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.
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.
Front Cover 1
Power Geometry in Algebraic and Differential Equations 4
Copyright Page 5
Contents 8
Preface 6
Chapter 0. Introduction 12
1. Concepts of Power Geometry 12
2. Historical remarks 15
3. A brief survey of the book 16
Chapter 1. The linear inequalities 20
1. Principal definitions and properties 20
2. The normal and tangent cones 25
3. Graphical solution of Problem 1 28
4. The Motzkin–Burger algorithm 30
5. Algorithmic solution of Problem 1 33
6. Cone of the problem 43
7. About the computer program 45
8. An infinite set S 50
9. Coherent boundary subsets 54
10. Comparison with the Bugaev–Sintsov method 57
11. Linear transformations 60
Chapter 2. Singularities of algebraic equations 66
1. Implicit function 66
2. Newton polyhedron 70
3. Power transformations 74
4. Asymptotic solution of an algebraic equation 76
5. Implicit functions 82
6. Truncated systems of equations 83
7. Linear transformations of power exponents 88
8. Asymptotic solution of a system of equations 92
9. Positional functions of mechanisms 99
10. Historical and bibliographical remarks 110
Chapter 3. Asymptotics of solutions to a system of ODE 116
1. Local theorems of existence 116
2. The power transformation 121
3. The generalized power transformations 127
4. Truncated systems 133
5. The power asymptotics 139
6. Logarithmic asymptotics 146
7. The simplex systems 156
8. A big example 161
9. Remarks 169
Chapter 4. Hamiltonian truncations 172
1. The theory 172
2. The generalized Henon–Heiles system 182
3. The Sokol'skii cases of zero frequencies 185
4. The restricted three-body problem 197
Chapter 5. Local analysis of an ODE system 202
1. Introduction 202
2. Normal form of a linear system 206
3. The Newton polyhedron 207
4. The reduction of System (3.10) 214
5. The classification of System (4.2) 216
6. The normal form of a nonlinear system 225
7. Cases I and .1 230
8. System (4.2) in Cases II and IV 232
9. The non-resonant case III 234
10. The normal form in the resonant Case III 239
11. The resonances of higher order 247
12. The resonance 1:3 in Case III 251
13. The resonance 1:2 in Case III 255
14. The normal form in Case .2 258
15. The normal form in Cases .0 and .3 262
16. The review of the results for System (4.2) 270
17. The transference of results to the original system 272
18. The comparison with the Hamiltonian normal form 273
19. The case µ=0 275
20. The Belitskii normal form 275
21. The problem of surface waves 282
22. On the supernormal form 284
Chapter 6. Systems of arbitrary equations 288
1. Truncated systems 288
2. Power transformations 297
3. The logarithmic transformation 301
4. A big example 304
5. One partial differential equation 309
6. The viscous fluid flow around a plate 311
Chapter 7. Self-similar solutions 326
1. Supports of a function 326
2. Supports of a differential polynomial 327
3. The Lie operators 328
4. Self-similar solutions 330
5. The power transformation 336
6. The logarithmic transformation 340
7. The ordinary differential equation 343
8. The system of equations 347
Chapter 8. On complexity of problems of Power Geometry 352
1. The levels of complexity 352
2. The linear equalities 354
3. The linear transformations 358
4. Linear inequalities 360
5. On applications of Power Geometry 363
6. Historical remarks 364
Bibliography 370
Subject index 394
Introduction
Alexander D. Bruno Keldysh Institute of Applied Mathematics, Moscow
1 Concepts of Power Geometry
Many problems in mechanics, physics, biology, economics and other sciences are reduced to nonlinear equations or to systems of such equations. The equations may be algebraic, ordinary differential or partial differential; and systems may comprise the equations of one type, but may include equations of different types. The solutions to these equations and systems subdivide into regular and singular ones. Near a regular solution the implicit function theorem or its analogs are applicable, which gives a description of all neighboring solutions. Near a singular solution the implicit function theorem is inapplicable, and until recently there had been no general approach to analysis of solutions neighboring the singular one. Although different methods of such analysis were suggested for some special problems.
The purpose of the book is to supply a general purpose set of algorithms for analysis of singularities applicable to all types of equations. At present, the usual way of development of mathematical sciences may be depicted as the following sequence:
→ Theory → Algorithm →Software → Computation → Application,
(1.1)
where some elements may be in plural and feedbacks play a major part [Bruno 1998a]. This book comprises all elements of the sequence (1.1).
The main concept of Power Geometry is to study the properties of solutions to an equation through the power exponents of its monomials. For instance, to the polynomial
(X)=ΣfQXQ,Q∈S,
(1.2)
where X = (x1,…,xn), Q = (q1,…,qn), Q=x1q1…xnqn, there corresponds the set S ⊂ n of the vector power exponents Q, for which the coefficients fQ ≠ 0. Together with the set S (the support of Polynomial (1.2)), we consider its convex hull Γ (the Newton polyhedron of Polynomial (1.2)) and faces j(d) of the polyhedron Γ in the space n, and also their normal cones j(d) in the dual space *n that is the space of logarithms of coordinates xi. To each face j(d) there corresponds the truncation of Polynomial (1.2)
^j(d)(X)=ΣfQXQ,Q∈S∩Γj(d).
