In this paper, two subspaces of the state space of constrained equilibrium states for solids are proposed and addressed. One subspace, constrained affinity space, is conjugate-force space with fixed temperature and internal variable. It is revealed in this paper that the remarkable properties of the kinetic rate laws of scalar internal variables, established by Rice (1971, “Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19, pp. 433–455) and elaborated by Yang (2005, “Normality Structures With Homogeneous Kinetic Rate Laws,” ASME J. Appl. Mech., 72, pp. 322–329; 2007, “Normality Structures With Thermodynamic Equilibrium Points,” ASME J. Appl. Mech., 74, pp. 965–971), are all located in constrained affinity space. Furthermore, the flow potential function monotonically increases along any ray from the origin in constrained affinity space. Another subspace, constrained configuration space, is the state space with fixed external variables. It is shown that the specific free and complementary energies monotonically decrease and increase, respectively, along the path of motion of the thermodynamic system of the material sample in constrained configuration space. For conservative conjugate forces, Hamilton’s action principle is established in constrained configuration space, and the action is the entropy production of the thermodynamic system in a time interval. The thermodynamic processes in constrained configuration space are just creep or relaxation processes of materials. The Hamilton principle can be considered as a fundamental principle of rheology.