On the viability of higher order theories

TL;DR

This paper critically examines the traditional view that higher order theories in physics are inherently unstable, questioning the universality of Ostrogradski's theorem and proposing that some higher order theories can be stable and viable.

Contribution

The authors challenge the general applicability of Ostrogradski's no-go theorem, showing that certain higher order theories can be stable and suggesting a need to revise the conceptual framework for such theories.

Findings

Ostrogradski's theorem applies only to a specific class of higher order theories.

Some higher order theories can be asymptotically stable contrary to traditional beliefs.

The standard second order framework may need extension for fundamental higher order theories.

Abstract

In physics, all dynamical equations that describe fundamental interactions are second order ordinary differential equations in the time derivatives. In the literature, this property is traced back to a result obtained by Ostrogradski in the mid 19th 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 method 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.