When an Energy Optimum Stops the Hands
A Double Pendulum’s Inertia, Reachable Motion, and the Limits of Inferring a Golfer’s Objective
A simplified swing optimizer once returned almost still hands at impact. Which parts of that result came from mechanics? Which came from the simulated club and its controls? Which, if any, say something about golfers? Three separate problems need keeping apart.
An Energy Budget Gives a Bound
Picture straight arms and a club with all its mass in the head. If a fixed amount of motion energy can be shared freely, the fastest head leaves none in the moving arms. That is true only for this idealized instant. A real swing must still reach that instant with the available forces, time, and range of motion.
The Club Changes the Whole Motion
Cutting an excessive tip mass changed the reported simulations a lot: the hands kept moving through impact. Matching a real club's resistance to rotation about the grip is useful. It keeps only one part of the equations. Where the mass sits also changes coupling, gravity, and the energy needed to move the hands.
A Plausible Result Is a Start
Several simulated goals gave similar final speeds. That does not show what a golfer optimizes, or that the paths and muscle actions matched. The next step is to compare complete motions and independent measurements under clearly stated models and controls.
The 27 August 2026 correction identified an excessive lumped club mass in the original experiments. This technical review retains that history and its reported numerical results, while correcting the wider conclusions drawn from it. Matching one inertia does not make two clubs dynamically equivalent; failure to find a trajectory does not prove that no trajectory exists; and agreement with selected comparison bands does not identify a human objective.
The historical results below come from Tools #4785, the archived article, and the design contract at revision dcd4b2ae. They have not been independently rerun for this review. The algebra and illustrative inertia calculations have been independently checked. These are different evidence levels.
After this review, Tools #5431 (merged in Tools PR #5554) made the distributed club convention canonical: uniform shaft, centre of mass at \(L_2/2\), \(I_{\mathrm{cm}}=m_2L_2^2/12\), and a head point mass. All numbers in this article, including the 36.4 m/s and 0.36 m/s result and the tip-mass tables, are historical, pre-fix values and are kept unchanged. Tools records the lumped-club characterisations used here as the exact \(m_2 \rightarrow 0\) limit of the new model, so they remain valid as statements about that limit; they are not results for the shipped distributed model.
What changes: the shipped model is no longer tip-concentrated, so its impact-optimality coefficient is negative (hands move backward) rather than zero, and the zero survives only in the head-only limit. Golf presets were re-derived (Tools reports roughly 40% lower distal inertia about the wrist and 26 to 29% higher peak tip speed for the GUI preset). Re-running the swing-objective and impact-optimality results is tracked in Tools #5523.
1 The Observation and the Three Questions
The model workbench compares downswing objectives involving clubhead speed, centrifugal release impulse, Coriolis transfer, grip-force work, and grip-force impulse. An early calculation with 0.50 kg concentrated at the end of a 1.10 m club reported a clubhead speed of 36.4 m/s and hand speed of about 0.36 m/s. A small dynamics residual accompanied the solution. The original article treated the almost stopped hands as evidence of a structural defect, then treated a revised club as resolving that defect.
Three questions make the diagnosis more precise:
- Energy bound: at a specified pose and kinetic energy, which instantaneous velocity maximizes a specified clubhead output?
- Reachable swing: which poses and velocities can the model reach from its starting state under its actuator, duration, contact, and endpoint limits?
- Human explanation: which model and control hypotheses survive comparison with independently measured golf swings?
A correct answer to the first question constrains the second. It does not solve it, and neither answer alone settles the third. This distinction also explains why correcting a parameter can improve a finite-duration simulation without changing the energy-optimality theorem.
2 The Model and the Instantaneous Bound
Consider two rigid links moving in a plane about a fixed hub. The proximal link has length \(L_1>0\) and inertia \(A=I_1+m_1r_1^2\) about that hub. The distal body has mass \(m_2>0\), axial center-of-mass distance \(r_2\) from the wrist, and planar moment of inertia \(I_2\) about its center of mass. The point called the head lies at distance \(L_2>0\) from the wrist. These assumptions omit transverse COM offset, shaft flexibility, and a moving hub.
