Geometric Mechanics & Dynamical Systems

Lorenz Attractor

“Formal perfection, in mathematics, is never negligible; it is even a necessary ingredient for progress. However, it seems that the classical formalization of analytical mechanics has overlooked an important part of Lagrange's ideas. Without doubt this happened because the mathematics of the first part of the 19th century did not have the necessary scope; also because the successes of the theory (first in celestial mechanics and later in statistical mechanics) hid the necessity of questioning categories so classical that they seemed natural. One had to wait until the 20th century to learn that the words time, space, and matter do not have any direct physical meaning, but are only formal symbols of revisable physical theories, and nowadays are outdated.

Analytical mechanics is not an outdated theory, but it appears that the categories which one classically attributes to it such as configuration space, phase space, Lagrangian formalism, Hamiltonian formalism, are, simply because they do not have the required covariance; in other words, because these categories are in contradiction with Galilean relativity. A fortiori, they are inadequate for the formulation of relativistic mechanics in the sense of Einstein.”

   - J M Souriau, “Structure of Dynamical Systems”

The Domain

Geometric Mechanics is Classical Mechanics formulated in the language of modern Differential Geometry. Of course, while Lagrangian Mechanics, to a certain extent, retains the standard Differential Geometric form, Hamiltonian Mechanics alters it significantly enough to give it a special name, -Symplectic Geometry. Lagrangian Mechanics itself initially takes the form of Riemannian Geometry as was first clearly demonstrated by Hertz in his “Principles of Mechanics.” The departure from Riemannian to Symplectic takes place via the Legendre transform (or fibre derivative, in symplectic language) that maps the Lagrangian to the Hamiltonian. When the transform is regular (or rather, hyperregular), the map in a sense, takes the “velocities” to the “momenta.” Otherwise, there arise singular constraints and need to be treated quite differently.

John L. Synge was one of the first to cast Lagrangian Mechanics into a geometric framework. In his beautiful Handbuch der Physik encyclopedia article, he gave a treatment of Lagrangian Mechanics as Riemannian Geometry. But the modern, qualitative-quantitative form of Mechanics really took off with Henri Poincare’s introduction of the qualitative methods of Topological Analysis into Mechanics. His methods were characterised by the global geometric point of view. He treated a dynamical system as a vector field on the phase space of the system. A solution then corresponded to a smooth curve  tangent at each of its points to the vector based at that point. This global point of view was capable of giving the complete information of the dynamical system. The manifold or bundle that arose could then be studied and the necessary structures like the metric, connections, almost-tangent structures, almost-complex structures as have now been developed, could be imposed. After Poincare, George Birkoff contributed immensely to the development of Dynamical systems, with the publication of his American Mathematical Society monograph, “Dynamical Systems,” one of the most illuminating books on the subject.

But there was another immense source of ideas and methods infiltrating into Mechanics. It was that of Elie Cartan. Elie Cartan’s work revolutionised Mechanics like nothing before. His powerful Exterior Calculus and classification of Lie Algebras and Lie Groups changed the face of Mechanics forever.

These twin streams of ideas and methods, from Poincare and Cartan were taken up by a new generation of mathematicians and the very fertile field of Geometric Mechanics was born. Presently, Geometric Mechanics has matured to take the shape of one of the most extensively developed and developing fields of Mathematics and Mathematical Physics.

Source Books & Links

  1. V I Arnold - Mathematical Methods of Classical Mechanics: The most straightforward and enjoyable introduction to Geometric Mechanics, and indeed, to modern Differential Geometric and Symplectic methods in Physics.
  2. Abraham and Marsden - Foundations of Mechanics: The most extensive, rigorous, complete and beautiful treatment of Geometric Mechanics, Dynamical Systems and Topological Dynamics in the field.
  3. G Marmo, Salaten, Simoni, Vitale - Dynamical Systems, A Differential Geometric Approach to Symmetry and Reduction: The most insightful and beautiful treatment of the subject with an orientation towards physical intuition.

The Field

Henri Poincare: The last “Universalist,” as E. T. Bell calls him in his “Men of Mathematics,” and perhaps the greatest mathematician of the 20th Century sharing that position only with David Hilbert. Poincare is the real founder of modern Geometric Mechanics with his introduction of qualitative methods into the analysis of dynamical systems. In his researches on Celestial Mechanics, he was led to introduce the enormously insightful concept of the phase portrait. Almost in parallel, he initiated Analysis Situs, that developed and matured into modern Topology. As he writes, “As for me all all the various journeys, one by one I found myself engaged, were leading me to Analysis Situs.”

He explained what he meant by Analysis Situs as follows.

“L’ Analysis Situs est la science qui nous fait connaitre les proprettes qualitatives des figures géométriques non seulement dans l’ espace ordinaire, mais dans l’ espace a plus trois dimensions. L’ Analysis Situs a trois dimensions est pour nous une connaissance presque intuitive, L’ Analysis Situs a plus de trios dimensions au contraire des difficultés énormes; il faut pour tenter de les surmounter être bien persuade de l’ extreme importance de cette science.”

