We introduce a notion of conformal relative equilibria for vector fields that are conformally invariant with respect to an action of the multiplicative group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ \mathbb {R}^+ $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation> on a smooth manifold. This extends the classical concept of relative equilibria from invariant to conformally invariant dynamics and leads to several equivalent characterizations, highlighting qualitative differences from the classical theory. We then specialize this framework to Hamiltonian systems on exact symplectic manifolds <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ (M,\omega ,\theta ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>ω</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ \omega =-d\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo>=</mo> <mo>-</mo> <mi>d</mi> <mi>θ</mi> </mrow> </math></EquationSource> </InlineEquation>, admitting scaling symmetries. In this setting, we introduce conformally symplectic actions, conformally invariant Hamiltonians, and a natural analog of the momentum map, called the conformal momentum map. Using an augmented Hamiltonian formulation, we obtain a direct and practical characterization of conformal relative equilibria associated with scaling symmetries. For cotangent bundles, we define scaled cotangent-lifted actions and derive explicit formulas for the conformal momentum map. We also introduce a general notion of central configurations arising from scaling symmetries, extending classical ideas from celestial mechanics. The theory is illustrated with examples including the Newtonian <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-body problem, systems with non-flat and conformally flat metrics, and the Kepler problem.

Conformal Relative Equilibria for Scaling Symmetries and Central Configurations

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发表日期：

2026年3月26日

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Giovanni Rastelli,
Manuele Santoprete

摘要

We introduce a notion of conformal relative equilibria for vector fields that are conformally invariant with respect to an action of the multiplicative group $ \mathbb {R}^+ $ $R^{+}$ on a smooth manifold. This extends the classical concept of relative equilibria from invariant to conformally invariant dynamics and leads to several equivalent characterizations, highlighting qualitative differences from the classical theory. We then specialize this framework to Hamiltonian systems on exact symplectic manifolds $ (M,\omega ,\theta ) $ $(M, ω, θ)$ , with $ \omega =-d\theta $ $ω = - d θ$ , admitting scaling symmetries. In this setting, we introduce conformally symplectic actions, conformally invariant Hamiltonians, and a natural analog of the momentum map, called the conformal momentum map. Using an augmented Hamiltonian formulation, we obtain a direct and practical characterization of conformal relative equilibria associated with scaling symmetries. For cotangent bundles, we define scaled cotangent-lifted actions and derive explicit formulas for the conformal momentum map. We also introduce a general notion of central configurations arising from scaling symmetries, extending classical ideas from celestial mechanics. The theory is illustrated with examples including the Newtonian $n$ $n$ -body problem, systems with non-flat and conformally flat metrics, and the Kepler problem.