## Applications of Lie Groups to Differential EquationsSymmetry methods have long been recognized to be of great importance for the study of the differential equations arising in mathematics, physics, engineering, and many other disciplines. The purpose of this book is to provide a solid introduction to those applications of Lie groups to differential equations that have proved to be useful in practice, including determination of symmetry groups, integration of ordinary differential equations, construction of group-invariant solutions to partial differential equations, symmetries and conservation laws, generalized symmetries, and symmetry methods in Hamiltonian systems. The computational methods are presented so that graduate students and researchers in other fields can readily learn to use them. Following an exposition of the applications, the book develops the underlying theory. Many of the topics are presented in a novel way, with an emphasis on explicit examples and computations. Further examples, as well as new theoretical developments, appear in the exercises at the end of each chapter. This second edition contains a new section on formal symmetries and the calculus of pseudodifferential operators, simpler proofs of some theorems, new exercises, and a substantially updated bibliography. |

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### Contents

Introduction to Lie Groups | 1 |

11 Manifolds | 2 |

Change of Coordinates | 6 |

Maps Between Manifolds | 7 |

Submanifolds | 8 |

Regular Submanifolds | 11 |

Curves and Connectedness | 12 |

12 Lie Groups | 13 |

The Variational Derivative | 244 |

Null Lagrangians and Divergences | 247 |

Invariance of the Euler Operator | 249 |

42 Variational Symmetries | 252 |

Infinitesimal Criterion of Invariance | 253 |

Symmetries of the EulerLagrange Equations | 255 |

Reduction of Order | 257 |

43 Conservation Laws | 261 |

Lie Subgroups | 17 |

Local Lie Groups | 18 |

Local Transformation Groups | 20 |

Orbits | 22 |

13 Vector Fields | 24 |

Flows | 27 |

Action on Functions | 30 |

Differentials | 32 |

Lie Brackets | 33 |

Tangent Spaces and Vector Fields on Submanifolds | 37 |

Frobenius Theorem | 38 |

14 Lie Algebras | 42 |

OneParameter Subgroups | 44 |

Subalgebras | 46 |

The Exponential Map | 48 |

Structure Constants | 50 |

Infinitesimal Group Actions | 51 |

15 Differential Forms | 53 |

PullBack and Change of Coordinates | 56 |

The Differential | 57 |

The de Rham Complex | 58 |

Lie Derivatives | 60 |

Homotopy Operators | 63 |

Integration and Stokes Theorem | 65 |

Notes | 67 |

Exercise | 69 |

Symmetry Groups of Differential Equations | 75 |

21 Symmetries of Algebraic Equations | 76 |

Invariant Functions | 77 |

Infinitesimal Invariance | 79 |

Local Invariance | 83 |

Invariants and Functional Dependence | 84 |

Methods for Constructing Invariants | 87 |

22 Groups and Differential Equations | 90 |

23 Prolongation | 94 |

Systems of Differential Equations | 96 |

Prolongation of Group Actions | 98 |

Invariance of Differential Equations | 100 |

Prolongation of Vector Fields | 101 |

Infinitesimal Invariance | 103 |

The Prolongation Formula | 105 |

Total Derivatives | 108 |

The General Prolongation Formula Theorem | 110 |

Properties of Prolonged Vector Fields | 115 |

24 Calculation of Symmetry Groups | 116 |

25 Integration of Ordinary Differential Equations | 130 |

First Order Equations | 131 |

Higher Order Equations | 137 |

Differential Invariants | 139 |

Multiparameter Symmetry Groups | 145 |

Solvable Groups | 151 |

Systems of Ordinary Differential Equations | 154 |

26 Nondegeneracy Conditions for Differential Equations | 157 |

Invariance Criteria | 161 |

The CauchyKovalevskaya Theorem | 162 |

Characteristics | 163 |

Normal Systems | 166 |

Notes | 172 |

Exercises | 176 |

GroupInvariant Solutions | 183 |

31 Construction of GroupInvariant Solutions | 185 |

32 Examples of GroupInvariant Solutions | 190 |

33 Classification of GroupInvariant Solutions | 199 |

Classification of Subgroups and Subalgebras | 203 |

Classification of GroupInvariant Solutions | 207 |

34 Quotient Manifolds | 209 |

Dimensional Analysis | 214 |

35 GroupInvariant Prolongations and Reduction | 217 |

