Rotational Equilibrium: Theory and Applications

• Rotational equilibrium is a state where net torques cancel, ensuring constant rotational motion in physical and engineered systems.
• It underpins diverse fields such as astrophysics, fluid dynamics, statistical mechanics, and neural network training by balancing force, pressure, and gravitational effects.
• Practical modeling employs iterative methods and variational principles to accurately describe layered, baroclinic, and stochastic rotational phenomena.

Rotational equilibrium refers to the state in which a system's rotational motion remains constant over time due to exact balance among all applied torques or the statistical conditions underpinning the effective cancellation of mean angular change. This concept is central to classical mechanics, astrophysics, statistical mechanics, and even neural network optimization, where it manifests as a balance of rotationally relevant quantities. In each context, rotational equilibrium is distinguished by conserved or stabilized rotational observables—be it angular momentum, angular velocity, rotational free energy, or an effective angular step.

1. Classical and Astrophysical Formulation of Rotational Equilibrium

In axisymmetric self-gravitating fluids or stellar models, rotational equilibrium is achieved when centrifugal, pressure, and gravitational forces balance locally, ensuring the system can persist in steady rotation. The equilibrium is described by the force balance equation

$$\nabla P + \rho \nabla \Phi = \rho \frac{j^2}{\varpi^3} \, e_\varpi$$

where $P$ is the pressure, $\rho$ the density, $\Phi$ the gravitational potential, $j$ the specific angular momentum, and $\varpi$ the distance from the rotation axis. For barotropic equations of state ($P = P(\rho)$), this yields a first integral

$H(\rho) + \Phi + \Psi_\text{cen} = C$

with enthalpy $H(\rho) = \int (dP/\rho')$, centrifugal potential $\Psi_\text{cen} = -\tfrac12(j/\varpi)^2$, and $C$ a Bernoulli constant. This provides the basis for classical Maclaurin spheroids and their differentially rotating generalizations (Yasutake et al., 2016, Kiuchi et al., 2010).

For multi-layered, differentially rotating bodies—such as massive stars with compositional interfaces—more complex configurations arise. Each layer admits its rotation law (e.g., $\Omega$-constant, $j$-constant), equation of state, and its own Bernoulli constant, with pressure continuity at interfaces being essential. Iterative schemes (e.g., extensions of Hachisu's self-consistent field method) are employed to solve the coupled system for rotation, gravitational potential, pressure, and density, producing equilibria relevant for advanced stellar evolution studies (Kiuchi et al., 2010).

2. Variational and Numerical Approaches

The Lagrangian variational principle offers a unified formalism for constructing multi-dimensional equilibria, including baroclinic (non-barotropic) stars. The energy functional

$$J[\xi] = \int\left[ \epsilon(\rho,s)\rho + \frac{1}{2}\rho\phi(\rho) + \frac{1}{2}\rho \left(\frac{j}{\varpi}\right)^2 \right]dV$$

is minimized over fluid element displacements $\xi$, preserving each element's mass, specific entropy, and angular momentum. This approach naturally captures both shellular (nearly constant angular velocity on isobars) and more general rotational equilibria without explicit enforcement of first integrals. Finite-element discretization and Monte Carlo minimization methods provide robust numerical solutions, with pressure, density, and isobaric surfaces emerging self-consistently (Yasutake et al., 2016).

Barotropic and baroclinic stars are both tractable within this workflow. For barotropic cases, rotation is constrained to be cylindrical; for baroclinic cases, generalized entropy distributions (e.g., $P/\rho^\gamma$ having nontrivial latitude dependence) directly yield shellular equilibrium, as often assumed in two-dimensional stellar evolution.

3. Rotational Equilibrium in Statistical Mechanics

In the context of Brownian particles and open systems with rotation, rotational equilibrium is reached when the angular degrees of freedom are statistically stationary despite continual stochastic input from a thermal bath. In system-bath models with Caldeira-Leggett-type oscillators, the resulting Langevin equation features noise with long-range temporal correlations due to bath rotation. The stationary state is described by a rotational Gibbs distribution over angular momentum $L$,

$$P(L) \propto \exp[-\beta(H_\text{rot}(L)-\Omega L_z)]$$

provided that coupling to the bath (characterized by friction $\gamma$) is weak ($\gamma \ll \omega_0$) and no magnetic field is present (Matevosyan et al., 3 Jul 2025). The presence of a uniform magnetic field disrupts this equilibrium, violating the Bohr–van Leeuwen theorem's assumptions; the stationary distribution is then non-Gibbsian.

