# Power Geometry in Algebraic and Differential Equations

**Edited by**

- A.D. Bruno, Keldysh Institute of Applied Mathematics, RAS, Miusskaja sq.4, Moscow 125047, Russia

The geometry of power exponents includes the Newton polyhedron, normal cones of its faces, power and logarithmic transformations. On the basis of the geometry universal algorithms for simplifications of systems of nonlinear equations (algebraic, ordinary differential and partial differential) were developed.

The algorithms form a new calculus which allows to make local and asymptotical analysis of solutions to those systems.

The efficiency of the calculus is demonstrated with regard to several complicated problems from Robotics, Celestial Mechanics, Hydrodynamics and Thermodynamics. The calculus also gives classical results obtained earlier intuitively and is an alternative to Algebraic Geometry, Differential Algebra, Lie group Analysis and Nonstandard Analysis.

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### Book information

- Published: August 2000
- Imprint: ELSEVIER
- ISBN: 978-0-444-50297-1

### Table of Contents

Preface. 0.Introduction. 1. Concepts of Power Geometry. 2. Historical remarks. 3. A brief survey of the book. 1. The linear inequalitites. 1. Principal definitions and properties. 2. The normal and tangent cones. 3. Graphical solution of Problem 1. 4. The Motzkin-Burger algorithm. 5. Algorithmic solution of Problem 1. 6. Cone of the problem. 7. About the computer program. 8. An infinite set S. 9. Coherent boundary subsets. 10. Comparison with the Bugaev-Sintsov method. 11. Linear transformations.2. Singularities of algebraic equations. 1. Implicit function. 2. Newton polyhedron. 3. Power transformations. 4. Asymptotic solution of an algebraic equation. 5. Implicit functions. 6. Truncated systems of equations. 7. Linear transformations of power exponents. 8. Asymptotic solution of a system of equations. 9. Positional functions of mechanisms. 10. Historical and bibliographical remarks. 3. Asymptotics of solutions to a system of ODE. 1. Local theorems of existence. 2. The power transformation. 3. The generalized power transformations. 4. Truncated systems. 5. The power asymptotics. 6. Logarithmic asymptotics. 7. The simplex systems. 8. A big example. 9. Remarks. 4. Hamiltonian truncations. 1. The theory. 2. The generalized Henon-Heiles system. 3. The Sokol'skii cases of zero frequencies. 4. The restricted three-body problem. 5. Local analysis of an ODE system. 1. Introduction. 2. Normal form of a linear system. 3. The Newton polyhedron. 4. The reduction of System (3.10) 5. The classification of System (4.2) 6. The normal form of a nonlinear system. 7. Cases I and &ggr;1. 8. System (4.2) in Cases II and IV. 9. The non-resonant case III. 10. The normal form in the resonant Case III. 11. The resonances of higher order. 12. The resonance 1:3 in Case III. 13. The resonance 1:2 in Case III. 14. The normal form in Case &ggr;2. 15. The normal form in Cases &ggr;0 and &ggr;3. 16. The review of the results for System (4.2). 17. The transference of results to the original system. 18. The comparison with the Hamiltonian normal form. 19. The case &mgr;=0. 20. The Belitskii normal form. 21. The problem of surface waves. 22. On the supernormal form. 6. Systems of arbitrary equations. 1. Truncated systems. 2. Power transformations. 3. The logarithmic transformation. 4. A big example. 5. One partial differential equation. 6. The viscous fluid flow around a plate. 7. Self-similar solutions. 1. Supports of a function. 2. Supports of a differential polynomial. 3. The Lie operators. 4. Self-similar solutions. 5. The power transformation. 6. The logarithmic transformation. 7. The ordinary differential equation. 8. The system of equations. 8. On complexity of problems of Power Geometry. 1. The levels of complexity. 2. The linear equalities. 3. The linear transformations. 4. Linear inequalities. 5. On applications of Power Geometry. 6. Historical remarks. Bibliography. Subject index.