Let \(q=(\theta,\phi)\): \(\theta\) is the proximal absolute angle and \(\phi\) is the club’s angle relative to it. Then \(\omega_1=\dot\theta\) and the club’s absolute angular rate is \(\omega_2=\dot\theta+\dot\phi\). Define
\[ \delta=I_2+m_2r_2^2,\qquad \mu=m_2L_1r_2. \]
The kinetic energy is \(T=\tfrac12\dot q^{\mathsf T}M(\phi)\dot q\), with
\[ M(\phi)= \begin{bmatrix} M_{11} & M_{12}\\ M_{12} & \delta \end{bmatrix}. \tag{1}\]
Its entries are
\[ \begin{aligned} M_{11}&=A+m_2L_1^2+\delta+2\mu\cos\phi,\\ M_{12}&=\delta+\mu\cos\phi. \end{aligned} \]
At the assumed collinear pose \(\phi=0\), choose the common forward tangent. The signed head velocity along it is
\[ \begin{aligned} v_h&=(L_1+L_2)\omega_1+L_2\dot\phi\\ &=c^{\mathsf T}\dot q, \qquad c=\begin{bmatrix}L_1+L_2\\L_2\end{bmatrix}. \end{aligned} \]
The signed hand velocity is \(L_1\omega_1\); its speed is \(L_1|\omega_1|\). Collinearity is a model condition, not a general description of a golfer at impact. Away from it, head velocity is a two-component Jacobian output, and maximizing its magnitude is a different optimization problem.
Assume \(M=M(0)\) is positive-definite and \(T=E>0\). The energy-metric Cauchy–Schwarz inequality gives
\[ (c^{\mathsf T}\dot q)^2 \leq (c^{\mathsf T}M^{-1}c)(\dot q^{\mathsf T}M\dot q) =2E\,c^{\mathsf T}M^{-1}c. \]
Equality in the forward direction gives the unique maximizing rate vector:
\[ \begin{aligned} \dot q^*&=\sqrt{\frac{2E}{c^{\mathsf T}M^{-1}c}}\,M^{-1}c,\\ v_h^*&=\sqrt{2E\,c^{\mathsf T}M^{-1}c}. \end{aligned} \tag{2}\]
This is a fixed-pose, fixed-energy, signed-velocity result. It contains no initial condition, muscle model, torque limit, swing duration, or collision law. At \(E=0\) the only velocity is zero when \(M\) is positive-definite. A singular mass matrix requires a separate treatment; the displayed inverse does not exist.
2.1 Why a Point Mass at the Head Stops the Hands
For a tip mass \(m_e\), set \(r_2=L_2\) and \(I_2=0\). At collinearity,
\[ T=\frac12 A\omega_1^2+\frac12 m_e v_h^2. \]
With \(A>0\), maximizing forward \(v_h\) at fixed \(E\) sets \(\omega_1^*=0\) and \(v_h^*=\sqrt{2E/m_e}\). The wrist is stationary at this ideal instant because any nonzero proximal motion consumes some of the specified energy.
The positive-inertia condition matters. If \(A=0\), the matrix is singular and many rate combinations can give the same head velocity and energy. Thus the unique zero-hand-rate conclusion is not valid for every conceivable parameter value. Nor does a zero terminal rate require that the optimizer apply a large braking torque: the theorem says nothing about the route to that state.
2.2 What Distributed Inertia Changes
At \(\phi=0\), direct inversion gives
\[ \begin{aligned} \det M&=A\delta+m_2L_1^2I_2,\\ (M^{-1}c)_1&=\frac{L_1 B}{\det M},\\ B&=I_2-m_2r_2(L_2-r_2). \end{aligned} \tag{3}\]
The sign of \(B\) determines the sign of the energy-optimal proximal rate under these assumptions. For the original article’s illustrative tuple \((m_2,r_2,L_2,I_2)=(0.31,0.89,1.143,0.043)\) in SI units,
\[ \begin{aligned} B&=-0.0268027\;\mathrm{kg\,m^2},\\ I_{2,\mathrm{threshold}}&=0.0698027\;\mathrm{kg\,m^2}. \end{aligned} \]
The formal zero-rate threshold is about 1.62 times the specified COM inertia, not five times. Holding the other values fixed instead gives the outer COM root \(r_2\approx1.005\) m, or about 88% of \(L_2\). These are algebraic thresholds; independently varied mass, COM, and inertia need not describe a physically realizable body.
