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[QUE/CM-10009]

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  1. Let \(x\)denote a column matrix of the canonical variables \(q_1,q_2,p_1,p_2\) for a system with two degrees of freedom, and consider a linear transformation \[ x \longrightarrow x^\prime= M x\] where \(M\) is \(4\times 4\) matrix with constant elements.Use the Poisson bracket conditions to find the conditions on the elements of \(M\) in order that this be a canonical transformation.
  2. Show that these are equivalent to requiring that\(M\) satisfy the conditions \[ M J \widetilde{M} = J\] Here \(\widetilde{M}\) is the transpose of \(M\), and \(J\) is the matrix \[\begin{vmatrix} 0 & 1\\-1& 0\end{vmatrix}\] \end{AlphaList1} where ``0''stands for \(2\times2\) zero matrix aand ``1'' stands for \(2\times2\) unit matrix. Matrices which satisfy the above condition are called symplectic matrices.

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