Contact Geometry and Nonlinear Differential Equations (Encyclopedia of Mathematics and its Applications, Series Number 101)

The most relevant FirstSearch result that mentions Lie, Monge, and Ampère! [Kumei](https://isidore.co/calibre/#panel=book_list&search=authors:%22%3DKumei%2C%20Sukeyuki%22)'s [1981 dissertation](https://open.library.ubc.ca/cIRcle/collections/ubctheses/831/items/1.0080150) mentions Lie, Legendre, and Ampère. ([Monge-Ampère equations](https://www.encyclopediaofmath.org/index.php/Monge-Amp%C3%A8re_equation) are invariant under [Legendre transform](https://www.encyclopediaofmath.org/index.php/Legendre_transform)s.) mentions symplectic geometry: * Frauenfelder, Urs, and Joa Weber. “[The Fine Structure of Weber’s Hydrogen Atom: Bohr–Sommerfeld Approach](https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/LVWRAVB4/Frauenfelder%20and%20Weber%20-%202019%20-%20The%20fine%20structure%20of%20Weber%E2%80%99s%20hydrogen%20atom%20Bohr%E2%80%93.pdf).” *Zeitschrift Für Angewandte Mathematik Und Physik* 70, no. 4 (June 21, 2019): 105. [Cantwell](https://isidore.co/calibre#panel=book_details&book_id=6578) PDF p. 148 gives a good definition of **symplectic manifolds** , in the context of classical mechanics, Hamilton's equations, and Poisson brackets. ([ *OED*](https://oed.com/view/Entry/196304#eid19343811): " *συμπλεκτικός* twining or plaiting together, copulative") > The conserved elements of a Hamiltonian system ... define a vector space. The rules of algebra in this space are given by the skew-symmetry of the composition operator (Poisson bracket) (4.43), the additive properties in (4.44), and the Jacobi identity (4.47). A vector space with these special properties is called a *symplectic* space, and the solution of the Hamiltonian system is said to lie on a symplectic manifold. This odd word comes from the greek *symplektikos* meaning “twining together,” from *syn* (together) and *plekein* (to twine). It is an apt description of the solution trajectories of a periodically forced Hamiltonian system, which can be visualized as a family of spiraling curves on a torus in a three-dimensional phase space where the third dimension is the phase angle of the forcing function. * * * Methods from contact and symplectic geometry can be used to solve highly non-trivial nonlinear partial and ordinary differential equations without resorting to approximate numerical methods or algebraic computing software. This book explains how it's done. It combines the clarity and accessibility of an advanced textbook with the completeness of an encyclopedia. The basic ideas that Lie and Cartan developed at the end of the nineteenth century to transform solving a differential equation into a problem in geometry or algebra are here reworked in a novel and modern way. Differential equations are considered as a part of contact and symplectic geometry, so that all the machinery of Hodge-deRham calculus can be applied. In this way a wide class of equations can be tackled, including quasi-linear equations and Monge-Ampere equations (which play an important role in modern theoretical physics and meteorology). * * * [MR2352610](https://mathscinet.ams.org/mathscinet-getitem?mr=2352610) The aim of this book is to explain how methods from symplectic and contact geometry, and the use of the theory of symmetries, can be helpful in solving even highly nontrivial ordinary and partial differential equations without the need to have recourse to approximate numerical techniques or computer algebra. In doing so, the authors have accomplished a great job: the book excels in clarity and accessibility on the one hand, and in completeness on the other hand. The basic ideas and pioneering work of Lie and Cartan concerning the transformation of solving differential equations into a geometric or algebraic problem are treated and interpreted in a novel and modern way. In particular, it is shown that a large class of differential equations can be naturally embedded into the framework of symplectic or contact geometry so that one can apply, among other things, the whole machinery of the Hodge-de Rham calculus to it. Throughout, the authors seek geometric transparency of the ideas they expound upon and combine mathematical rigor with the search for exact solutions