Swing Surface Calculation Overview

• Swing Surface Calculation is a mathematical framework that defines and computes surfaces generated by rotating curves, phase boundaries, and entanglement prescriptions using integrals and eigenmode analysis.
• It integrates techniques from classical differential geometry, hybrid control systems, galactic dynamics, and quantum holography to provide precise and robust surface quantifications.
• The methodology combines analytical integrations, uncertainty quantification via thick sets, and eigen decompositions to yield actionable insights in stability analysis and entropy computations.

Swing Surface Calculation encompasses a range of precise mathematical techniques for quantifying, characterizing, or prescribing the geometry of surfaces associated with swing phenomena in both classical differential geometry and applied systems. The term is encountered in the computation of surfaces of revolution (where a curve is "swung" about an axis), in phase-space or state-space boundaries in control and power grid theory, in stellar and galactic dynamics (swing amplification), and in holographic entanglement entropy beyond standard AdS/CFT. While the notion of "swing" can vary according to context—from literal rotation around an axis, to regions where phase synchronization occurs, to boundaries associated with sliding or chattering—rigorous calculation often involves integrating local geometric, analytic, or set-valued properties across a given domain.

1. Geometric Foundations: Surface of Revolution via General Swing Axis

The classical computation of a swing surface in differential geometry pertains to the area generated when a curve is revolved about an arbitrary line in the plane. For a differentiable curve defined parametrically by $x(t)$, $y(t)$ for $t\in[t_0, t_1]$, the surface area $A$ produced by swinging this curve about the line $L: Ax + By = C$ is

$$A = 2\pi \int_{t_0}^{t_1} \frac{|A x(t) + B y(t) - C|}{\sqrt{A^2+B^2}} \sqrt{[x'(t)]^2 + [y'(t)]^2} \, dt$$

This formula generalizes classical surfaces of revolution, recovering the standard area expressions for revolutions around the $x$ or $y$ axis as special cases. The radius function $r(t)$ is the perpendicular distance from $(x(t), y(t))$ to $L$, and the arc length element $ds$ is the Euclidean differential for the curve. When the function is $y=f(x)$ and the swing axis is $y=mx+k$, the formula specializes to

$$A = 2\pi \int_{a}^{b} \frac{|f(x) - m x - k|}{\sqrt{1 + m^2}} \sqrt{1 + [f'(x)]^2} dx$$

Smoothness and absence of self-intersections are required, with particular attention to points where $r(t)$ vanishes (the curve crosses the axis) (Goins et al., 2011).

2. Swing Surfaces in Hybrid Systems and Sliding Mode Control

In control systems—especially those featuring non-continuous (hybrid) controllers—the "sliding surface" denotes the locus in state space where trajectories exhibit indefinite mode-switching (chattering). Consider a system governed by two vector fields $f_a(x)$ (active in region $\mathcal{A}$) and $f_b(x)$ (in $\mathcal{B} = \overline{\mathcal{A}}$), with $\mathcal{A}$ defined by a constraint $c(x) \leq 0$ and boundary $\partial \mathcal{A} : c(x) = 0$. The sliding surface $\mathcal{S}(\mathcal{A})$ is characterized analytically as

$$\mathcal{S}(\mathcal{A}) = \partial \mathcal{A} \cap \{ x \mid \mathcal{L}_{f_a}^c(x) \geq 0 \wedge \mathcal{L}_{f_b}^c(x) \leq 0 \}$$

where

$$\mathcal{L}_{f_a}^c(x) = \nabla c(x) \cdot f_a(x), \quad \mathcal{L}_{f_b}^c(x) = \nabla c(x) \cdot f_b(x)$$

Under parametric and sensing uncertainties, the sets $$\mathcal{A}, \mathcal{L}_{f_a}^c, \mathcal{L}_{f_b}^c$$ themselves become uncertain, prompting the use of thick sets—set-theoretic intervals in the powerset of $\mathbb{R}^n$ defined as $$[\![\mathbb{X}]\!] = [\mathbb{X}^{\subset}, \mathbb{X}^{\supset}]$$ with inner and outer bounds. The sliding surface enclosure $[\![\mathcal{S}]\!]$ is computed via interval propagation as

$$[\![\mathcal{S}]\!] = \partial[\![\mathcal{A}]\!] \cap \overline{[\![\mathcal{L}_{f_a}^c]\!]} \cap [\![\mathcal{L}_{f_b}^c]\!]$$

This technique yields conservative but efficient outer approximations for sliding surfaces under arbitrary uncertainty, which is critical for verifying controller behavior in practical cyber-physical systems such as the child swing example (Jaulin et al., 2021).

