Summary: In this work the problem of the consistent treatment of multidimensional quadratic Hamiltonians is analyzed and developed from the geometric point of view. To this end, two approaches to the treatment for the problem are studied and developed: the pure matrix representation that involves Madelung type transforms (maps), and the evolution type based in a group manifold endowed with the symplectic groups and their coverings. Some of our goals is to introduce important geometric and group theoretical tools in the proposed approaches, such as Fermi normal coordinates in the first and a generalization of the method of nonlinear realizations in the second one. Several interesting results appear and some examples of application of these concepts in different physical scenarios are developed and presented such as the relationship with the Zeldovich approximation for the dynamic of large-scale cosmological structure, the classic case of Gertsner waves or the evolution problem with an inverted Hamiltonian of Caldirola-Kanai type.