The two-thirds power law derived from an higher-derivative action

TL;DR

This paper derives the two-thirds power law in human movement from a class of higher-derivative Lagrangians, linking movement cost functions to physical principles through Hamiltonian analysis.

Contribution

It introduces a broader class of higher-derivative Lagrangians that explain the two-thirds power law and connects squared jerk to action variables in a Hamiltonian framework.

Findings

Higher-derivative Lagrangians lead to the two-thirds power law.

Squared jerk corresponds to an action variable in the Hamiltonian analysis.

Minimization of squared jerk may relate to energy-efficient movement.

Abstract

The two-thirds power law is a link between angular speed $\omega$ and curvature $\kappa$ observed in voluntary human movements: $\omega$ is proportional to $\kappa^{2/3}$. Squared jerk is known to be a Lagrangian leading to the latter law. We propose that a broader class of higher-derivative Lagrangians leads to the two-thirds power law and we perform the Hamiltonian analysis leading to action-angle variables through Ostrogradski's procedure. In this framework, squared jerk appears as an action variable and its minimization may be related to power expenditure minimization during motion. The identified higher-derivative Lagrangians are therefore natural candidates for cost functions, i.e. movement functions that are targeted to be minimal when one individual performs a voluntary movement.