(1.3)
It is the first approximation to the Polynomial (1.2) in the part of X-space that corresponds to the normal cone j(d) in the (ln X) -space *n. In that part of the X-space, the first approximation to a solution to the equation f(X) = 0 is the solution to the truncated equation ^j(d)(X)=0. The power transformations of coordinates, which are linear transformations of logarithms of coordinates, induce the linear transformations in spaces n and *n, which allows to reduce the number of coordinates in a problem, especially for the truncated equations. The linear transformations in n and *n make Power Geometry a geometry in the Klein’s sense (see [Klein 1872]).
Example. Consider a plane (i.e. n = 2) algebraic curve, which is called the folium of Descartes and is defined by the equation
(X)=defx13+x23−3x1x2=0.
(1.4)
Let us study solutions to this equation near the origin x1 = x2 = 0 and at infinity, where the implicit function theorem can not be applied. The set S of power exponents of the equation consists of three points: Q1 = (3, 0), Q2 = (0, 3), Q3 = (1, 1). Their convex hull Γ is the triangle (Fig. 0.1, a). The points Q1 and Q3 lie in its edge 1(1). Hence, to it there corresponds the truncated equation
^1(1)(X)=defx13−3x1x2=0.
(1.5)
Its nontrivial solution 2=x12/3 is the first approximation to the branch 1 of solutions to Equation (1.4) (see Fig. 0.2). The branch 1 passes through the point x1 = x2 = 0. The points Q2 and Q3 lie on the edge 2(1); to it there corresponds the truncated equation 23−3x1x2=0. Its nontrivial solution 2=x12/3 the first approximation to the second branch 2 passing through the origin. To obtain subsequent approximations to the branch 1, let us make the power transformation
y1=x12x2−1,y2=x1−1x2,with inverse{x1=y1/y2,x2=y1y22.
(1.6)
Then Equations (1.4) and (1.5) are transformed into
=defy12y23(y1+y1y23−3)=0,f^1(1)=defy12y23(y1−3)=0.
(1.7)
Cancelling 12y23, we obtain the complete equation
1+y1y23−3=0
(1.8)
and its truncated equation y1 − 3 = 0. For Equation (1.8), the support and polyhedron are shown in Fig. 0.1, b. The root y1 = 3 of the truncated equation is simple. Hence in the neighborhood of the point y1 = 3,y2 = 0 the implicit function theorem is applicable to the complete Equation (1.8), which allows to obtain y1 − 3 as a power series of y2. Here it may be done explicitly: 1=3/(1+y23). Substituting this expression in right Formulae (6), we obtain the parametric representation for the branch 1:
1=3y2/(1+y23),x2=3y22/(1+y23).
The points Q1 and Q2 lie on the edge 3(1), and to it there corresponds the truncated equation 13+x23=0. Its only real solution x1 + x2 = 0 is the first approximation to the asymptote x1 + x2 = −1 of the folium of Descartes, shown in Fig. 0.2. To obtain the asymptotic expansion for its branches that go into infinity we need to apply the power transformation 1=x1,x2=x1−1x2.
Exercise. Plot the support and the Newton polyhedron for the left Equation (1.7).
2 Historical remarks
Power Geometry is based upon the three concepts: the Newton polyhedron, the power transformation and the logarithmic transformation. The crucial points of their development are as follows.
I. The Newton polyhedron. For n = 2, approximately in 1670 Newton [1711] suggested to use one edge of the “Newton open polygon” [Bruno 1979a] of a polynomial f(x, y) to find the branches of solutions to the equation f(x, y) = 0 near the origin x = y = 0, where the polynomial f has no constant or linear terms. Puiseux [1850] was already using all the edges of the Newton open polygon and had given a rigorous substantiation to the solution of the problem by this method. Liouville [1833] was using this approach to find the rational solutions y = y(x) to the linear ordinary differential equation
0(x)dny/dxn+…+an−1(x)dy/dx+an(x)=0,
where αi(x) are polynomials. Briot and Bouquet [1856] were using an analog to the Newton open polygon to find solutions y(x) to the nonlinear ordinary differential equation dy/dx = f(x, y)/g(x, y) near the point x = y = 0, where polynomials f and g vanish. A survey of other applications of the Newton (open) polygon was made by Chebotarev [1943]. The survey should be completed by mentioning the dual open polygons [Bruno 1979a] (see for example the Frommer open polygon [Frommer [1928]) and by attempts to reduce the solution of a system of algebraic equations of three or more coordinates to the plane Newton open polygon (see Section 10 of Chapter 2).
Sintsov [1898] to obtain expansions y(x), z(x) of the branches of solutions to an algebraic system of equations f1 (x, y, z) = f2...
Erscheint lt. Verlag | 3.8.2000 |
---|---|
Sprache | englisch |
Themenwelt | Mathematik / Informatik ► Informatik ► Theorie / Studium |
Mathematik / Informatik ► Mathematik ► Analysis | |
Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
Technik | |
ISBN-10 | 0-08-053933-5 / 0080539335 |
ISBN-13 | 978-0-08-053933-1 / 9780080539331 |
Haben Sie eine Frage zum Produkt? |
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