On parametric representations of orthogonal and symplectic matrices

Symplectic matrices are subject to certain conditions that are inherent to the Jacobian matrices of transformations preserving the Hamiltonian form of differential equations. A formula is derived that parameterizes symplectic matrices with symmetric matrices. An analogy is drawn between the obtained formula and the Cayley formula that connects orthogonal and antisymmetric matrices. It is shown that orthogonal and antisymmetric matrices are transformed by the covariant law when replacing the Cartesian coordinate system. Analogously, the covariance of transformations of symplectic and symmetric matriсes is proved.