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By Maciej Blaszak

This is a contemporary method of Hamiltonian structures the place multi-Hamiltonian platforms are provided in e-book shape for the 1st time. those structures let a unified remedy of finite, lattice and box structures. Having multiple Hamiltonian formula in one coordinate method for a nonlinear process is a estate heavily on the topic of integrability. therefore, the ebook provides an algebraic conception of integrable structures. it's written for scientists and graduate students.

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9) are equivalent. 3 For a symplectic operator J the following identity holds v E ,(1,0). 12) Proof. 14) Proof. (()) = O. Then we have Now we can pass to covector fields. Let () be an implectic operator. 15) by Co. 5 () is a homomorphism from C e into the vector field Lie algebra C. 46 3. The Theory of Hamiltonian and Bi-Hamiltonian Systems Proof. 6 £:;; is the covector field Lie algebra. Proof. 15) and 0- 1 lifts the Jacobi identity from £: to £: ;;. e. 5 we only know that the Jacobi identity in £:;; is satisfied up to an element of the ker O.

124) where the summation convention over repeating indices is used. The general rule can be guessed now very easily but it is not difficult to derive it by applying the chain rule 38 2. 124) we find .. (LvT(r,S)(X)f 1"' Jr = r 0 j OX~l ... ls Vk - 'L,Tjl ... jr OVX: oTj1 ... jr h ... ls 0<=1 s . + 'L, TJl ... Jr 11 ... IIl-lklll+1 ... l s ,6=1 OV k __ OX IIl . 126) We finish this section with various properties of the Lv-derivative which are useful in the further considerations. 129) Proof. (i) We proceed by induction On k.

54) where the covector field 'V f means the gradient of f(x). More interesting for our considerations is the infinite dimensional case. Let u be an m-component function u = (u l , ... , Um)T. Then an arbitrary scalar field on M is represented by the formula 24 2. 55) f(u(x))dx, where the density f(u) is a differential function of the ui components. In this case the gradient operator V' turns into a variational operator b (b/bul, ... , b/bum) T . :O kx where bF =(bF/bul, ... 18 n=+oo F(u, w) = 2:: [u(n)w(n + 2) + u 2 (n + 1)], n=-oo bF = ( w(n + 2) + E- 1 2u(n + 1) ) = ( w(n + 2) + 2u(n) ) .