a limit cycle.

To generalize, let υ be a smooth vector field on a two-dimensional compact manifold M. To each isolated equilibrium p of υ, Poincaré associated an integer Ind(υ, p), called the index of υ at p, which is defined as follows. Let J be a small loop surrounding the isolated equilibrium p, and let Δθ be the total change of the angle θ that the vector of υ makes with some fixed direction when one runs counterclockwise along the loop J. This number is independent of the choice of loop. The index of p is the number Δθ/2π which is always an integer. (The index of a focus, center, or node is +1, and the index of a saddle point is −1.)

If υ has only a finite number of equilibria p₁,…, p_r, then Poincaré showed that

(1)

where χ(M) is the Euler-Poincaré characteristic of M and no restrictions are imposed on the nature of equilibria p₁,…, p_r. This result was generalized later by Heinz Hopf to the case of vector fields on compact manifolds of arbitrary dimension and is now called the Poincaré-Hopf theorem.

For a sphere, we have χ(M)=2, that is, an arbitrary smooth vector field on a two-dimensional sphere must vanish in at least one point—one cannot evenly comb the hair on a sphere! Another consequence of this theorem is that, for example, on the two-dimensional sphere, one cannot have a smooth vector field having as equilibria only two saddle points or having only three centers.

Poincaré’s investigations of celestial mechanics led him to the study of Hamiltonian systems with n degrees of freedom

(2)

with an analytic Hamiltonian function H(q, p), His researches in this area were summarized in his epoch-making three-volume treatise “Les méthodes nouvelles de la mécanique céleste” (1892, 1893, 1899).

The study of Hamiltonian systems close to integrable ones was called the “general problem of dynamics” by Poincaré. Specifically, he studied Hamiltonian equations (2) with a perturbed Hamiltonian of the form

H(q, p, ε)=H₀(p)+ε H₁(q, p) +ε² H₂(q, p)+…,

(3)

where H₁, H₂,… are periodic functions with respect to q of the same period. The system obtained by setting ε=0 is integrable, and the phase space is foliated by n-dimensional invariant tori p=const. For small nonzero ε, system (2) usually becomes non-integrable. In the case of two degrees of freedom, this means that the Hamiltonian function H is the only first integral of Equations (2) that is an analytic and uniform function of variables q, p and ε.

Poincaré’s participation in a mathematical competition organized in 1885 by Oskar II, King of Sweden and Norway, led him to the discovery of homoclinic orbits and related phenomena explaining how a very complicated behavior now called dynamical chaos, occurs in nonlinear dynamical systems. In his prize-winning work (1889), Poincaré discusses mainly the restricted three-body problem, where one mass is negligible compared with the other two, executing circular Keplerian motion.

Within the three-dimensional constant-energy manifold of this problem, he considered a two-dimensional surface Π transversal to most of the orbits. An orbit starting on this surface at a point x will pierce it again for the first time at some point y. The map x→y is the Poincaré return map induced on a surface of section Π.

It was in this framework that he discovered the existence of homoclinic orbits, that is, orbits which are asymptotically attracted by some periodic orbit γ when t→+∞ and t→−∞.

During his 1912 investigations on the restricted three-body problem, Poincaré conjectured that if one has a closed plane annulus Ω bordered by two concentric circles Γ₁ and Γ₂, then any area-preserving homeomorphism of Ω, such that and rotating these circles in opposite directions, has in Ω at least two different fixed points. This assertion, known as Poincaré’s last geometric theorem, was proved in 1913 by George D.Birkhoff and is known also as the Poincaré-Birkhoff theorem.

Inspired by the analogy between a flow induced by a vector field and a flow of an incompressible fluid, Poincaré developed a theory of integral invariants, which was later refined by Elie Cartan.

Considering time t as an independent variable, we can study Hamiltonian system (2) in the extended phase space (q, p, t). A tube of trajectories is a two-dimensional cylindrical surface formed by the segments of trajectories of the vector field defined by (2) and bounded by two disjoint smooth closed curves. According to the Poincaré-Cartan theorem, in the extended phase space (q, p, t), the action integral has the same value for two different closed paths γ₁ and γ₂ encircling the same tube of trajectories and lying on it.

Poincaré also proved an important property of the long-time behavior of dynamical systems. In contemporary formulation, this proof shows that for any measure-preserving mapping of a measure space with a finite total measure, almost all trajectories starting from a given subset of positive measure eventually return to it. This is known as the Poincaré recurrence theorem, and it lies at the foundations of ergodic theory.

In his works on celestial mechanics, Poincaré provided the first formal definition of asymptotic series: divergent series giving nevertheless good numerical approximations for functions they represent.

Poincaré is the founder of the concept of normal forms in the theory of ordinary differential equations and of the contemporary bifurcation theory. His work was also at the beginning of modern variational methods in mathematics, in particular of the Morse theory which strongly links mathematical analysis to geometry and topology.

Other nonlinear problems studied by Poincaré include the problem of the existence of geodesics on convex surfaces, the problem of tides, and the stability of rotating fluid bodies.

Poincaré’s impact on the theory of ordinary differential equations and dynamical systems is described in the books by Birkhoff (1927), Nemytskii & Stepanov (1960), Coddington & Levinson (1955), and Guckenheimer & Holmes (1990).

JEAN-MARIE STRELCYN AND ALEXEI TSYGVINTSEV

Poincaré, H. 1904. L’état actuel et l’avenir de la physique mathématique. Conférence lue le 24 Septembre 1904 au Congrés d’arts et de sciences de Saint-Louis. Bulletin des Sciences Mathématiques, 28:302–324; translation in Bulletin of the American Mathematical Society, 12 (1905–1906): 240–260, also published in Poincaré’s book La valeur de la science, Paris: Flammarion, 1905, Chapters VII, VIII, and IX

Poincaré, H. 1905. Sur la dynamique de l’élZctron, Comptes Rendus de l’Académie des Sciences, 140:1504–1508; also in Œuvres, vol. 9, pp. 489–493

Poincaré, H. 1906. Sur la dynamique de l’électron, Rendiconti del Circolo matematico di Palermo, 21:129–176; also in Œuvres, vol. 9, pp. 494–550

Poincaré, H. 1916–1956. Œuvres de Henri Poincaré, 11 vols, Paris: Gauthier-Villars (This paper (Poincaré, 1904), which is not included in Poincaré (1916–1956), is in fact the first Poincaré’s paper anticipating the special theory of relativity. The annoucement (Poincaré, 1905) and its full presentation in Poincaré (1906) is Poincaré’s contribution to what is now called the special theory of relativity.)

Symposium on the Mathematical Heritage of Henri Poincaré. 1983. The Mathematical Heritage of Henri Poincaré, edited by Felix E.Browder, 2 vols., Providence, RI: American Mathematical Society

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