Relative Equilibria for Scaling Symmetries and Central Configurations

Abstract

In this paper, we explore scaling symmetries within the framework of symplectic geometry. We focus on the action $\Phi$ of the multiplicative group $G = \mathbb{R}^+$ on exact symplectic manifolds $(M, \omega,\theta)$, with $\omega = -d\theta$, where $ \theta $ is a given primitive one-form. Extending established results in symplectic geometry and Hamiltonian dynamics, we introduce conformally symplectic maps, conformally Hamiltonian systems, conformally symplectic group actions, and the notion of conformal invariance. This framework allows us to generalize the momentum map to the conformal momentum map, which is crucial for understanding scaling symmetries. Additionally, we provide a generalized Hamiltonian Noether's theorem for these symmetries. We introduce the (conformal) augmented Hamiltonian $H_{\xi}$ and prove that the relative equilibria of scaling symmetries are solutions to equations involving $ H _{ \xi } $ and the primitive one-form $\theta$. We derive their main properties, emphasizing the differences from relative equilibria in traditional symplectic actions. For cotangent bundles, we define a scaled cotangent lifted action and derive explicit formulas for the conformal momentum map. We also provide a general definition of central configurations for Hamiltonian systems on cotangent bundles that admit scaling symmetries. Applying these results to simple mechanical systems, we introduce the augmented potential $U_{\xi}$ and show that the relative equilibria of scaling symmetries are solutions to an equation involving $ U _{ \xi } $ and the Lagrangian one-form $\theta_L$. Finally, we apply our general theory to the Newtonian $n$-body problem, recovering the classical equations for central configurations.