## Symmetries and Conservation Laws for Differential Equations of Mathematical Physics |

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### Contents

Ordinary Differential Equations | 1 |

2 Ordinary differential equations of arbitrary order | 6 |

3 Symmetries of distributions | 10 |

4 Some applications of symmetry theory to integration of distributions | 17 |

5 Generating functions | 29 |

an example of using symmetries | 33 |

FirstOrder Equations | 37 |

2 Infinitesimal contact transformations and characteristic fields | 50 |

2 The Cartan distribution on J𝛑 and its infinitesimal automorphisms | 138 |

3 infinitelyprolonged equations and the theory of higher symmetries | 154 |

4 Examples of computation | 164 |

Conservation Laws | 185 |

2 The Cspectral sequence | 187 |

3 Computation of conservation laws | 206 |

4 Symmetries and conservation laws | 214 |

Nonlocal Symmetries | 221 |

3 Complete integrals of firstorder differential equations | 60 |

The Theory of Classical Symmetries | 69 |

2 Jet manifolds and the Cartan distribution | 72 |

3 Lie transformations | 83 |

4 Classical symmetries of equations | 92 |

5 Examples of computations | 96 |

6 Factorization of equations by symmetries | 108 |

7 Extrinsic and intrinsic symmetries | 116 |

Higher Symmetries | 123 |

coverings | 238 |

3 Nonlocal symmetries | 249 |

nonlocal symmetries of the Burgers equation | 251 |

5 The problem of symmetry reconstruction | 257 |

6 Symmetries of integrodifferential equations | 266 |

From Symmetries of Partial Differential Equations Towards Secondary Quantized Calculus | 301 |

323 | |

329 | |

### Common terms and phrases

arbitrary automorphism bundle Burgers equation C-spectral sequence calculus called Cartan distribution Cartan plane Chapter classical symmetries coefficients coincides completely integrable computations condition conservation laws Consider construction contact transformations coordinates corresponding covering defined definition denoted depend describe diffeomorphism differential operators diffiety elements equality equivalent evolutionary derivations Example Exercise fiber finite first-order formula geometric given graph Hamiltonian heat equation Hence higher symmetries homomorphism independent variables infinite jets infinitely prolonged integral manifolds invariant isomorphism Jacobi bracket Jk(n Jk{n Jx(n Korteweg-de Vries equation Lie algebra Lie field Lie transformation lifting linear maximal integral manifolds neighborhood Noether nonlinear nonlocal symmetries nontrivial obtain one-dimensional one-parameter group ordinary differential equations partial differential equations projection Proof Proposition Prove recursion operator respect restriction satisfying singular smooth functions solution space subalgebra submanifold tangent vector Theorem theory tion total derivatives trivial vector field vertical