For example, if all mass lies on the line segment \(0\leq s\leq L_2\), then
\[ \begin{aligned} B&=\int s^2\,dm-L_2\int s\,dm\\ &=-\int s(L_2-s)\,dm\leq0. \end{aligned} \]
Only endpoint-supported mass can attain equality; positive interior mass makes \(B<0\). A body with transverse extent or mass outside those endpoints requires a different realizability assessment. This prevents interpreting the formal positive-\(B\) regime as an arbitrary design change to a slender club.
The negative optimal hand velocity is a property of the declared instantaneous problem. It does not establish that distributed inertia worsens a realistic swing, or rule it out as a useful model refinement. It changes both the energy metric and the trajectory dynamics through which the final state is reached.
3 Matching One Inertia and Preserving the Dynamics
A tip mass chosen as \(m_e=\delta/L_2^2\) preserves the wrist-row diagonal inertia \(M_{22}=\delta\). This matches one coefficient. The real coupling is \(\mu=m_2L_1r_2\), while the replacement gives \(\mu_e=m_eL_1L_2\). Their difference is tied to exactly the same bracket as the preceding theorem:
\[ \begin{aligned} \mu_e-\mu&=\frac{L_1}{L_2} \left[I_2-m_2r_2(L_2-r_2)\right]\\ &=\frac{L_1}{L_2}B. \end{aligned} \tag{4}\]
This connects the energy bound to model reduction: a nonzero \(B\) means the tip replacement cannot preserve both \(\delta\) and \(\mu\). Even \(B=0\) does not by itself preserve the \(m_2L_1^2\) term in \(M_{11}\), gravitational loading, or total mass. Endpoint-supported mass can be represented by masses at the wrist and head, but discarding the wrist mass loses its contribution to proximal motion.
The provider’s separate driver example uses \(m_2=0.310\) kg, \(r_2=0.867\) m, \(I_2=0.0551\) kg·m² and overall length 1.143 m. These are the provider’s declared example properties, not new measurements made for this article. Its club specification measures COM distance from the butt of the grip. The model uses a 1.10 m second link. A butt-based inertia must not silently become an inertia about a different hand or wrist point.
To make the arithmetic explicit, identify the example’s reference point with the model wrist, use \(L_1=0.65\) m and model \(L_2=1.10\) m, and compare:
| Quantity | Distributed Example | Old Tip Mass | Inertia-Matched Tip Mass |
|---|---|---|---|
| Represented mass (kg) | 0.310 | 0.500 | 0.238119 |
| COM distance from declared origin (m) | 0.867 | 1.100 | 1.100 |
| \(\delta\) (kg·m²) | 0.288124 | 0.605000 | 0.288124 |
| \(\mu\) (kg·m²) | 0.174701 | 0.357500 | 0.170255 |
The old tip inertia is about 2.10 times the example’s inertia. The equivalent mass substantially reduces that discrepancy, but its coupling is still about 2.54% lower than the distributed example’s, and the physical/model head lengths also differ. A 0.310 kg tip mass would give 0.3751 kg·m², not 0.605 kg·m²; the latter belongs to the 0.500 kg preset.
If the grip reference moves distance \(a\) from the butt toward the head, use \(r_{\mathrm{grip}}=r_{\mathrm{butt}}-a\) and
\[ \begin{aligned} \delta_{\mathrm{grip}}&=I_2+m_2(r_{\mathrm{butt}}-a)^2\\ &=\delta_{\mathrm{butt}}-2am_2r_{\mathrm{butt}}+m_2a^2. \end{aligned} \]
The head lever arm must change consistently too. Choosing the origin, club geometry, and preserved coefficients is part of the model specification.
