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Hamilton–Jacobi equation and action-angle variables
Published in Bijan Kumar Bagchi, Advanced Classical Mechanics, 2017
where the entries stand for the Poisson brackets. Using their properties as furnished in (9.8) our assertion is seen to hold. We refer to J as a symplectic matrix and the canonical transformations induced by (Q, P) are said to be symplectic.
Decision-Oriented Two-Parameter Fisher Information Sensitivity Using Symplectic Decomposition
Published in Technometrics, 2023
The Williamson’s theorem provides us with a symplectic variant of the results above. Let be a symmetric and positive definite matrix, the Williamson’s theorem says that can be diagonalized using symplectic matrices (Gosson 2006; Nicacio 2021): where is a diagonal matrix with positive entries ( maybe zero if is semidefinite). The are said to be the symplectic eigenvalues of matrix (Bhatia and Jain 2015) and are in general not equal to the eigenvalues given in (5). The matrix is a real symplectic matrix, and it is called the symplectic eigenvector matrix of . Each symplectic eigenvalue corresponds to a pair of eigenvectors :
Geometry of symplectic partially hyperbolic automorphisms on 4-torus
Published in Dynamical Systems, 2020
If the dimension of the torus is even, n = 2m, then one can introduce the standard symplectic structure on using coordinates in : and consider symplectic automorphisms of the torus which preserve this symplectic structure. A symplectic automorphism is then defined by a symplectic matrix A with integer entries. Such matrices satisfy the identity , where a skew-symmetric matrix I has the form The identity above implies the product of two symplectic matrices and the inverse matrix of a symplectic matrix be symplectic, i.e. symplectic matrices form a group denoted by w.r.t. the operation of matrix multiplication. This group is one of the standard matrix Lie groups [10].
Reducibility of a class of 2k-dimensional Hamiltonian systems with quasi-periodic coefficients
Published in Dynamical Systems, 2019
Jia Li, Youhui Su, Yanling Shi
Below we will define (39) and prove it is symplectic. In fact, by (37), define by where Let We define the mapping (39) by φ. Now we first prove are all symplectic. By (41), we have where and . It follows that for and or else, . Then by is a symplectic matrix and (41), we get is a symplectic matrix. So are all symplectic mappings. Then by (42), we get that (39) is a symplectic mapping.