Homogeneous cosmology dynamics revealed by Hamiltonian ADM formalism
  We study the homogeneous but anisotropic cosmological models of Bianchi in presence of a massive scalar field using the ADM Hamiltonian formalism. We begin to describe the main steps to find the ADM Hamiltonian of the General Relativity with a massive scalar field and then we study the dynamics of the flat Bianchi type $I$ anisotropic Universe according to initial and final values of this Hamiltonian and sign of the potential. After a brief recall of the conditions necessary to isotropise an anisotropic Bianchi class A model with such a field, we extend them to a non minimally coupled scalar field by using a conformal transformation of the metric which casts the General Relativity with a scalar field into a scalar-tensor theory. The new line element then corresponds to the so-called Brans-Dicke frame, the former one being the Einstein frame. Then, we study the isotropisation process of the Bianchi class A model when we consider the low energy form of the string theory without its antisymmetric tensor and the Brans-Dicke theory with some exponential or power laws of the scalar field for the potential. Finally, assuming an isotropic Universe such as all the metric functions behave as some power or exponential laws of the proper time, we find the conditions such that the gravitation function and the potential of the scalar field are bounded as it is observed today, and compare them with the necessary conditions for isotropy. 
Stephane Fay