to nontrivial problems. Many examples, simple as well as advanced, serve to illustrate the theoretical part, and from the physical applications that are treated we may mention, for instance, an analysis of laser beams and the dynamics of cyclones. The book is divided into five parts, each consisting of several chapters. Part I is devoted to the theory of symmetries and integrals of distributions and of ordinary differential equations on a manifold. First, a general discussion of distributions and their Lie algebra of symmetries is given. One of the highlights here is the Lie-Bianchi theorem, which gives a condition and a constructive algorithm for the integrability by quadratures of a distribution in terms of a solvable Lie algebra of symmetries. The treatment continues with, among others, the theories of symmetries of ordinary differential equations, linear symmetries of self-adjoint differential operators with application to the Schrödinger operator, reduction by symmetries and the Lie superposition principle. Part II deals with symplectic algebra. It starts with a review of the basic elements from the theory of symplectic vector spaces and exterior algebra on symplectic vector spaces. Next, the symplectic classification of exterior $2$-forms in four and in higher dimensions and the symplectic classification of $3$-forms in six dimensions are treated. Part III is then devoted to the study of the Monge-Ampère equations. After a general discussion of symplectic and contact manifolds, Monge-Ampère differential operators are introduced. Symmetries and contact transformations, as well as conservation laws of the Monge-Ampère equations, are treated. Some geometric structures related to Monge-Ampère equations on a two-dimensional manifold are described. This part ends with a treatment of first-order partial differential equations in two dimensions, namely the class of Jacobi equations. Part IV is completely devoted to physical applications of the theory, which are taken from nonlinear acoustics (symmetries, conservation laws and exact solutions of the Khokhlov-Zabolotskaya equation, which describes the propagation of a sound beam in a nonlinear medium), nonlinear thermal conductivity (symmetries and invariant solutions of a version of the Kolmogorov-Petrovskiĭ-Piskunov equation with nonlinear diffusion) and also from meteorology (Monge-Ampère equations in semi-geostrophic models), which is based upon some ongoing research by one of the authors with I. Roulstone from the University of Surrey (UK). Finally, Part V is devoted to the classification problem of Monge-Ampère equations in the framework of symplectic and contact geometry for which the most complete solution exists in two and three dimensions. The two-dimensional case is related to the classical equivalence problem of Sophus Lie, and a modern version of Lie's results are presented. A complete proof of these results was in fact first obtained by the authors of this book in some papers from the 1980's. Reviewed by [Frans Cantrijn](https://mathscinet.ams.org/mathscinet/search/author.html?mrauthid=44940) Cover 1 Title 4 Copyright 5 Contents 6 Preface 14 Part I Symmetries and Integrals 23 1 Distributions 25 1.1 Distributions and integral manifolds 25 1.1.1 Distributions 25 1.1.2 Morphisms of distributions 26 1.1.3 Integral manifolds 27 1.2 Symmetries of distributions 33 1.3 Characteristic and shuffling symmetries 37 1.4 Curvature of a distribution 40 1.5 Flat distributions and the Frobenius theorem 42 1.6 Complex distributions on real manifolds 45 1.7 The Lie--Bianchi theorem 46 1.7.1 The Maurer--Cartan equations 46 1.7.2 Distributions with a commutative symmetry algebra 49 1.7.3 Lie--Bianchi theorem 52 2 Ordinary differential equations 54 2.1 Symmetries of ODEs 54 2.1.1 Generating functions 54 2.1.2 Lie algebra structure on generating functions 59 2.1.3 Commutative symmetry algebra 60 2.2 Non-linear second-order ODEs 62 2.2.1 Equation y=y+F(y) 65 2.2.2 Integration 68 2.2.3 Non-linear third-order equations 70 2.3 Linear differential equations and linear symmetries 72 2.3.1 The variation of constants method 72 2.3.2 Linear symmetries 73 2.4 Linear symmetries of self-adjoint operators 76 2.5 Schrödinger operators 78 2.5.1 Integrable potentials 80 2.5.2 Spectral problems