Euler's Disk Dynamics: Dissipation and Impacts

• Euler’s Disk is a spinning rigid disk on a flat surface that exhibits a finite-time singularity as its tilt angle decreases and precession rate diverges.
• The dynamics are characterized by competing dissipation mechanisms—air drag, rolling friction, and impact losses—each yielding distinct power-law behaviors.
• Experiments and analyses demonstrate regime-dependent behaviors, with boundary-layer air drag dominating late stages and rolling friction governing early periods, while impacts become significant under certain conditions.

Euler's Disk denotes the motion of a rigid disk spinning on a flat support after being tipped on its side. In the idealized frictionless limit, the disk rolls without slip on its rim while the tilt angle decreases and the precession rate diverges, so the motion approaches a dissipation-induced finite-time singularity. The principal question in the modern literature has been which dissipation mechanism controls the terminal phase: air drag, rolling friction, or impact losses generated by geometric imperfections. Analyses of polygonal-disk impact dynamics and recent stereoscopic experiments on disks with varying mass and radius have made that question more precise by distinguishing regular precession-like motion from chatter-like impact regimes and by identifying a late-time boundary-layer air-drag regime with a characteristic power law (Baranyai et al., 2017, Thorne et al., 15 Mar 2026).

1. Singular motion and asymptotic variables

A standard small-angle description uses the tilt angle $\phi$ and the precession rate $\Omega$. In the frictionless limit, a disk of mass $m$ and radius $R$ rolling without slip on its rim satisfies the adiabatic relation

$\Omega^2 \sin\phi = \frac{4g}{R}.$

In the small-angle limit, the total mechanical energy is

$$E \approx \tfrac{1}{2} I_{\rm cm}\Omega_{\rm spin}^2 + mgR(1-\cos\phi) \approx \tfrac{3}{2} m g R \phi,$$

and energy balance gives

$$-\frac{dE}{dt}=\Phi \quad\Longrightarrow\quad \frac{d\phi}{dt}=-\frac{2}{3m g R}\,\Phi(\phi,\Omega).$$

If the dissipation rate diverges as $\Phi\propto \phi^{-p}$ as $\phi\to 0$, then

$$\phi(t)\sim (t_0-t)^{1/(p+1)}, \qquad \Omega(t)\sim (t_0-t)^{-1/[2(p+1)]},$$

so the terminal state is reached in finite time (Thorne et al., 15 Mar 2026).

The impact-based analysis uses a different but related asymptotic description. For self-similar impact sequences on a polygonal disk, the mechanical energy just before the $\Omega$0-th impact satisfies

$\Omega$1

while the impact times accumulate at

$\Omega$2

Consequently,

$\Omega$3

so impacts alone produce

$\Omega$4

In that regime the singularity remains finite-time, with

$\Omega$5

The two descriptions use different state variables and exponents, but both characterize the same terminal phenomenon as a singular approach to rest under dissipation (Baranyai et al., 2017).

2. Rigid-body and small-tilt formulations

The impact analysis models an imperfect disk as a regular $\Omega$6-gon with unilateral point contacts. A global frame $\Omega$7 is fixed to the ground plane $\Omega$8, and a body frame $\Omega$9 is attached to the disk center $m$0. Vertex positions in the body frame are

$m$1

and in space

$m$2

The corresponding vertex velocities are

$m$3

The mass is $m$4, and the inertia tensor in local coordinates is

$m$5

with $m$6 for a thin homogeneous disk (Baranyai et al., 2017).

Under frictionless unilateral contacts with normal forces $m$7, the Newton–Euler equations are

$m$8

$m$9

The unilateral constraints impose, whenever $R$0,

$R$1

In free flight, when all $R$2,

$R$3

For asymptotic analysis, the same work introduces a linearized small-tilt model with generalized coordinates

$R$4

At vertex $R$5,

$R$6

Free flight becomes

$R$7

while a single sustained contact at $R$8 satisfies

$R$9

with $\Omega^2 \sin\phi = \frac{4g}{R}.$0 chosen so that $\Omega^2 \sin\phi = \frac{4g}{R}.$1. This linearization provides the platform for the stability and chatter analyses developed later in the paper (Baranyai et al., 2017).