4 What the Historical Simulations Establish
The original mass sweep reported the following outputs. They are retained as historical simulation results, not as newly validated human measurements:
| Lumped Tip Mass | Head Speed (m/s) | Hand Speed (m/s) | Club/Arm Rate Ratio | Reported Braking Fraction |
|---|---|---|---|---|
| 0.500 kg | 36.4 | 0.36 | 59.4 | 33% |
| 0.320 kg | 45.3 | 6.01 | 3.86 | 24% |
| 0.238 kg | 50.8 | 7.95 | 3.18 | 24% |
The source reports dynamics defects no larger than \(2\times10^{-13}\) for this sweep. The substantial change in terminal hand speed demonstrates sensitivity to the modeled mass. Braking did not disappear: the source still reports 24% for the two lighter cases. Other historical tables round or configure the old case differently, giving a ratio of 59.2 and braking fraction of 32%; those figures should not be treated as one precisely identified run.
The reported hand-speed-floor experiment used the old club, a 0.36 s duration, and torque limits of ±250 N·m:
| Hand-Speed Floor | Reported Solver Outcome | Head Speed (m/s) | Club/Arm Rate Ratio |
|---|---|---|---|
| None | Successful candidate | 36.4 | 59.2 |
| 3 m/s | Successful candidate | 34.1 | 6.1 |
| 5 m/s | Described as marginal | 29.6 | 2.9 |
| 6 m/s | No feasible candidate reported | — | — |
A local solver’s failure to find a feasible trajectory is not an infeasibility certificate. The archived analysis predicate classifies reachability using a dynamics-defect threshold. That predicate alone does not verify every endpoint or inequality constraint. A small collocation residual also does not establish global optimality, accuracy between grid points, independent reintegration agreement, or validity as a golfer model.
The archived Hill-limit example reported 6.8 m/s hand speed, a rate ratio of 2.8, and zero modeled braking, but also dynamics defects near 1.0. Those numbers cannot be presented as a feasible swing or as measured confirmation of the actuation model. The available narrative and implementation preserve useful leads, but a complete run manifest, solver settings, states, controls, and constraint residuals are needed before these tables support reproducible quantitative comparisons.
5 Actuation, Deceleration, and Muscle Inference
A symmetric torque clamp allows driving and opposing moments of equal magnitude. Replacing it with velocity-dependent actuator capacity can change the reachable motions. Hill’s force–shortening relation concerns active muscle under specified experimental conditions; converting it into a joint torque–velocity law adds assumptions about moment arms, muscle lengths, activation, tendon mechanics, and the contributing muscle groups (Hill 1938, 160–63).
Active muscle can absorb mechanical energy while lengthening. Such eccentric behavior is not ruled out by a shortening-force relation. The provider’s actuation implementation explicitly allows a braking branch; its chosen capacity is a model parameter, not a universal physiological constant.
For generalized hub and wrist actuator moments \(\tau_h,\tau_w\), actuator power is \(\tau_h\dot\theta+\tau_w\dot\phi\). A negative hub-power interval identifies energy absorbed through that modeled input. An anatomical joint’s net moment, a segment’s angular deceleration, and an individual muscle’s activation are three different quantities. Antagonists can be active while their net moment is small; interaction forces can decelerate a segment while muscles remain active. The work-and-power study of four amateur golfers used kinematically driven models to estimate net joint work and club energy transfer. It does not establish zero active arm braking in humans (Nesbit and Serrano 2005, 521–23, 527–31).
MacKenzie and Sprigings used four aggregate torque generators with activation and force–velocity behavior in a three-dimensional forward model. Their comparison involved one male golfer with a three handicap, followed by a separate optimization. This provides a modeling precedent, rather than a universal joint law or proof that a particular two-link simulation reproduces human muscle action (MacKenzie and Sprigings 2009, secs. 2.1, 2.3–2.5).