3. Swing Amplification and Surface Calculation in Galactic Dynamics

In disk galaxies, swing amplification describes the temporary growth and decay of spiral arms driven by the phase synchronization of stellar epicyclic motions. Analytically, the relevant dynamical surface is governed by equations

$$x = x_g - x_a \cos\phi,\qquad y = y_g + \frac{2\Omega}{\kappa} x_a \sin\phi$$

where $x_g, y_g$ are the guiding center coordinates, $x_a$ is the epicycle amplitude, $\phi=\kappa t - \phi_0$ is the phase, $\Omega$ the angular frequency, and $\kappa$ the epicyclic frequency.

Simulations using a shearing box approximation reveal that swing surfaces are transient regions in phase space where the local density autocorrelation $\bar{\Psi}(s)$ peaks within an epicyclic period, modulated by $\kappa/\Omega$ and the critical wavelength for instability $\lambda_{cr}$. Calculations of swing surfaces must therefore take into account both the temporal evolution (phase synchronization) and the spatial modulation (density enhancement) given by the joint action of differential rotation, self-gravity, and epicyclic motion (Michikoshi et al., 2020).

4. Swing Surfaces in Holographic Entanglement Beyond AdS/CFT

In holographic dualities extending beyond AdS/CFT—namely flat$_3$/BMSFT and (W)AdS$_3$/WCFT—the swing surface prescription generalizes the Ryu–Takayanagi formula for entanglement entropy. A swing surface comprises "ropes" (null geodesics propagated from boundary entangling points) and a "bench" (spacelike extremal surface connecting the ropes). Its area determines the entanglement entropy by

$$S_\mathcal{A} = \min_{\mathcal{X}_\mathcal{A} \sim \mathcal{A}} \operatorname{ext} \left\{ \frac{\operatorname{Area}(\mathcal{X}_\mathcal{A})}{4G} \right\}$$

with the surface composed as $\mathcal{X}_\mathcal{A} = X \cup \gamma_b$, where $X$ is the bench and $\gamma_b$ the ropes.

This prescription, supported by an extended Lewkowycz–Maldacena argument, accurately accounts for the modular flow structure of non-Lorentz-invariant boundary theories and provides nontrivial entropic contributions in flat and warped holography, including explicit entropy formulas for symmetric intervals and vacuum states (Apolo et al., 2020).

5. Swing Surface Boundaries in Power System Dynamics

In electric power grids, swing surface calculation emerges from the analytical solution to the nonlinear swing equation governing generator rotor stability. By transforming the swing equation from polar coordinates $\delta$ to Cartesian $(x,y)$:

$x = E\cos\delta,\qquad y = E\sin\delta$

with magnitude constraint $x^2 + y^2 = E^2$,

the problem is recast as a constrained linear differential equation. The "swing surface" here corresponds to the stability boundary separating synchronized (stable) and desynchronized (unstable) states. The analytical solution involves eigenvalue decomposition of the system matrix and projection onto the constraint manifold:

$z(t) = z_G(t) + \sum_i \beta_i \Psi_i u_i$

where $z_G(t)$ is the base solution, $\Psi_i$ are null-space basis vectors, $u_i$ are eigenmodes, and coefficients $\beta_i$ are adjusted for initial conditions.

These surfaces are pivotal for transient stability analysis, with real-time discrimination of fault-induced phenomena and potential for early warning applications. Limitations arise in very large systems due to computational costs of eigenvalue decompositions, and the validity region is bounded by parametric constraints (Oh, 2019).

6. Assumptions, Limitations, and Generalizations

Across contexts, swing surface calculation is built upon a variety of structural assumptions, including differentiability and non-intersecting curves for geometric revolution, bounded uncertainty intervals for thick set calculations, linearized phase-space models in galactic dynamics, extremality of bench surfaces in holography, and manageable system size in power system applications. Limitations often arise from singularities (e.g., vanishing radius during revolution, phase drift in galactic dynamics, invalidation of bench extremality) and computational tractability (high-dimensional eigenmode calculations, multi-parameter thick set enclosure).

Special cases are typically well-understood—the classical axis of revolution, constant-parameter hybrid systems, idealized Toomre $Q$-parameter disks, highly symmetric boundary entangling regions, and single-machine swing equations—while generalizations require adaptive integration of uncertainty, symmetry-breaking, and nonlinearities.

7. Significance and Interdisciplinary Impact

Swing surface calculation serves as a nexus of geometry, analysis, control theory, dynamical systems, and modern theoretical physics. It provides rigorous tools for quantifying physical and informational boundaries (area, entropy, stability threshold), designing and verifying controllers against harmful oscillatory or chattering behaviors, understanding transient phenomena in astrophysical and engineered systems, and extending quantum gravity techniques to non-standard holographic regimes. The adoption of thick set and interval-based methods, modular flow-based prescriptions, and eigenmode decompositions exemplifies the modern trend towards robust, uncertainty-inclusive, and structurally generalized analytical approaches.