3. Geometric imperfections, unilateral impacts, and self-similarity

The distinctive contribution of the impact study is to treat dissipation generated by geometric imperfections of the disk and of the underlying flat surface. In that model, impacts occur at single vertices, and the collision law is Newtonian with restitution coefficient $\Omega^2 \sin\phi = \frac{4g}{R}.$2. Denoting pre-impact and post-impact generalized velocities by $\Omega^2 \sin\phi = \frac{4g}{R}.$3 and $\Omega^2 \sin\phi = \frac{4g}{R}.$4, one has

$\Omega^2 \sin\phi = \frac{4g}{R}.$5

$\Omega^2 \sin\phi = \frac{4g}{R}.$6

$\Omega^2 \sin\phi = \frac{4g}{R}.$7

which can be written compactly as

$\Omega^2 \sin\phi = \frac{4g}{R}.$8

The parameters of the model are $\Omega^2 \sin\phi = \frac{4g}{R}.$9 and $$E \approx \tfrac{1}{2} I_{\rm cm}\Omega_{\rm spin}^2 + mgR(1-\cos\phi) \approx \tfrac{3}{2} m g R \phi,$$0 (or $$E \approx \tfrac{1}{2} I_{\rm cm}\Omega_{\rm spin}^2 + mgR(1-\cos\phi) \approx \tfrac{3}{2} m g R \phi,$$1) as the inertia parameter, and there is no friction in the impact nor continuous phases (Baranyai et al., 2017).

A precession-free analogue is obtained through a self-similar cyclic sequence of impacts on the vertices $$E \approx \tfrac{1}{2} I_{\rm cm}\Omega_{\rm spin}^2 + mgR(1-\cos\phi) \approx \tfrac{3}{2} m g R \phi,$$2. For $$E \approx \tfrac{1}{2} I_{\rm cm}\Omega_{\rm spin}^2 + mgR(1-\cos\phi) \approx \tfrac{3}{2} m g R \phi,$$3, there exists a scaling factor $$E \approx \tfrac{1}{2} I_{\rm cm}\Omega_{\rm spin}^2 + mgR(1-\cos\phi) \approx \tfrac{3}{2} m g R \phi,$$4 such that, if $$E \approx \tfrac{1}{2} I_{\rm cm}\Omega_{\rm spin}^2 + mgR(1-\cos\phi) \approx \tfrac{3}{2} m g R \phi,$$5 is the $$E \approx \tfrac{1}{2} I_{\rm cm}\Omega_{\rm spin}^2 + mgR(1-\cos\phi) \approx \tfrac{3}{2} m g R \phi,$$6-th impact time and $$E \approx \tfrac{1}{2} I_{\rm cm}\Omega_{\rm spin}^2 + mgR(1-\cos\phi) \approx \tfrac{3}{2} m g R \phi,$$7 denotes rotation by $$E \approx \tfrac{1}{2} I_{\rm cm}\Omega_{\rm spin}^2 + mgR(1-\cos\phi) \approx \tfrac{3}{2} m g R \phi,$$8 about $$E \approx \tfrac{1}{2} I_{\rm cm}\Omega_{\rm spin}^2 + mgR(1-\cos\phi) \approx \tfrac{3}{2} m g R \phi,$$9,

$$-\frac{dE}{dt}=\Phi \quad\Longrightarrow\quad \frac{d\phi}{dt}=-\frac{2}{3m g R}\,\Phi(\phi,\Omega).$$0

$$-\frac{dE}{dt}=\Phi \quad\Longrightarrow\quad \frac{d\phi}{dt}=-\frac{2}{3m g R}\,\Phi(\phi,\Omega).$$1

This implies finite-time accumulation at

$$-\frac{dE}{dt}=\Phi \quad\Longrightarrow\quad \frac{d\phi}{dt}=-\frac{2}{3m g R}\,\Phi(\phi,\Omega).$$2

Within this regular regime, impact dissipation yields the exponent $$-\frac{dE}{dt}=\Phi \quad\Longrightarrow\quad \frac{d\phi}{dt}=-\frac{2}{3m g R}\,\Phi(\phi,\Omega).$$3 in the energy law $$-\frac{dE}{dt}=\Phi \quad\Longrightarrow\quad \frac{d\phi}{dt}=-\frac{2}{3m g R}\,\Phi(\phi,\Omega).$$4 (Baranyai et al., 2017).

The importance of this construction is methodological as much as physical. It supplies a controlled asymptotic regime analogous to the precession-free motion of a smooth rolling disk, allowing impact losses to be compared directly with air-drag and rolling-friction models on the same singular timescale.

