Normal forms of weakly dissipative dynamical systems

 

Normal forms of dynamical systems allow to investigate special solutions and their nearby phase space geometry in an easy and concise way. Dynamical systems usually follow from the statement of a physical problem in some set of coordinates. The choice of coordinates often resembles the point of view of the observer of the problem. From the point of view of the theoretician those coordinates are usually not the best ones to investigate the problem. Better coordinates are chosen to transform the dynamical system into a form that allows to better understand the underlying structure of possible solutions and the phase space geometry. Canonical variables are the right choice to tackle dynamical problems that can be derived from a potential or admit a Hamiltonian function. Sophisticated mathematical tools have been invented over the last centuries to understand the nature of such systems, and to reveal the hidden secrets common to most dynamical problems found in physics. One major topic concerns the stability of motions, i.e. the stability problem. Roughly speaking, one seeks to find conditions on the parameters space and initial state of motions that allow stable solutions over long times. I have used normal forms to investigate stability of motions in various research studies over the past years, close to the triangular Lagrange points in a mapping of the restricted three-body problem or in the spin-orbit problem (1,2). For a review of the stability problem in generic dynamical systems, see Chapter 4 of my book.

In my work (I, II) on normal forms of weakly dissipative dynamical systems I generalized normal form theory to a class of dynamical systems that cannot be derived from a Hamiltonian function. The new theory was successfully applied to special classes of dynamical systems and allowed to state theorems about the stability of motions for different categories of initial states and system parameters for dynamical problems.

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