This paper addresses the dynamics and quasi-statics of floating flexible structures as well as extensions to unconstrained substructures and partitions of coupled mechanical systems. The principal solution is defined as the state of self-equilibrated forces obtained as the particular solution of the rigid motion and interface equilibrium equations. This solution is independent of the stress-strain constitutive properties as well as of the compatibility equations. For statically determinate systems, the principal solution is the final force solution. For statically indeterminate systems, the correction due to flexibility and compatibility is orthogonal to the principal solution. The formulation is done in the context of d' Alembert' s principle, which supplies the d' Alembert-Lagrange principal equations for floating bodies. These are obtained by summation of virtually working forces and moments acting on the floating systems. Applications of this approach are demonstrated on a set of dynamic and quasi-static example problems of increasing generality. Linkage to variational principles with an interface potential is eventually discussed as providing the theoretical foundation for handling interacting semi-discrete subsystems linked by node-collocated Lagrange multipliers.