The Description of Integrable Quadratic Hamiltonians

Not to burden the presentation, we dwell only on one series of such examples associated with various equations of motion of a rigid body (in the multidimensional situation). To demonstrate our general method, we elucidate more or less comprehensively the procedure of studying the maximal linear commutative algebra of polynomials performed in Theorem 4.1.1 for a complex semisimple Lie algebra G. 

Let V be the maximal linear commutative algebra of polynomials, on orbits of a semisimple Lie algebra G, presented in Theorem 4.1.1 and in Proposition 4.1.1 (Sec. 1.2). We will now describe the subspace of quadratic Hamiltonians contained in 7. 

Comment Here dH () (that is, the differential of the function H) is given as the vector (element) of the Lie algebra G because H () is a function on the dual space G and G = G. 

Here grad H X) may be treated as the element of the Lie algebra G identified with G by means of the Killing form. 

Proof of Theorem 4.2.1 The Hamiltonian function H ) must depend functionally on functions of the form /( + Aa), where A are some numbers, a covector, a G, f functions constant on the orbits of the coadjoint representation. Thus, H( ) = + Aia)./j ( + Ajs a)). We suppose that all the func- 

---