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I-form 26,27

accessible stress path 367,369 affine group 111 affine transformations 103,104 affine vector field 56,104,290

INDEX

almost contact metric structure 227-228,263-264 asymptotically stable equilibrium point 118, 121,

129,136,139,140,143 attractor 125,127

Bertrand algorithm 237 bidimensional Hamiltonian systems 312,313 bifurcation point 202,210 bifurcation set 156,202,357,361,365 Biot-Savart-Laplace vector field 49, 101, 303, 314,

323,347 biscalar vector field 42,58,236,238,270 Bobbio-Marrucci vector field 295-296 bracket 21,26 butterfly potential 157,166

catastrophe manifold 155 Cauchy stress tensor 358 centre 125,128 centre manifold 207 ,215 chart 17 Chow theorem 242 coframe 28 cylindrical hypersurfaces 187 complete vector field 88-93,100,153 completely integrable Pfaff equations 230,233,

247,253 conformal transformations 105,113 conformal vector field 53,104,105,288 conical hypersurfaces 189 conservative differential systems of order

two 141,286,287,290-293,307,355 constant level set 2,42 constitutive equations 357 convex function 32 cotangent space 26

393

covariant derivative 20 critical points 2,14,146,147,268,284,285,287,

288,290-292,296,299,336-338,342, 344 curve 17 curve defined by a sequence of points 255 curvilinear integral 38 cusp potential 157,161 cuspidal catastrophes 157

derivation 23,26 derivative with respect to a vector 18,19,23 diffeomorphism 14,110,114,182 differential systems with n degrees of

freedom 95,128,280,286,287,290-293,307 differentiability theorem 70 distribution orthogonal to a vector field 240,247 divergence 30

electromagnetic dynamical systems 100,116,350,352 electrostatic field 12,45,46,78 elliptic umbilic potential 157,167 energy of a magnetic field 336-346 energy of a vector field 130,274,289 equation of continuity 359 equilibrium point 65,117,121,130,133,135,142,

155,167,202,204,205,211,213,221,324,328, 329,332,334

equilibrium points after interaction 263,268 Euler equations 119,120,185,186,302 Euler theorem 80 Euler-Lagrange equations 279 Euler potentials 48 extension theorem 69 extrema constrained by Pfaff system 253,258

field hypersurface 177,198 field of velocities 13 field lines (orbits) 63-70,74,82,244,245,274,277,

296,299 first integral 72,120 flow 96,98,99,110,116,251,273,298-299

394

flow bifurcation 210-224 flux of a vector field 232 focus 126,211,213 fold potential 157,158 frame 8,24,25,28 Frobenius theorem 233

Gauss-Kronecker curvature 156,263 Gibbs-Pfaff equation 261 Goodwin differential system 132,218,301 Grad conjecture 320 gradient 13,18,29 gradient line 148,150,151,158,161,164,166,167,

171,173,176,297,300 granular material 357,360,384 gravitational field 12,44 grid numerical method 196 Gronwall lemma 99 group of roto-translations 110

Hamiltonian 93,116,141,176,279,280,281,293, 294,307,313,321,347,349,351,352

Hamiltonian vector field 73,93,101,116,138, 141,279,293,307,313,321,347,349

harmonic function 31,42 harmonic polynomials 33 harmonic vector field 50,81 Hessian 29 heteroclinic structure 332-335 homogeneous function 184,186 Hopf theorem of bifurcation 214,215,253 Hurwitz criterion 123,134,135 hyperbolic umbilic potential 157,171 hypersurface of revolution 190 hypoelastic materials 357

immersion 14 implicit function theorem 15 infinitesimal transformations 96 integrable combinations 75,76 integrable distribution 241 integral manifolds 226,253 integrant factor 230 invariant set 99,206 inverse function theorem 14

Index

irrotational map 108 irrotational vector field 29,36,37,58,108,229,284 isometries 103

Killing vector field 51,73,81,92,102,103,191,192, 194,286

kinetic energy 93,267 knot 126,213

Lagrange theorem 142 Lagrangian 278 Laplacian 30 Legendre transformation 279 Lie algebra 26,61,82 linear approximation 129,333,335,363,366 linear vector field 81 Liouville theorem 97 Lorentz group 111 Lorentz manifold 352 Lorentz force law 307,308,349 Lorentz-Udriste world-force law 281,308 Lorenz system 92,101,217,272,301 Lyapunov functions 137,141-144

magnetic field 41,101,303,314 magnetic line 305,315-332 magnetic surface 305-322 magnetic trap 305,306 magnetics dynamics 305,307 Maxwell equations 101,264-266,350,353 mixed product 9,10 Monge potentials 49,60,295,301 Morse I-saddle 156,157

Newtonian field 44,45,60,101 nonholonomic manifolds 242,244,249 nonlinear vector field 81 normal space 17 normal vector 17

parabolic umbilic potential 157,173 parametrization 17 particle dynamics 280 Pfaff equation 75,226,229,232,233,253,270,271

Geometric Dynamics

potential differential systems of order two 141,286,287,307,355

potential differential systems of order one 145,148, 153,155,158,161,164,166,167,171,173,295

potential energy 93 potential vector field 35,37,39,58,81 projective transformations 107,112 projective vector field 56,106,107,291 proper values of a vector field 193

rectification theorem 71 Riemann-Jacobi-Lagrange structure 282,283,308 Riemann-Jacobi metric 283,286,307,333 rotation point 125 rotor (curl) 29,30 ruled hypersurface 187 Runge-Kutta formulas 86,87

Sabba Steranescu conjecture 314,316-320 scalar product 8,10 scalar field 1 saddle point 127 simple thermodynamic system 264 singular point 2,14,243 singularity set 156 solenoidal map 109 solenoidal vector field 30,46-49,58,81,101,109,

195,278,304 spacetime 352 spherical symmetry 43,67 stability domains 357

395

stable equilibrium point 118,121,137,140,142,295 Stokes potentials 50,60 stress-energy tensor 354 stress history 357,358 submanifold 15-17 submersion 14 swallowtail potential 157,164 symmetric differential system 75,178 symplectic manifold 347,352

tangent bundle 17 tangent space 7,17,24 tangent vector 6,17,23,32 thermodynamic systems 236,261-264 torse forming map 109 torse forming vector field 58,66,92,109,187,236,

292 total energy 93

umbilic catastrophes 157 unstable equilibrium point 118,121,129,140-144,

295 uniformly constrained extrema by a Pfaff system

258

Van der Pol equation 216 vector field 9,10,17,25,26,43 vector potential 47,304 vector product 8-11

waves 353