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Flows on Surfaces
Poincare began work on differential equations of celestial mechanics and announced preliminary results about the curves defined by a first-order equation of the form X=x(x,y) and Y=y(x,y). Poincare emphasized coefficients X and Y were real variables whose solutions were real curves. Attention to the real instead of the complex case reversed the priorities of a generation of pure mathematicians. His actions established new ideas on the subject.
Henri Poincare -
Rotating Fluid Masses
In notes,Poincare sought to establish the unproven claim of Thomson and Tati's that it is probable that for moment of momentum greater than one definite limit and less than another, there is just one annular figure of equilibrium, consisting of a single ring.
He proved if an arbitrary system is stable under the action of certain forces and subjected to infinitesimally small perturbing forces, it will always occupy a new stable equilibrium position infinitely closer to the new one. -
Residues of Double Integrals
Poincare's paper on the residues of double integrals generalized Cauchy's theory to the new situation. Poincare admits because the problem involves two complex variables and four real ones if one uses geometrical language about a four-dimensional space this would drive away most people.
Poincare's alternative was to remove the use of geometry and regard curves and surfaces in a three-dimensional space. He proposed to focus on closed surfaces where S is bound by C, to double integrals. -
Thermodynamics
It was while preparing for a lecture, Poincare came across HelmholtzHel's theory of how the subject could be squared with mechanics known as reversible phenomena. Poincare agreed with this theory. However, when Helmholtz turned to irreversible phenomena, Poincare was not satisfied with that account. He argued when a system of particles is formulated in Hamiltonian dynamics, it will be time-reversible if and only if the coordinates enter not only in even powers but also odd powers.