  Time dependent quadratic Hamiltonians are well known as well in classical
mechanics and in quantum mechanics. In particular for them the correspondance
between classical and quantum mechanics is exact. But explicit formulas are non
trivial (like the Mehler formula). Moreover, a good knowlege of quadratic
Hamiltonians is very useful in the study of more general quantum Hamiltonians
and associated Schr\"{o}dinger equations in the semiclassical regime. Our goal
here is to give our own presentation of this important subject. We put emphasis
on computations with Gaussian coherent states. Our main motivation to do that
is application concerning revivals and Loschmidt echo.