for KdV potentials 87 2.5.3 Lagrange integrals 95 3 Model differential equations and the Lie superposition principle 98 3.1 Symmetry reduction 98 3.1.1 Reductions by symmetry ideals 98 3.1.2 Reductions by symmetry subalgebras 99 3.2 Model differential equations 100 3.2.1 One-dimensional model equations 102 3.2.2 Riccati equations 104 3.3 Model equations: the series Ak,Dk,Ck 105 3.3.1 Series Ak 105 3.3.2 Series Dk 108 3.3.3 Series Ck 109 3.4 The Lie superposition principle 111 3.4.1 Bianchi equations 114 3.5 AP-structures and their invariants 116 3.5.1 Decomposition of the de Rham complex 116 3.5.2 Classical almost product structures 118 3.5.3 Almost complex structures 120 3.5.4 AP-structures on five-dimensional manifolds 120 Part II Symplectic Algebra 123 4 Linear algebra of symplectic vector spaces 125 4.1 Symplectic vector spaces 125 4.1.1 Bilinear skew-symmetric forms on vector spaces 125 4.1.2 Symplectic structures on vector spaces 126 4.1.3 Canonical bases and coordinates 129 4.2 Symplectic transformations 130 4.2.1 Matrix representation of symplectic transformations 132 4.3 Lagrangian subspaces 135 4.3.1 Symplectic and Kähler spaces 139 5 Exterior algebra on symplectic vector spaces 141 5.1 Operators and 141 5.2 Effective forms and the Hodge--Lepage theorem 147 5.2.1 sl2-method 154 6 A symplectic classification of exterior 2-forms in dimension 4 157 6.1 Pfaffian 157 6.2 Normal forms 159 6.3 Jacobi planes 164 6.3.1 Classification of Jacobi planes 165 6.3.2 Operators associated with Jacobi planes 167 7 Symplectic classification of exterior 2-forms 169 7.1 Pfaffians and linear operators associated with 2-forms 169 7.2 Symplectic classification of 2-forms with distinct real characteristic numbers 171 7.3 Symplectic classification of 2-forms with distinct complex characteristic numbers 174 7.4 Symplectic classification of 2-forms with multiple characteristic numbers 176 7.5 Symplectic classification of effective 2-forms in dimension 6 182 8 Classification of exterior 3-forms on a 6D 184 8.1 A symplectic invariant of effective 3-forms 184 8.1.1 The case of trivial invariants 187 8.1.2 The case of non-trivial invariants 189 8.1.3 Hitchin's results on the geometry of 3-forms 195 8.2 The stabilizers of orbits and their prolongations 197 8.2.1 Stabilizers 197 8.2.2 Prolongations 200 Part III Monge--Ampère Equations 203 9 Symplectic manifolds 205 9.1 Symplectic structures 205 9.1.1 The cotangent bundle and the standard symplectic structure 206 9.1.2 Kähler manifolds 208 9.1.3 Orbits and homogeneous symplectic spaces 209 9.2 Vector fields on symplectic manifolds 211 9.2.1 Poisson bracket and Hamiltonian vector fields 211 9.2.2 Canonical coordinates 213 9.3 Submanifolds of symplectic manifolds 214 9.3.1 Presymplectic manifolds 214 9.3.2 Lagrangian submanifolds 216 9.3.3 Involutive submanifolds 219 9.3.4 Lagrangian polarizations 220 10 Contact manifolds 223 10.1 Contact structures 223 10.1.1 Examples 224 10.2 Contact transformations and contact vector fields 230 10.2.1 Examples 231 10.2.2 Contact vector fields 237 10.3 Darboux theorem 241 10.4 A local description of contact transformations 243 10.4.1 Generating functions of Lagrangian submanifolds 243 10.4.2 A description of contact transformations in R3 244 11 Monge--Ampère equations 246 11.1 Monge--Ampère operators 246 11.2 Effective differential forms 248 11.3 Calculus on (C) 252 11.4 The Euler operator 255 11.5 Solutions 258 11.6 Monge--Ampère equations of divergent type 263 12 Symmetries and transformations of Monge--Ampère equations 265 12.1 Contact transformations 265 12.2 Lie equations for contact symmetries 273 12.3 Reduction 278 12.4 Examples 281 12.4.1 The boundary layer equation 281 12.4.2 The thermal conductivity equation 283 12.4.3 The Petrovsky--Kolmogorov--Piskunov equation 284 12.4.4 The Von Karman equation 286 12.5 Symmetries of the reduction 289 12.6 Monge--Ampère equations in symplectic geometry 292 13 Conservation laws 295 13.1 Definition and examples 295 13.2 Calculus for conservation laws 296 13.3 Symmetries and conservations laws 301 13.4 Shock waves and the Hugoniot--Rankine condition 302 13.4.1 Shock Waves for ODEs 302 13.4.2 Discontinuous solutions 303 13.4.3 Shock waves 305 13.5 Calculus of variations and the Monge--Ampère equation 307 13.5.1 