An effective free energy exists as long as rotational symmetry is preserved; thus, no work can be extracted from cyclic processes that respect this symmetry. However, symmetry-breaking protocols (e.g., rotating an external anisotropic potential) permit work extraction, showcasing the role of rotational equilibrium as a condition for non-exploitability in cyclic thermodynamic protocols (Matevosyan et al., 3 Jul 2025).

4. Rotational Equilibrium in Neural Network Training

In deep neural networks, especially those with normalization layers, the concept of rotational equilibrium emerges as a steady-state for the angular update of weight vectors. The per-step angular update $\eta_r$ of a neuron's weight vector $w_t$ converges to an equilibrium value $\widehat{\eta}_r$ due to the competing effects of stochastic gradient growth and shrinkage from weight decay:

$w_{t+1} = w_t - \eta g_t - \lambda w_t.$

At equilibrium, the expected radial components of updates cancel, and the mean angular step becomes stationary. The specific forms of $\widehat{\eta}_r$ and the equilibrium norm $\widehat{\|w\|}$ depend on the optimizer (e.g., SGDM, AdamW, Lion), weight decay parameter $\lambda$, and normalization scheme. Homogeneous angular velocities across neurons and layers can be analytically predicted and achieved with appropriate normalization (BatchNorm, Weight Standardization) and optimizer configuration (e.g., AdamW vs. Adam+$\ell_2$) (Kosson et al., 2023). This has direct implications for scheduler choices and underpins the modern understanding of weight decay's efficacy.

<table> <tr> <th>Context</th> <th>Rotational Variable</th> <th>Equilibrium Condition</th> </tr> <tr> <td>Fluid/stellar mechanics</td> <td>Angular velocity, angular momentum</td> <td>Force balance: pressure, gravity, centrifugal</td> </tr> <tr> <td>Statistical mechanics</td> <td>Probability distribution over $L$</td> <td>Stationary Gibbs over $L$, or generalized if symmetry-broken</td> </tr> <tr> <td>Neural networks</td> <td>Mean angular update $\eta_r$</td> <td>Steady state from balance of decay and stochastic growth</td> </tr> </table>

5. Multi-Layered and Baroclinic Rotational Equilibria

Rotational equilibrium in astrophysical and fluid systems often involves compositional stratification and differential rotation. Multi-layered configurations are constructed by solving for each layer's equilibrium under its own equation of state and rotation law, enforcing pressure continuity at interfaces. The system of equations includes separate Bernoulli integrals per layer and constraints from equilibrium shapes and boundary values. The iterative solution delivers self-consistent, non-spherical equilibrium sequences suitable for modeling massive stars with cores and envelopes of differing rotational and thermodynamic properties (Kiuchi et al., 2010).

Baroclinic equilibria, not restricted to cylindrical rotation, arise naturally in the variational framework and enable modeling of more realistic stellar interiors, where entropy and angular momentum are advected with mass elements (Yasutake et al., 2016).

6. Rotational Equilibrium in Molecular Systems

In molecular spectroscopy, equilibrium rotational constants are obtained by correcting experimental ground-state rotational constants for vibration–rotation interactions. The semi-empirical equilibrium constants ($B_{i,e}^{\rm SE}$) include first-order vibrational corrections ($\alpha_j^{(B_i)}$) derived from high-level quantum-chemical calculations. These corrections enable precise determination of molecular geometry at equilibrium and are essential for astronomical identification of molecules via their rotational spectra (Müller et al., 2021).

7. Practical and Theoretical Significance

Rotational equilibrium underpins both the static structure and the evolution of systems across scales—from planetary rotation and atmospheric resonance to stellar interiors, Brownian motors, and the training dynamics of neural networks. Its mathematical formulation varies with context, but always encodes an invariance or steady-state of rotation-relevant quantities under specified dynamical laws and constraints. In all cases, the analysis and numerical realization of rotational equilibrium are fundamental to predictive modeling, efficient simulation, and the interpretation of observed behavior within both physical and engineered systems (Yasutake et al., 2016, Kiuchi et al., 2010, Kosson et al., 2023, Matevosyan et al., 3 Jul 2025, Müller et al., 2021).