On the viability of higher-order theories

In physics, all dynamical equations that describe fundamental interactions are second-order differential equations in the time derivatives. In the literature, this property is traced back to a result obtained by Ostrogradski in the mid-nineteenth century, which is the technical basis of a no-go theorem for higher-order theories. In this work, we review the connection of symmetry properties with the order of dynamical equations, before reconsidering Ostrogradski's result. Then, we show how Ostrogradski's conclusion is reached by applying to higher-order theories concepts and methods that have been specifically developed for second-order theories. We discuss a potential lack of consistency in this approach, to support the claim that Ostrogradski's result applies to a class of higher-order theories that is nowhere representative of generic ones: we support this claim by giving an example of a higher-order Lagrangian that is asymptotically stable, but that would be unstable under Ostrogradski's criterion. We also conclude that, when considering higher-order theories as fundamental, we may need to reconsider and extend the conceptual framework on which our standard treatment of second-order theories is based. This article is part of the theme issue 'Newton, Principia, Newton Geneva Edition (17th-19th) and modern Newtonian mechanics: heritage, past & present'.