The Three-Body Problem: Covarinace of Lagarangian Derivatives.- Canonical Transformation.- The Hamilton-Jacobi Equation.- The Cauchy-Existence Theorem.- The n-Body Poblem.- Collision.- The Regularizing Transformation.- Application to the Three-Bdy Problem.- An Estimate of the Perimeter.- An Estimate of the Velocity.- Sundman's Theorem.- Triple Collision.- Triple-Collision Orbits.- Periodic Solutions: The Solutions of Lagrange.- Eigenvalues.- An Existence Theorem.- The Convergence Proof.- An Application to the Solution of Lagrange.- Hill's Problem.- A Generalization of Hill's Problem.- The Continuation Method.- The Fixed-Point Theorem.- Area-Preserving Analytic Transformations.- The Birkhoff Fixed-Point Theorem.- Stability: The Function-Theoretic Center Problem.- The Convergence Proof.- The Poincare Center Problem.- The Theorem of Liapunov.- The Theorem of Dirichlet.- The Normal Form of Hamiltonian Systems.- Area-Preserving Transformations.- Existence of Invariant Curves.- Proof of Lemma.- Application to the Stability Problem.- Stability of Equilibrium Solutions.- Quasi-Periodic Motion and Systems of Several Degrees of Freedom.- The Recurrence Theorem.

Lectures on celestial mechanics

This is a review paper on the formulation of nonrelativistic Quantum Mechanics on Phase Space. In an overlook to the traditional approach, we present the basic concepts including the Weyl mapping, the Wigner function, the Moyal propagator and the Spectral projections. We discuss the advantages and the disadvantages of this formulation. We apply it to the study of quadratic Hamiltonians. The Stratonovich-Weyl correspondence is presented as the natural framework in which the theory can be extended to include spin, relativity and constraints. The theory of identical particles is also considered.

Moyal Formulation of Quantum Mechanics

Versal Deformations and Normal Forms for Reversible and Hamiltonian Linear Systems

Normal form procedure This is the stepwise ‘simplification’ by changes of coordinates, of the Taylor series at an equilibrium point, or of similar series at periodic or quasi-periodic solutions. Preservation of structure The normal form procedure is set up in such a way that all coordinate changes preserve a certain appropriate structure. This applies to the class of Hamiltonian or volume preserving systems, as well as to systems that are equivariant or reversible with respect to a symmetry group. In all cases the systems may also depend on parameters. Symmetry reduction The truncated normal form often exhibits a toroidal symmetry that can be factored out, thereby leading to a lower dimensional reduction. Perturbation theory The attempt to extend properties of the (possibly reduced) normal form truncation, to the full system.

/pdf/normal-forms-in-perturbation-theory-13q6sbcpno.pdf

Normal forms in perturbation theory

Let R be a fixed linear involution (R2=id) of the spaceR n . A linear operator M is said to bereversible with respect to R if RM R=M−1 and infinitesimally reversible with respect to R if M R=−RM. A linear differential equation dx/dt=B(t)x is said to be reversible with respect to R if V(t)R ≡−RV(−t). We construct normal forms and versal deformations for reversible and infinitesimally reversible operators. The results are applied to describe the homotopy classes of strongly stable reversible linear differential equations with periodic coefficients. The analogous theory for linear Hamiltonian systems was developed by J. Williamson, M. G. Krein, I.M. Gel'fand, V. B. Lidskii, D. M. Galin, and H. Kocak.

Reversible linear systems and their versal deformations

Normal forms and versal deformations of linear Hamiltonian systems

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Normal forms and versal deformations of linear Hamiltonian systems