In thermodynamics, a cyclical process is one in which the body undergoes a series of changes such that it is finally brought back to its initial condition. [1] The essential example, as visualized in the concept of "re-establishment of equilibrium in the caloric" as described by French physicist Sadi Carnot in his 1824 Reflections on the Motive Power of Fire, being that in which a body of gas, enclosed in cylinder and piston, at a specific pressure and volume, is expanded, due to the input of heat, and then contracted, due to the removal of heat, back to its original position (and assumed molecular arrangement) quantified by the values of the original pressure and volume.

Internal energy
The integral of an indefinitely small change in internal energy is given by the following expression:

where

U1 = H1 + J1 (initial values)
U2 = H2 + J2 (final values)

each signifying, respectively, the initial and final values of the sums of the heat contents H and ergal contents J of the body. For a cyclical process, according to German physicist Rudolf Clausius, the integral of the change in internal energy is zero, being that initial and final conditions of the body are assumed, by definition, to be the same, or U1 equals U2. [1]

References
1. Clausius, Rudolf. (1879). The Mechanical Theory of Heat, (pg. 33). London: Macmillan & Co.