Symplectic matrix

We shall see now that the eigenvalues of a symplectic matrix have some special properties. We express property (39) as... 

The implicit-midpoint (IM) scheme differs from IE above in that it is symmetric and symplectic. It is also special in the sense that the transformation matrix for the model linear problem is unitary, partitioning kinetic and potential-energy components identically. Like IE, IM is also A-stable. IM is (herefore a more reasonable candidate for integration of conservative systems, and several researchers have explored such applications [58, 59, 60, 61]. 

These various techniques were recently applied to molecular simulations [11, 20]. Both of these articles used the rotation matrix formulation, together with either the explicit reduction-based integrator or the SHAKE method to preserve orthogonality directly. In numerical experiments with realistic model problems, both of these symplectic schemes were shown to exhibit vastly superior long term stability and accuracy (measured in terms of energy error) compared to quaternionic schemes. 

Newton, the limit h —> 0 is singular. The symmetries underlying quantum and classical dynamics - unitarity and symplecticity, respectively - are fundamentally incompatible with the opposing theory s notion of a physical state quantum-mechanically, a positive semi-definite density matrix classically, a positive phase-space distribution function. 

Furthermore, under symplectic transformations, it is relatively easy to show, using the Hessian formula for calculating the Fisher information matrix, that the measurement covariance matrix transforms as... 

To examine this exchange behavior for an arbitrary one-electron operator we start by rewriting the general expression of Eq. 6 in a sum over symplectic pairs. In the following and t] are assumed to be compound indices for resp. (mS, mi ) and Also — refers to (—m.v — m,-). The matrix element... 

A discrete scheme (3) is a symplectic scheme if the transformation matrix S is symplectic. 

The product of symplectic matrices is also symplectic. Hence, if each matrix M is symplectic the transformation matrix S is symplectic. Consequently, the discrete scheme (2) is symplectic if each matrix M is symplectic. 

An important property of the monodromy matrix of a Hamiltonian system is the symplectic property that we shall prove now. Let and be two solutions of the variational equations. The following property holds ... 

In a Hamiltonian system the monodromy matrix is a 2n x 2n sym-plectic matrix, and the eigenvalues are in reciprocal pairs (because of the symplectic property), and in complex conjugate pairs (because the elements of the matrix are real). 

Here, M is a symplectic Id x Id matrix which is chosen in such a way that the second-order term of fhe fransformed Hamilton function... 

Thus A is an eigenvalue of which, in turn, implies that 1 /A is an eigenvalue of T. The matrix being real implies that the conjugates of A and 1 /A are also eigenvalues, thus we have the same eigenvalue quadruplets as for a linear symplectic map. 

A class of methods that do provide the necessary features can be found in the work of Feng Kang [133], referred to as J-splitting by McLachlan and Quispel [261]. Let J be the skew-symmetric canonical symplectic structure matrix. The idea is to consider a splitting of J into a finite number K of skew-symmetric matrices 7 , i=, K. This induces a splitting of the Hamiltonian vector field into K vector... 

---