Extended Jet Bundles | 218 |

Differential Equations | 222 |

Group Actions | 223 |

The Invariant Jet Space | 224 |

Connection with the Quotient Manifold | 225 |

The Reduced Equation | 227 |

Local Coordinates | 228 |

Notes | 235 |

Exercise | 238 |

Symmetry Groups and Conservation Laws | 242 |

41 The Calculus of Variations | 243 |

Trivial Conservation Laws | 264 |

Characteristics of Conservation Laws | 266 |

44 Noethers Theorem | 272 |

Divergence Symmetries | 278 |

Notes | 281 |

Exercise | 283 |

Generalized Symmetries | 286 |

51 Generalized Symmetries of Differential Equations | 288 |

Generalized Vector Fields | 289 |

Evolutionary Vector Fields | 291 |

Equivalence and Trivial Symmetries | 292 |

Computation of Generalized Symmetries | 293 |

Group Transformations | 297 |

Symmetries and Prolongations | 300 |

The Lie Bracket | 301 |

Evolution Equations | 303 |

52 Recursion Operators Master Symmetries and Formal Symmetries | 304 |

Frechet Derivatives | 307 |

Lie Derivatives of Differential Operators | 308 |

Criteria for Recursion Operators | 310 |

The Kortewegde Vries Equation | 312 |

Master Symmetries | 315 |

Pseudodifferential Operators | 318 |

Formal Symmetries | 322 |

53 Generalized Symmetries and Conservation Laws | 328 |

Characteristics of Conservation Laws | 330 |

Variational Symmetries | 331 |

Group Transformations | 333 |

Noethers Theorem | 334 |

Selfadjoint Linear Systems | 336 |

Action of Symmetries on Conservation Laws | 341 |

Abnormal Systems and Noethers Second Theorem | 342 |

Formal Symmetries and Conservation Laws | 346 |

54 The Variational Complex | 350 |

The DComplex | 351 |

Vertical Forms | 353 |

Total Derivatives of Vertical Forms | 355 |

Functionals and Functional Forms | 356 |

The Variational Differential | 361 |

Higher Euler Operators | 365 |

The Total Homotopy Operator | 368 |

Notes | 374 |

Exercise | 379 |

FiniteDimensional Hamiltonian Systems | 389 |

61 Poisson Brackets | 390 |

Hamiltonian Vector Fields | 392 |

The Structure Functions | 393 |

The LiePoisson Structure | 396 |

62 Symplectic Structures and Foliations | 398 |

Rank of a Poisson Structure | 399 |

Symplectic Manifolds | 400 |

Maps Between Poisson Manifolds | 401 |

Poisson Submanifolds | 402 |

Darboux Theorem | 404 |

The Coadjoint Representation | 406 |

63 Symmetries First Integrals and Reduction of Order | 408 |

Hamiltonian Symmetry Groups | 409 |

Reduction of Order in Hamiltonian Systems | 412 |

Reduction Using Multiparameter Groups | 416 |

Hamiltonian Transformation Groups | 418 |

The Momentum Map | 420 |

Notes | 427 |

EXercise | 428 |

Hamiltonian Methods for Evolution Equations | 433 |

71 Poisson Brackets | 434 |

The Jacobi Identity | 436 |

Functional Multi vectors | 439 |

72 Symmetries and Conservation Laws | 446 |

Conservation Laws | 447 |

73 BiHamiltonian Systems | 452 |

Recursion Operators | 458 |

Notes | 461 |

Exercise | 463 |

467 | |

489 | |

497 | |

501 | |

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### Common terms and phrases

adjoint characteristic coefficient computation conservation law constant coordinate chart corresponding defined Definition dependent variables determined differential functions differential operator equivalent Euler-Lagrange equations evolution equations evolutionary vector field Example formal symmetry formula geometric given group action group G group of transformations group-invariant solutions Hamiltonian system Hamiltonian vector field heat equation hence independent infinitesimal criterion integral invariant Jacobi identity jet space Korteweg-de Vries equation Lemma Let G Lie algebra Lie bracket Lie derivative Lie group linear matrix maximal rank n-th order Noether's theorem one-parameter group open subset orbits order derivatives order equation ordinary differential equations partial differential equations Poisson bracket pr(n prolongation proof Proposition Prove pseudo-differential operator quadrature quotient manifold recursion operator reduced system rotations satisfy second order smooth function solvable subalgebras subgroup submanifold subvariety symmetry group symplectic system of differential theory tion total derivative vanishes variational problem