Chapter 7 Applications of Lie Groups to Differential Equations Peter J. Olver Email :: olver@umn.edu Home Page :: http://www.math.umn.edu/~olver Vladimir Igorevich Arnold (1237–2010) was a Soviet and Russian mathematician. Presented by Mehdi Nadjafikhah Email :: m_nadjafikhah@iust.ac.ir Home Page :: webpages.iust.ac.ir/m_nadjafikhah

Hamiltonian Methods for Evolution Equations Introduction to Lie Groups 1.1. Manifolds Change of Coordinates, Maps Between , Manifolds, The Maximal Rank Condition, Submanifolds, Regular Submanifolds, Implicit Submanifolds, Curves and Connectedness 1.2. Lie Groups Lie Subgroups, Local Lie Groups, Local Transformation Groups, Orbits 1.3. Vector Fields Flows, Action on Functions, Differentials, Lie Brackets, Tangent Spaces and Vectors Fields on Submanifolds, Frobenius' Theorem 1.4. Lie Algebras One-Parameter Subgroups, Subalgebras, The Exponential Map, Lie Algebras of Local Lie Groups, Structure Constants, Commutator Tables, Infinitesimal Group Actions 1.5. Differential Forms Pull-Back and Change of Coordinates, Interior Products, The Differential, The de Rham Complex, Lie Derivatives Homotopy Operators, Integration and Stokes' Theorem CHAPTER 7 Hamiltonian Methods for Evolution Equations 433 7.1. Poisson Brackets :: Poisson Bracket, The Jacobi Identity, Functional Multi-vectors 7.2. Symmetries and Conservation Laws :: Distinguished Functionals, Lie Brackets 7.3. Bi-Hamiltonian Systems :: Recursion Operators

435 7.1. Poisson Brackets

436 The Jacobi Identity

Functional Multi-vectors In finite dimensions, multivectors are the dual objects to differential forms.

Except for this distinction these objects are the same. Here we introduce the analogous objects for infinite-dimensional Hamiltonian systems of evolution equations. Since we are working with open subsets of Euclidean space M, the theory of functional multi-vectors is identical with that of functional forms developed in Section 5.4. The only reason that we employ different terminology and notation is that, from a more global standpoint, the transformation rules for these objects under changes of variables are not the same; functional forms transform like Euler-Lagrange expressions whereas functional multi-vectors are more like evolutionary vector fields. Except for this distinction these objects are the same.

7.2. Symmetries and Conservation Laws We need to investigate the "distinguished functionals" arising from degeneracies of the Poisson bracket itself; these will provide conservation laws for any system having the given Hamiltonian structure. Further conservation laws, particular to the symmetry properties of the individual Hamiltonian functionals, can then be deduced from generalized symmetries which are themselves Hamiltonian.

Distinguished Functionals 446 Distinguished Functionals

446 Lie Brackets

447 Conservation Laws

7.3. Bi-Hamiltonian Systems 492 7.3. Bi-Hamiltonian Systems

Thus our results on bi-Hamiltonian systems will provide ready-made proofs of the existence of infinitely many conservation laws and symmetries for the Korteweg-de Vries equation.

458 Recursion Operators

There is thus a whole hierarchy of conservation laws and commuting flows for the Boussinesq equation.

May 24, 2014