The Euler operator 307 13.5.2 Symmetries and conservation laws in variational problems 308 13.5.3 Classical variational problems 309 13.6 Effective cohomology and the Euler operator 310 14 Monge--Ampère equations on 2D manifolds and geometric structures 316 14.1 Non-holonomic geometric structures associated with Monge--Ampère equations 317 14.1.1 Non-holonomic structures on contact manifolds 317 14.1.2 Non-holonomic fields of endomorphisms generated by Monge--Ampère equations 317 14.1.3 Non-degenerate equations 320 14.1.4 Parabolic equations 324 14.2 Intermediate integrals 326 14.2.1 Classical and non-holonomic intermediate integrals 326 14.2.2 Cauchy problem and non-holonomic intermediate integrals 329 14.3 Symplectic Monge--Ampère equations 330 14.3.1 A field of endomorphisms A on TM 330 14.3.2 Non-degenerate symplectic equations 332 14.3.3 Symplectic parabolic equations 334 14.3.4 Intermediate integrals 335 14.4 Cauchy problem for hyperbolic Monge--Ampère equations 335 14.4.1 Constructive methods for integration ofCauchy problem 336 15 Systems of first-order partial differential equations 340 15.1 Non-linear differential operators of first order on two-dimensional manifolds 341 15.2 Jacobi equations 343 15.3 Symmetries of Jacobi equations 350 15.4 Geometric structures associated withJacobi's equations 352 15.5 Conservation laws of Jacobi equations 354 15.6 Cauchy problem for hyperbolic Jacobi equations 356 Part IV Applications 359 16 Non-linear acoustics 361 16.1 Symmetries and conservation laws of the KZ equation 362 16.1.1 KZ equation and its contact symmetries 362 16.1.2 The structure of the symmetry algebra 364 16.1.3 Classification of one-dimensional subalgebras of sl(2,R) 367 16.1.4 Classification of symmetries 369 16.1.5 Conservation laws 370 16.2 Singularities of solutions of the KZ equation 371 16.2.1 Caustics 371 16.2.2 Contact shock waves 373 17 Non-linear thermal conductivity 378 17.1 Symmetries of the TC equation 378 17.1.1 TC equation 378 17.1.2 Group classification of TC equation 379 17.2 Invariant solutions 385 18 Meteorology applications 393 18.1 Shallow water theory and balanced dynamics 394 18.2 A geometric approach to semi-geostrophic theory 396 18.3 Hyper-Kähler structure and Monge--Ampère operators 398 18.4 Monge--Ampère operators with constant coefficients and plane balanced models 402 Part V Classification of Monge--Ampère equations 405 19 Classification of symplectic MAOs on two-dimensional manifolds 407 19.1 e-Structures 408 19.2 Classification of non-degenerate Monge--Ampère operators 410 19.2.1 Differential invariants of non-degenerate operators 410 19.2.2 Hyperbolic operators 414 19.2.3 Elliptic operators 423 19.3 Classification of degenerate Monge--Ampère operators 428 19.3.1 Non-linear mixed-type operators 428 19.3.2 Linear mixed-type operators 438 20 Classification of symplectic MAEs on 2D manifolds 444 20.1 Monge--Ampère equations with constant coefficients 444 20.1.1 Hyperbolic equations 445 20.1.2 Elliptic equations 447 20.1.3 Parabolic equations 448 20.2 Non-degenerate quasilinear equations 450 20.3 Intermediate integrals and classification 451 20.4 Classification of generic Monge--Ampère equations 452 20.4.1 Monge--Ampère equations and e-structures 452 20.4.2 Normal forms of mixed-type equations 458 20.5 Applications 462 20.5.1 The Born--Infeld equation 462 20.5.2 Gas-dynamic equations 464 20.5.3 Two-dimensional stationary irrotational isentropic flow of a gas 467 21 Contact classification of MAEs on two-dimensional manifolds 469 21.1 Classes Hk,l 469 21.2 Invariants of non-degenerate Monge--Ampère equations 476 21.2.1 Tensor invariants 476 21.2.2 Absolute and relative invariants 478 21.3 The problem of contact linearization 481 21.4 The problem of equivalence for non-degenerate equations 486 21.4.1 e-Structure for non-degenerate equations 486 21.4.2 Functional invariants 492 22 Symplectic classification of MAEs on 3D manifolds 494 22.1 Jets of submanifolds and differential equations on submanifolds 495 22.2 Prolongations of contact and symplectic manifolds and overdetermined Monge--Ampère equations 498 22.2.1 Prolongations of symplectic manifolds 498 22.2.2 Prolongations of contact manifolds 501 22.3 Differential equations for symplectic equivalence 504 References 509 Index 515