By Derek F. Lawden

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**Extra resources for A Course in Applied Mathematics, Vols 1 & 2**

**Example text**

Let 80 be a fixed eigenvalue with respect to Zo such that -rmax S; 80 S; rmax. From each chain ero ... 5) where the a's are constants and sum runs over all picked eso's. 4)]. 5) a linear relation between elements of the form ermax ' Since the latter ones are linearly independent, the corresponding values of a's should be zero. , we get that all a's are zero. Now suppose that elements of (e) are linearly dependent. Then so must be elements of the form eso for some So since polynomials corresponding to different eigenvalues are linearly independent; but independence of eso 's was just established.

20). In the sequel several problems of the kind will be discussed. 8. Hamiltonian Systems 43 where 1j; and 7r are canonically conjugate and the Hamiltonian is H = ho + ch} + ... , then the corresponding operator is X = Xo + cX} + ... , where L Xd = k (8h i ~ _ 8hi ~) 87r k 81j; k 81j; k 87rk , or, in the notations used in mechanics, Xd = {hi, J}, where {f,g} = (~ 8g _ 8f 87rk 81j; k ~) 81j; k 87r k is the Poisson bracket of f and g. , apply only canonical transformations. This technique is well known.

Then Z'h 2 = (Xo - A)Zh2 - Zh 1 - (ZA)h2 and we find similarly that Z h2 corresponds to A. Continuing this process we see that Zhp = Zc,o corresponds to A. This completes the proof. 1') is true only for basic functions and corresponding eigenvalues, provided that for any basic c,o the function (XoA)8c,o for s ~ 1 is also basic or zero. Since this condition is verified by an extended Jordan basis by its construction, the following theorem holds. 4. 1') holds only for functions from the extented Jordan basis of Xo· The following theorem gives one of the practical methods for constructing S (see the proof).