5.1 Why a Single Rollout Does Not Prove a Release Mechanism
At a fixed state, a hub-moment increment alone contributes \(-M_{12}\Delta\tau_h/\det M\) to relative angular acceleration. That local coupling helps explain why increasing proximal drive can initially increase lag under a chosen sign convention. The complete acceleration also includes wrist moment, gravity, velocity-dependent terms, and configuration.
The historical full-drive rollout increased its cock-angle measure from 100° to 184°. This is an example for its stated input, not proof that every fixed-hub release requires hub-torque reversal. In particular, an independently actuated wrist changes the argument. Releasing the club, decelerating the proximal link, and applying a negative proximal actuator power need to be measured separately in each trajectory.
6 Comparing the Model With Golfers
The provider’s six intervals are useful heuristic comparison bands until their exact numerical provenance and observable mappings are established. Calling all six directly measured ranges overstates the evidence. For example, Nesbit’s study included 84 male and one female amateur golfers; Table 3 reports aggregate-group impact grip speed of 7.1–8.0 m/s, with mean 7.75 m/s, and impact head speed of 43–50 m/s. Those are not the implementation’s 6–9 and 45–55 m/s intervals, and they are not a professional-tour cohort (Nesbit 2005, 503–4).
| Implementation Observable | Historical Band | Evidence and Mapping Needed |
|---|---|---|
| Head speed at impact | 45–55 m/s | Distinguish resultant speed from a target-direction component; the cited Table 3 reports 43–50 m/s. |
| Hand speed at impact | 6–9 m/s | Define the measured grip point and compare it with the model wrist; Table 3 reports 7.1–8.0 m/s. |
| Downswing duration | 0.23–0.32 s | The exact claimed interval has not been verified in the cited sources; define start and impact events consistently. |
| Absolute club/proximal angular-rate ratio | 2.5–4 | This is a derived planar-model quantity, not an interchangeable ratio of arbitrary anatomical Euler rates; specify denominator behavior near zero. |
| Wrist cock at impact | −5° to 20° | Match the angle convention, joint representation, and club reference; the exact empirical interval remains unverified here. |
| Fraction of downswing before half release | 0.55–0.80 | The implementation uses the first grid index where cock angle falls to half its initial value, divided by \(N-1\); the exact cited empirical interval remains unverified here. |
The historical duration and release bands cite Jorgensen and Sprigings–Neal, respectively (Jorgensen 1970; Sprigings and Neal 2000). Those citations are retained as provenance of the implementation, not as certification that these exact intervals were measured in those papers. A discretized crossing index also needs interpolation and time-grid checks before being compared to a continuous-time release measurement.
In the implementation, deviation outside each interval is normalized by its half-width, then summed. This is a declared scoring rule: it encodes relative weights and supplies neither a statistical likelihood nor uncertainty bounds. Being inside five intervals is therefore a limited descriptive result. It is not validation of all six observables, the complete trajectory, tissue loads, or a golfer’s control objective.
7 What the Objective Comparison Can Tell Us
Tools #4785 describes the revised objective comparison as using the inertia-matched club and a 0.28 s duration, distinct from the 0.36 s old-club frontier. The article’s historical values are:
| Objective | Deviation Score | Head Speed (m/s) | Hand Speed (m/s) | Club/Arm Rate Ratio |
|---|---|---|---|---|
| Head speed | 0.80 | 49.72 | 7.26 | 3.46 |
| Coriolis transfer | 0.80 | 49.65 | 7.10 | 3.54 |
| Grip-force energy transfer | 0.80 | 49.72 | 7.26 | 3.46 |
| Grip-force impulse transfer | 0.80 | 49.72 | 7.26 | 3.46 |
| Centrifugal release impulse | 0.93 | 48.71 | 6.14 | 4.10 |
These terminal summaries show close agreement for four objectives and a reported difference for the fifth. They do not show identical whole trajectories: even the listed Coriolis case differs in its endpoint values. Nor is maximizing centrifugal release impulse equivalent to deliberately holding a wrist angle. The roughly 1 m/s difference is a reported simulation comparison, not evidence that a coaching strategy costs a golfer that speed.