4. Competing dissipation mechanisms and reported exponents

The literature summarized in the impact study reports several exponents for dissipation-only models under precession-free drift. For air drag, specifically viscous squeeze-film models associated with Moffatt, the reported values are $$-\frac{dE}{dt}=\Phi \quad\Longrightarrow\quad \frac{d\phi}{dt}=-\frac{2}{3m g R}\,\Phi(\phi,\Omega).$$5 or $$-\frac{dE}{dt}=\Phi \quad\Longrightarrow\quad \frac{d\phi}{dt}=-\frac{2}{3m g R}\,\Phi(\phi,\Omega).$$6. For rolling friction, various laws yield $$-\frac{dE}{dt}=\Phi \quad\Longrightarrow\quad \frac{d\phi}{dt}=-\frac{2}{3m g R}\,\Phi(\phi,\Omega).$$7, $$-\frac{dE}{dt}=\Phi \quad\Longrightarrow\quad \frac{d\phi}{dt}=-\frac{2}{3m g R}\,\Phi(\phi,\Omega).$$8, or $$-\frac{dE}{dt}=\Phi \quad\Longrightarrow\quad \frac{d\phi}{dt}=-\frac{2}{3m g R}\,\Phi(\phi,\Omega).$$9, the last corresponding to no singularity. Against this background, the impact-only self-similar $\Phi\propto \phi^{-p}$0-gon gives $\Phi\propto \phi^{-p}$1, and is therefore subdominant near $\Phi\propto \phi^{-p}$2 in that regular regime (Baranyai et al., 2017).

The 2026 experimental and scaling analysis gives a detailed late-time air-drag mechanism based on viscous shear in the boundary layer beneath the disk. The boundary-layer thickness is

$\Phi\propto \phi^{-p}$3

and the viscous drag torque scales as

$\Phi\propto \phi^{-p}$4

The corresponding power dissipation is

$\Phi\propto \phi^{-p}$5

which, after using $\Phi\propto \phi^{-p}$6, gives

$\Phi\propto \phi^{-p}$7

Substitution into the energy balance yields

$\Phi\propto \phi^{-p}$8

in precise agreement with the experimentally measured late-time exponent $\Phi\propto \phi^{-p}$9 (Thorne et al., 15 Mar 2026).

At earlier times, the same experiments find a rolling-friction regime with

$\phi\to 0$0

consistent with $\phi\to 0$1. On glass, however, $\phi\to 0$2 is nearly independent of the normal load $\phi\to 0$3, which contradicts Coulomb-type rolling models of the form

$\phi\to 0$4

and instead points to an adhesion-dominated torque

$\phi\to 0$5

so that

$\phi\to 0$6

This separates two experimentally distinct regimes: an early-time rolling-friction regime and a late-time boundary-layer air-drag regime (Thorne et al., 15 Mar 2026).

5. Stability, chatter, and irregular terminal dynamics

Regular self-similar impact motion is not generically stable. The impact study defines a Poincaré-type map $\phi\to 0$7 on normalized, rotated states, so that self-similar motion corresponds to a fixed point of $\phi\to 0$8. The Jacobian of the linearized map has two nonzero eigenvalues. For $\phi\to 0$9 and large $$\phi(t)\sim (t_0-t)^{1/(p+1)}, \qquad \Omega(t)\sim (t_0-t)^{-1/[2(p+1)]},$$0—for example $$\phi(t)\sim (t_0-t)^{1/(p+1)}, \qquad \Omega(t)\sim (t_0-t)^{-1/[2(p+1)]},$$1 with $$\phi(t)\sim (t_0-t)^{1/(p+1)}, \qquad \Omega(t)\sim (t_0-t)^{-1/[2(p+1)]},$$2—one finds $$\phi(t)\sim (t_0-t)^{1/(p+1)}, \qquad \Omega(t)\sim (t_0-t)^{-1/[2(p+1)]},$$3, implying asymptotic stability of the self-similar precession-free mode. For $$\phi(t)\sim (t_0-t)^{1/(p+1)}, \qquad \Omega(t)\sim (t_0-t)^{-1/[2(p+1)]},$$4 and a homogeneous disk with $$\phi(t)\sim (t_0-t)^{1/(p+1)}, \qquad \Omega(t)\sim (t_0-t)^{-1/[2(p+1)]},$$5, numerical eigenanalysis gives $$\phi(t)\sim (t_0-t)^{1/(p+1)}, \qquad \Omega(t)\sim (t_0-t)^{-1/[2(p+1)]},$$6 for all $$\phi(t)\sim (t_0-t)^{1/(p+1)}, \qquad \Omega(t)\sim (t_0-t)^{-1/[2(p+1)]},$$7, so arbitrarily small perturbations grow and precession or wobble develops (Baranyai et al., 2017).

