D'Alembert's Direct and Inertial Forces Acting on Populations: The Price Equation and the Fundamental Theorem of Natural Selection

I develop a framework for interpreting the forces that act on any population described by frequencies. The conservation of total frequency, or total probability, shapes the characteristics of force. I begin with Fisher’s fundamental theorem of natural selection. That theorem partitions the total evolutionary change of a population into two components. The first component is the partial change caused by the direct force of natural selection, holding constant all aspects of the environment. The second component is the partial change caused by the changing environment. I demonstrate that Fisher’s partition of total change into the direct force of selection and the forces from the changing environmental frame of reference is identical to d’Alembert’s principle of mechanics, which separates the work done by the direct forces from the work done by the inertial forces associated with the changing frame of reference. In d’Alembert’s principle, there exist inertial forces from a change in the frame of reference that exactly balance the direct forces. I show that the conservation of total probability strongly shapes the form of the balance between the direct and inertial forces. I then use the strong results for conserved probability to obtain general results for the change in any system quantity, such as biological fitness or energy. Those general results derive from simple coordinate changes between frequencies and system quantities. Ultimately, d’Alembert’s separation of direct and inertial forces provides deep conceptual insight into the interpretation of forces and the unification of disparate fields of study.