The source also reports half-release near 90% of the downswing, later than its chosen 55–80% interval. Until the interval and observable are reconciled with measurements, this is a mismatch with the implementation’s reference, not a quantified population error.
If different objectives generate indistinguishable predictions under a given model and experiment, the experiment cannot identify which objective a person uses. Discrimination requires observations on which the hypotheses differ, matched constraints and initial conditions, numerical convergence assessment, and independent data. A plausible endpoint is valuable, but its evidential role is smaller than the original article claimed.
8 Hand Path, Moving Hubs, and Energy Transfer
Miura’s modified model used a moving pivot driven by a linear actuator, while keeping the two link lengths fixed. An imposed 0.08 m pull over the final 0.04 s increased modeled head speed from 46.8 to 50.1 m/s in the reported example. The paper distinguishes this emulation from direct simulation of a golfer’s actual pull motion (Miura 2001, 82–83, Fig. 7 and Table 1). It does not require \(L_1\) to shorten, nor does it establish a unique cause of human arm deceleration.
The mechanical connection is power at a common grip reference, \(P=\mathbf F\cdot\mathbf v_{\mathrm{grip}} +\mathbf M_{\mathrm{grip}}\cdot\boldsymbol\omega\) for a rigid club. A force that is radial relative to one chosen center can do work when its application point moves. Changing the hand path changes the force–velocity alignment, while moving the model base adds a source of motion and potentially work that must be included in the energy budget. Inward motion is not, by itself, an energy source.
A torso segment, prescribed base motion, and variable proximal length represent different model extensions. Each changes geometry, inertia, admissible forces, and the interpretation of the inputs. Their contributions cannot be inferred merely by adding one more coordinate or by transferring a fixed-hub result. Distributed club inertia likewise remains a legitimate refinement: the instantaneous theorem does not exclude its effects on reachable trajectories.
9 A Controlled Route to a Stronger Explanation
First specify the club and grip origin, then compare the distributed model with the inertia-matched tip approximation under identical start states, durations, actuator laws, and endpoint requirements. Inspect the full mass matrix, gravity, energy balance, and resulting trajectories. This isolates the model reduction from changes in the experiment.
Next compare actuator choices using their declared positive and negative power capacities. Report all constraint residuals, mesh refinement, multistart sensitivity, and independent forward integration. Distinguish a feasible candidate, a locally optimized result, and any certified global statement.
Then test geometric extensions with complete power accounting and observables that can be mapped to actual measurements. A moving hub may improve a hand path or release timing, but that is a hypothesis to test. It should be compared with parameter uncertainty, distributed inertia, actuator changes, and the measurement definition itself.
Finally compare candidate objectives on held-out motions and outputs that actually discriminate between them. The connected argument is that geometry sets the velocity map; inertia sets energy and coupling; actuation and history set reachability; measurement determines what can be tested; and the experiment determines what can be identified. The simplified model is most useful when those relationships remain visible.
10 Interpretation Boundary
The exact zero-hand-rate result applies to a collinear, fixed-hub, planar model with a point mass at the head, positive proximal inertia, and an instantaneous fixed kinetic-energy budget. The historical optimization outputs concern particular two-link implementations and settings. Neither supplies anatomical attribution, a muscle-activation measurement, a ball-flight optimum, or coaching advice. Impact performance also depends on delivered face orientation, contact location, effective collision dynamics, and ball properties.
11 Implementation and Review Evidence
The historical provider implementation is pinned to revision dcd4b2ae9e0e3aa665280efb20769ea2b5d33948. Its impact_optimality.py, club_equivalence.py, actuation.py, reference_kinematics.py, model_adequacy.py, and objective_realism.py preserve the implementation context. Their prose and reference labels are not independent scientific evidence. The current development branch may change. The review record and independent checks are tracked under AffineDrift #4714.
13 References
Citation
@online{olson2026,
author = {Olson, Dieter},
title = {When an {Energy} {Optimum} {Stops} the {Hands}},
date = {2026-08-27},
url = {https://affinedrift.com/articles/impact-optimality-and-model-limits.html},
langid = {en}
}