Once regular motion is unstable, the dynamics enter an irregular chatter-like regime. For bouncing on a three-point imperfect support, the paper reports a richer range $$\phi(t)\sim (t_0-t)^{1/(p+1)}, \qquad \Omega(t)\sim (t_0-t)^{-1/[2(p+1)]},$$8. In the complete-chatter regime, with gravity absent or negligible, a rod ($$\phi(t)\sim (t_0-t)^{1/(p+1)}, \qquad \Omega(t)\sim (t_0-t)^{-1/[2(p+1)]},$$9) or triangle ($\Omega$00) accumulates infinitely many impacts with

$\Omega$01

where $\Omega$02 are real eigenvalues of $\Omega$03. As $\Omega$04 or $\Omega$05, one gets $\Omega$06. This implies that sufficiently small $\Omega$07 or sufficiently small $\Omega$08 can make impact-driven dissipation asymptotically stronger than the air-drag exponents $\Omega$09 or $\Omega$10 (Baranyai et al., 2017).

The numerical phase structure is correspondingly nontrivial. In the region denoted $\Omega$11, meaning complete chattering without partial chatter, triangle-bounce simulations with gravity $\Omega$12 and random initial perturbations exhibit power-law energy decay with $\Omega$13, often with $\Omega$14 for small $\Omega$15. Outside complete chatter, or near its boundary, gravity remains important and $\Omega$16. These results establish that impacts need not be negligible in principle, but their relevance depends sharply on restitution, inertia, and the qualitative type of contact sequence (Baranyai et al., 2017).

6. Experiments, parameter regimes, and the present state of the debate

The recent experimental study uses stereoscopic high-speed imaging at $\Omega$17 of a checkerboard printed on the disk to reconstruct the three-dimensional plane of the disk and thereby obtain $\Omega$18 and $\Omega$19, with $\Omega$20. Several controls discriminate among candidate dissipation mechanisms. When $\Omega$21 is varied at fixed $\Omega$22, the prefactor $\Omega$23 in

$\Omega$24

scales as $\Omega$25, as predicted by the boundary-layer theory, whereas rolling-friction-only models predict mass-independent dynamics. In a partial vacuum at $\Omega$26, reducing $\Omega$27 by a factor of ten increases the late-time $\Omega$28 in a manner quantitatively consistent with the $\Omega$29 scaling of $\Omega$30. A geometric control using a steel ring with identical outer radius but open center shows no crossover to $\Omega$31; instead the motion remains at $\Omega$32 throughout, indicating that removing the solid underside eliminates the relevant boundary-layer drag (Thorne et al., 15 Mar 2026).

The full trajectory can be synthesized by coupling air drag and rolling friction through

$\Omega$33

Numerical integration reproduces the observed crossover: early times at large $\Omega$34 are governed by rolling friction, while late times for $\Omega$35 are governed by boundary-layer air drag. At $\Omega$36, contact is lost through the condition $\Omega$37, which terminates the singular motion (Thorne et al., 15 Mar 2026).

Set against these measurements, the impact theory gives a more conditional conclusion. It shows that there exists a range of parameters—small radii of gyration or small restitution coefficients—in which absorption by impacts dominates all previously investigated mechanisms during the last phase of motion. Yet the parameter values associated with a homogeneous disk on a hard surface are typically not in that range. Specifically, for $\Omega$38 and $\Omega$39, the self-similar solution has $\Omega$40 and $\Omega$41; for the $\Omega$42 bounce, simulations give $\Omega$43–$\Omega$44; and for a homogeneous disk on a hard surface one finds $\Omega$45. Under those conditions, impacts remain subdominant relative to the late-time air-drag mechanism identified experimentally (Baranyai et al., 2017).

A plausible implication is that the debate is best understood as regime-dependent rather than binary. The late-time motion of a homogeneous disk on a hard smooth surface is consistent with viscous air drag in the boundary layer beneath the disk, while rolling friction dominates earlier stages, and impact losses become decisive only in parameter regions associated with low restitution, small radius of gyration, or strongly irregular chatter-like contact dynamics. The broader implication drawn in the experimental work is that these mechanisms are relevant not only to Euler's Disk itself but also to rolling-contact systems operating under low loads on smooth surfaces (Thorne et al., 15 Mar 2026).