Review of Lagrangian formalism in biology: recent advances and perspectives

The Lagrangian formalism has attracted the attention of mathematicians and physicists for more than 250 years because of its significant roles in establishing modern theoretical physics. The history of the Lagrangian formalism in biology is much shorter, spanning only the last 50 years. In this article, a broad review of the Lagrangian formalism in biology is presented in the context of both its historical and modern developments. Detailed descriptions of different methods to derive Lagrangians for five selected population dynamics models are given, and the resulting Lagrangians are presented and discussed. The procedure to use the obtained Lagrangians to gain new biological insights into the evolution of the populations without solving the equations of motion is described and applied to the models. Finally, perspectives of the Lagrangian formalism in biology are discussed.