Why an Optimized Double Pendulum Stops Its Hands

An Impact-Optimality Result, and the Structural Limit It Exposes

In a point-mass-clubhead double pendulum the clubhead-speed-optimal hand speed at impact is identically zero. This note derives that result, shows why distributed club inertia and actuation limits do not repair it, and identifies the model change that would.
Author

Dieter Olson

Published

August 27, 2026

Ask a computer to find the golf swing that produces the fastest possible clubhead, and at first it hands back something no golfer has ever done: it swings the arms hard and then slams them to a complete stop right at the ball, flinging the club through. Real golfers still have their hands moving at about 7 m/s at impact. The simulation arrived at 0.36 m/s — essentially parked. It turned out the simulated club was wrong, and once that was fixed the computer's answer looked like a real golf swing after all.

The optimizer is not broken

There is a real effect underneath. If you pretend the whole clubhead is a single weight at the very end of the shaft, then all the energy in the club is the energy of that weight moving — and any energy left in your moving arms is energy that failed to reach the ball. In the perfect, unlimited case, stopping the arms really is optimal, and we can prove it exactly for every choice of arm and club weight. What that proof does not say is how close a real, constrained swing gets to that extreme.

The club was twice as heavy as it should have been

A real driver weighs about 310 grams, and most of that sits three quarters of the way down the shaft, not right at the end. The simulation had put 500 grams at the very tip. To a swinging arm that feels like a club with roughly twice the resistance to being whipped through, so the simulated golfer had to fight it by slamming on the brakes. Give the model a club with the right feel and the braking mostly disappears.

What the corrected model says

With the right club, the simulated swing reaches 111 mph with the hands still travelling at 7 m/s through the ball, matching measured golfers on five of the six things we can check. And it answers the question it was built for: swinging for pure clubhead speed, for the whip of the kinetic chain, or for power through the grip all produce the same swing. The one strategy that measurably costs you is holding the wrist angle for its own sake. The model still releases the club later than real golfers do, and that is the next thing to chase.

WarningCorrection, 27 August 2026

The conclusions in the later sections of the first published version of this article were wrong, and the corrected results are folded in below.

The cause was a mis-specified club, not a limit of the model. The simulation lumped 0.50 kg at the tip of a 1.10 m shaft; a real driver is 0.310 kg with its centre of mass 76% down. In a point-mass-at-tip model the quantity that governs the swing is inertia about the wrist, and that error overstated it — and the arm/club coupling that fights the release — by 2.1×.

With an inertia-matched club the same model produces 49.7 m/s of clubhead speed with 7.26 m/s of hand speed and a club/arm rate ratio of 3.46: five of six measured observables inside their published bands, with no constraint imposed on the hands. The earlier claim that measured golfer hand speed was unreachable, and that a moving hub was required to fix it, was an artifact of the club.

The impact-optimality result in the next section is unaffected — it concerns an unconstrained ideal, not where a constrained optimum lands. Tracked as Tools#4785.

1 The Observation

The interactive workbench now carries a Swing Objective Lab, which optimizes one downswing against five competing objectives — clubhead speed, centrifugal release impulse, Coriolis kinetic-chain transfer, grip-force energy transfer, grip-force impulse — under a single shared torque budget.

Every objective converged. Every trajectory satisfied the equations of motion to roughly \(10^{-14}\). And every one of them was wrong in the same way:

Optimized Measured golfer
Hand speed at impact 0.36 m/s 6–9 m/s
Club / arm angular rate 59× 2.5–4×
Downswing spent braking the arms 32% ≈0 (deceleration is passive)

Measured values from Nesbit (2005) and Miura (2001).

The natural reading is that the optimizer is misbehaving. It is not.

2 The Impact-Optimality Result

At impact the club is in line with the arms, so both angular rates drive the clubhead along the same perpendicular:

\[v_{\text{head}} = (L_1 + L_2)\,\omega_1 + L_2\,\dot\phi\]

Maximising \(v_{\text{head}}\) subject to a fixed kinetic energy \(T = \tfrac12 \dot q^{\mathsf T} M \dot q\) is a linear objective on a quadratic form, so the optimum lies along \(\dot q^\ast \propto M^{-1} c\) with \(c = (L_1 + L_2,\; L_2)\). Writing \(\delta = I_2 + m_2 r_2^2\) for the club’s inertia about the wrist and \(\mu = m_2 L_1 r_2\) for the coupling, the mass matrix at \(\phi = 0\) has \(M_{22} = \delta\) and \(M_{12} = \delta + \mu\), and the arm component of \(M^{-1}c\) reduces to

\[\omega_1^\ast \;\propto\; L_1\left[\,I_2 - m_2 r_2 (L_2 - r_2)\,\right]\]

This bracket decides everything.

2.1 Three Regimes

A point-mass clubhead — \(r_2 = L_2\), \(I_2 = 0\) — gives \(0 - m_2 L_2 (L_2 - L_2) = 0\). Identically zero, for every parameter value. There is nothing to tune. With all of the club’s mass at the tip, the club’s kinetic energy is \(\tfrac12 m_e v_{\text{head}}^2\), so energy in the moving arm is energy that never reached the head. The speed-optimal swing stops the hands because stopping them is optimal.

A real driver — \(m_2 = 0.31\) kg, \(r_2 = 0.89\) m, \(L_2 = 1.143\) m, \(I_2 = 0.043\) kg·m² — gives \(-0.027\). Negative. The optimum wants the hands travelling backward through impact. Modelling the club properly makes the pathology worse, not better.

A forward optimum requires \(r_2 \approx 1.0\) m — a centre of mass at 88% of club length — or \(I_2 \gtrsim 0.2\) kg·m², about five times a driver’s. No club anyone swings lands there.

