1. Chapter 7 Applications of Lie Groups to Differential Equations
Peter J. Olver
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Vladimir Igorevich Arnold
(1237–2010) was a Soviet and Russian mathematician.
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2. 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

5. 435
7.1. Poisson Brackets

12. 436
The Jacobi Identity

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

21. 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.

38. 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.

39. Distinguished Functionals
446
Distinguished Functionals

41. 446
Lie Brackets

43. 447
Conservation Laws

56. 7.3. Bi-Hamiltonian Systems
492
7.3. Bi-Hamiltonian Systems

59. 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.

67. 458
Recursion Operators

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

74. May 24, 2014