(“Analysis Situs is a science which lets us learn the qualitative properties of geometric figures not only in the ordinary space, but also in the space of more than three dimensions. Analysis Situs in three dimensions is almost intuitive knowledge for us. Analysis Situs in more than three dimensions presents, on the contrary, enormous difficulties, and to attempt to surmount them, one should be persuaded of the extreme importance of this science. If this importance is not understood by everyone, it is because everyone has not sufficiently reflected upon it.”)

The modern theory of Dynamical Systems, singularities and bifurcations, Chaos, and a host of topological and differential geometric constructions are an offshoot of Poincare’s ideas.

George D Birkhoff: Birkhoff proved Poincare’s Geometric Theorem almost immediately as it was posed by Poincare. The theorem may be stated in a simple form as follows: Let us suppose that a continuous one-to-one transformation T takes the ring R, formed by concentric circles Ca and Cb of radii a and b respectively (a > b > 0), into itself in such a way as to advance the points of Ca in a positive sense, and the points of Cb in the negative sense, and at the same time to preserve areas. Then there are at least two invariant points.

Birkhoff’s American Mathematical Society monograph, “Dynamical Systems,” is a classic and one of the first books that is still worth reading for its elegant treatment of the subject.

Constantin Caratheodory: Was David Hilbert’s successor at Gottingen University and a versatile mathematician and one of the most insightful developers of the field of Partial Differential Equations, the Calculus of Variations and Complex Function Theory. His 2-Volume monograph, “Calculus of Variations and Partial Differential Equations of First Order,” is the finest book on the Calculus of Variations and Partial Differential Equations and develops especially the Hamilton-Jacobi theory in all its detail. His book on Function theory is a classic. His approach to the second law of thermodynamics, captured by the famous “Caratheodory’s Theorem,” (as presented for example, in S. Chandrasekhar’s Introduction to the Study of Stellar Structure) is elegant and is taken as the starting point in several treatments of the subject.

Jacques Hadamard: One of the greatest mathematicians of the 20th Century and perhaps the most influential next only to Poincare, Hilbert and Weyl, and in the same class as Elie Cartan, Hadamard inspired and guided an entire generation of mathematicians including Nicholas Bourbaki, Andre Weil, Jean Dieudonne, John Leray, Laurent Schwartz…Hadamard had an pervasive influence on modern mathematics and in Geometric Mechanics his name is associated with the Cartan-Hadamard theorem a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature.

Elie Cartan

Elie Cartan

Elie Cartan: One of the greatest mathematicians of all time, and unquestionably, the greatest geometer of the 20th Century, Elie Joseph Cartan’s influence on mathematics and mathematical thought has been all-pervading, from Differential Geometry, Topology, Lie Groups and Lie Algebras, Theory of Spinors (that he had introduced and developed earlier to Dirac), General Relativity (the famous Einstein-Cartan theory), the classification problem of Riemannian manifolds. His most important contribution being the exterior derivative that changed Calculus forever after Newton and Leibniz. The modern coordinate-free approach that pervades modern physics is due to him. He was also the first to cast Newtonian mechanics into geometric form. After Poincare, it was Elie Cartan who inspired an entire generation of mathematicians including that of Nicholas Bourbaki. In a sense, therefore, it is Elie Cartan who laid the structural foundations of Geometric Mechanics and Dynamical Systems. His obituary by Chern and Chevalier, “Elie Cartan and his Mathematical Work,” and biography by Akvis and Rosenfeld, Elie Cartan, is a most inspiring and instructive read. Cartan’s own works were not that easy to read.

In Robert Bryant's words, “You read the introduction to a paper of Cartan and you understand nothing. Then you read the rest of the paper and still you understand nothing. Then you go back and read the introduction again and there begins to be the faint glimmer of something very interesting.” Nevertheless, his books, “Leçons sur les invariants intégraux” and “Leçons sur la géométrie des espaces de Riemann” are masterpieces of mathematical writing.

Amalie Emmy Noether: One of the greatest mathematicians of the 20th Century and whose work forms the cornerstone of the foundations of Classical Mechanics, Field Theory and Gauge Theory. Indeed, it is impossible to imagine proceeding in any of these fields without encountering “Noether’s theorem.” Albert Einstein wrote of her, “In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant creative mathematical genius since the higher education of women began”. Noether’s first and second theorems are the starting point of field theory and of Singular Constraint Systems. Nina Byer’s two articles, “The Life and Times of Emmy Noether,” and “Emmy Noether’s Discovery of the Deep Connection between Symmetry and Conservation Laws,” are most instructive to read.

Stephen Smale

Stephen Smale

Vladmir I Arnold

Vladmir I Arnold

Rene Thom

Rene Thom