So the intuitive repair, distributed club inertia, is ruled out analytically before any of it is built.

3 Why Actuation Limits Are Necessary but Not Sufficient

The model gives the golfer a symmetric, velocity-independent torque budget, so braking the arms costs exactly what driving them costs. Two pieces of physiology are missing: torque capacity falls hyperbolically with joint speed (Hill 1938), and the muscles that decelerate the arms are not the ones that drive them.

Adding a Hill-type torque–velocity relation of the form used in golf forward-dynamics models since Sprigings & Neal (2000) and MacKenzie & Sprigings (2009) produces exactly the kinematics the literature reports — hands at 6.8 m/s, a club/arm ratio of 2.8, and zero active braking, with the arm deceleration arising passively from the club’s interaction torque, which is what Nesbit & Serrano (2005) find in real swings.

Those trajectories are also infeasible when the club is mis-specified. The club never reaches the ball, because the overstated coupling keeps driving the wrist open. With a correct club that tension largely disappears.

4 The Club Correction

physics.mass_matrix treats the club as a point mass at the tip. A real club is not that, so an equivalence is needed — and the invariant to preserve is inertia about the wrist, because it sets both the wrist-row mass term and the coupling \(\mu = m_e L_1 L_2\) in every centrifugal and Coriolis term.

Real driver As simulated Ratio
Club mass 0.310 kg 0.500 kg 1.61×
COM from wrist 0.867 m 1.100 m 1.27×
Inertia about wrist 0.288 kg·m² 0.605 kg·m² 2.10×
Coupling \(\mu\) 0.172 kg·m² 0.358 kg·m² 2.08×

Matching the inertia gives an equivalent tip mass of \(m_e = 0.238\) kg. Holding everything else fixed:

Lumped tip mass Clubhead Hands Club/arm
0.500 kg (as simulated) 36.4 m/s 0.36 m/s 59.4
0.320 kg 45.3 m/s 6.01 m/s 3.86
0.238 kg (real driver) 50.8 m/s 7.95 m/s 3.18

The equivalent mass is a modelling quantity, not a claim about club weight: it is the mass that, placed at the tip, swings like the real club.

5 The Structural Limit

Hub torque does not only turn the arms. Through the off-diagonal mass term \(M_{12}\) it also drives the wrist open. In a free rollout at full drive, the wrist cock angle grows from 100° to 184° — the club does not release, it lags further.

The only way this model brings the club through to \(\phi = 0\) at impact is to cut, and then reverse, the hub torque, which necessarily decelerates the arms.

Releasing the club costs some hand speed in a two-link fixed-hub model. With a correct club that cost is a few m/s; with the coupling overstated by 2.1× it consumed all of it.

Sweeping a floor on hand speed at impact measures the price directly:

Hand-speed floor Feasible Clubhead speed Club / arm ratio
none yes 36.4 m/s 59.2
3 m/s yes 34.1 m/s 6.1
5 m/s marginal 29.6 m/s 2.9
6 m/s (measured band) no — —

That table characterises the mis-specified club. With the corrected club the unconstrained optimum already sits at 7.26 m/s — inside the measured band, with no floor imposed at all.

6 What This Means for the Objective Question

The workbench was built to ask which mechanism a good golfer optimizes. With the club corrected, that question becomes answerable.

Objective Deviation Clubhead Hands Club/arm
Clubhead speed 0.80 49.72 m/s 7.26 3.46
Coriolis transfer 0.80 49.65 m/s 7.10 3.54
Energy transfer 0.80 49.72 m/s 7.26 3.46
Impulse transfer 0.80 49.72 m/s 7.26 3.46
Centrifugal impulse 0.93 48.71 m/s 6.14 4.10

Optimizing for clubhead speed, Coriolis transfer, energy transfer or grip-force impulse produces the same swing — and that swing matches measured golfers on five of six observables. The mechanisms and the outcome coincide.

Holding lag for its own sake is the one strategy that measurably costs you, about 1 m/s of clubhead speed, with the club/arm ratio pushed to the edge of its measured band.

The one observable still outside its band for every objective is the release timing: the model releases at roughly 90% of the downswing against a measured 55–80%. That is the next thing worth chasing.

7 What Would Actually Change the Answer

  1. Correct the club first. Done, and it was the dominant term — everything below is now an improvement rather than a prerequisite.
  2. A moving hub — a torso segment. Still the largest remaining gap, and the most likely route to fixing the late release. Both Nesbit (2005) and MacKenzie & Sprigings (2009) treat the torso as a driven segment for exactly this reason. The triple-pendulum model in the workbench is the natural home.
  3. A variable hand radius. Miura (2001) measured the inward hand pull and named the resulting gain parametric acceleration. It requires \(L_1\) to shorten through impact, and it is the measured mechanism for the hand deceleration this model fakes with torque reversal.
  4. Hill actuation, retained. Necessary — it removes the impossible braking — and sufficient only once the release no longer requires reversing the hub.
  5. Not distributed club inertia, for the reason derived above.

8 Interpretation Boundary

This is a planar two-link model with a point-mass arm and clubhead, a fixed hub, constant torque limits, no shaft flex, no ground reaction and no plane change. The proximal link’s angular rate is not an anatomical shoulder or thorax velocity. Everything above describes a two-link chain under a torque budget. None of it is anatomical attribution, and none of it is coaching advice.

The result that a point-mass-clubhead model has \(\omega_1^\ast = 0\) is exact and parameter-independent. Everything downstream of it — which repairs fail, what the frontier costs, how the objectives rank — is a statement about this model class, and is offered as a reason to change the model rather than as a claim about people.

9 Implementation

Derivation, measurements and regression tests live in the Tools pendulum_simulator package:

10 References