The principle of minimum energy can be generalized to apply to constraints other than fixed entropy. For other constraints, other state functions with dimensions of energy will be minimized. These state functions are known as thermodynamic potentials. Thermodynamic potentials are at first glance just simple algebraic combinations of the energy terms in the expression for the internal energy. For a simple, multicomponent system, the internal energy may be written:
where the intensive parameters (T, P, &mu;) are functions of the internal energy's natural variables  via the equations of state. As an example of another thermodynamic potential, the Helmholtz free energy is written:
where temperature has replaced entropy as a natural variable. In order to understand the value of the thermodynamic potentials, it is necessary to view them in a different light. They may in fact be seen as (negative) Legendre transforms of the internal energy, in which certain of the extensive parameters are replaced by the derivative of internal energy with respect to that variable (i.e. the conjugate to that variable). For example, the Helmholtz free energy may be written:
and the minimum will occur when the variable T  becomes equal to the temperature since
The Helmholtz free energy is a useful quantity when studying thermodynamic transformations in which the temperature is held constant. Although the reduction in the number of variables is a useful simplification, the main advantage comes from the fact that the Helmholtz free energy is minimized at equilibrium  with respect to any unconstrained internal variables for a closed system at constant temperature and volume. This follows directly from the principle of minimum energy which states that at constant entropy, the internal energy is minimized. This can be stated as:
where  and  are the value of the internal energy and the (fixed) entropy at equilibrium. The volume and particle number variables have been replaced by x which stands for any internal unconstrained variables.
The minimization is with respect to the unconstrained variables. In the case of chemical reactions this is usually the number of particles or mole fractions, subject to the conservation of elements. At equilibrium, these will take on their equilibrium values, and the internal energy  will be a function only of the chosen value of entropy . By the definition of the Legendre transform, the Helmholtz free energy will be:
The Helmholtz free energy at equilibrium will be:
where  is the (unknown) temperature at equilibrium. Substituting the expression for :
By exchanging the order of the extrema:
showing that the Helmholtz free energy is minimized at equilibrium.
The Enthalpy and Gibbs free energy, are similarly derived.