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Unformatted text preview: Normal form for a 2D diagonalizable system Sujit Nair Consider the following example. bracketleftbigg x y bracketrightbigg = bracketleftbigg 1 2 bracketrightbigg bracketleftbigg x y bracketrightbigg + bracketleftbigg f ( x, y ) g ( x, y ) bracketrightbigg (1) Consider the following change of coordinates. bracketleftbigg x y bracketrightbigg = bracketleftbigg u v bracketrightbigg + h ( u, v ) (2) The operator L J ( h ( u, v )) for this example is L J ( h ( u, v )) = J ( h ( u, v ))- Dh ( u, v ) J bracketleftbigg u v bracketrightbigg (3) Lets consider the homogeneous polynomial basis vector of degree k given by h ku ( u, v ) = bracketleftbigg u i v j bracketrightbigg (4) where i, j are integers such that i + j = k and i, j 0. For this case, we have L J ( h ku ( u, v )) = bracketleftbigg 1 2 bracketrightbigg bracketleftbigg u i v j bracketrightbigg- bracketleftbigg iu i- 1 v j ju i v j- 1 bracketrightbigg bracketleftbigg 1 2 bracketrightbigg bracketleftbigg u v...
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This note was uploaded on 01/04/2012 for the course CDS 140A taught by Professor Marsden during the Fall '09 term at Caltech.
- Fall '09