aa r X i v : . [ phy s i c s . pop - ph ] J un Forces and Torques Near to Impact in the Golf Swing
Robert D. Grober ∗ Nissequogue, NY, 11780 (Dated: June 23, 2020)Motivated by MacKenzie’s observation of a negative force couple near to impact [1, 2], this paperexplores a model for how the golf club moves near to impact. It assumes the golf club is moving asthe distal arm of a double pendulum. At impact the club head is moving straight down the targetline, at its maximum speed, on a path with a specified radius of curvature. From this model theforces and torques required to move the club near to impact are calculated. The results are shownto be quantitatively consistent with data reported by MacKenzie to within a few percent. Thenegative couple near to impact is a robust feature of this model, balancing the torque associatedwith the force that drives the center of mass of the golf club. The negative couple allows the golferto maintain a larger radius of curvature of the path of the club head as it moves through impact.Because the negative couple can also serve to reduce the rotational speed of the club, the presenceof a negative couple at impact in the golf swing manifests a trade between distance and direction.
CONTENTS
I. Acknowledgments 1II. Summary for Golfers 1III. Introduction 2IV. Geometry of the Double Pendulum Near toImpact 3V. Dynamics of the Double Pendulum Near toImpact 5VI. The Lagrangian of the Double Pendulum 7A. Kinetic Energy 7B. Potential Energy 8C. Equations of Motion 8D. Solving for Couples 8E. Whence art thou, K α < K α < K α is Generated 13A. The Hands 13B. Aerodynamic Drag of the Club head 14C. Inertia of the squaring of the club face 14D. Speculation Summary 15X. Summary 15A. Model Parameters 15References 16 ∗ [email protected] I. ACKNOWLEDGMENTS
This paper has benefited from the conversations, sug-gestions, and thoughtful critique of many colleagues, in-cluding Grant Waite, Chris Como, Sasho MacKenzie,Young-Hoo Kwon, Michael Finney, Bill Greenleaf, ShawnCox, Phil Cheetham, and Paul Wood. The geometry de-fined in Fig. 3, which is the organizing principle of themodel explored in this paper, was inspired by ideas ex-pressed during a lecture by Michael Hebron.
II. SUMMARY FOR GOLFERS
This section summarizes several of the salient pointsdiscussed in this paper which may be of general interestto golfers. They are presented in this summary withoutthe mathematical detail provided in the paper. • This paper explores the forces and torques thatmove the club at impact. It assumes the club ismoving as the distal arm of a double pendulum,as depicted in Fig. 2. At impact the club head ismoving straight down the target line at maximumspeed. • There is a geometry particular to the double pen-dulum which allows the club head to access pointsalong the target line, as is shown in Fig. 3. Thelength of the target line that is accessible dependson how far the golfer stands from the ball, butis typically 8-16 inches long, covering the distancefrom the middle of the stance to the forward foot.In this geometry the hands are always ahead of theclub head. The path of the hands through this re-gion is up and in. This geometry is the organizingprinciple for the model explored in this paper. • In practice it is not possible to keep the club headmoving on a straight line for an extended distanceas it moves through impact at speed. Rather, theclub head moves on an arc, as depicted in Fig. 10.It is possible to make the radius of curvature of theclub head path sufficiently large that the deviationfrom a straight line is negligible for several inchesbefore and after impact, Fig. 11. This allows somemargin for error in the golf swing. • At impact the rotational speed of the proximal armof the double pendulum (i.e. the shoulders, arms,and hands) is decreasing while the rotational speedof the distal arm (i.e. the club) is increasing, as canbe inferred from Fig 4. This happens in a balancedway so as to allow the club head to move at maxi-mum speed in a direction straight down the targetline at impact. The deceleration of the proximalarm in vicinity of impact is consistent with previ-ous studies of the kinematic sequence [3]. • As is shown in Fig. 12, the force applied to theclub by the golfer at impact is oriented in the gen-eral direction of the hub (i.e. the fixed pivot aboutwhich the proximal arm of the double pendulumrotates, which corresponds roughly to the middleof the sternum). It is is dominated by the cen-tripetal force needed to keep the center of mass ofthe club moving on an arc. Both the magnitudeand orientation of the force are consistent with theinverse dynamics measurements of MacKenzie [2].The direction of the applied force at impact is animportant result, and could be an organizing themearound which a golfer’s biomechanics at impact areoptimized. • This force applied by the golfer at impact resultsin a torque applied to the club which serves to in-crease the rotational speed of the club. However,this torque also serves to decrease the radius of cur-vature of the path of the club head. To compensatefor this, an additional torque is applied to the clubso as to moderate the total torque without applyingany additional net force. The details of the balanc-ing of these two torques are shown in Fig. 14. Thisadditional torque takes the form of a force couple[4], which can be though of as two forces, equal inmagnitude, opposite direction, separated througha distance. A force couple generates a torque, butdoes not accelerate the center of mass.This force couple has been measured by MacKen-zie [1, 2] throughout the entire swing. It is negativewithin a few tens of milliseconds of impact, where italso acquires its largest magnitude. This large, neg-ative force couple in the vicinity of impact is ubiq-uitous among the golfers that have been measured.It is surprising because a negative couple would re-duce the rotational speed of the club, which seemscontrary to the goals of most golfers.This paper shows that this negative couple in thevicinity of impact is a robust feature of the dou-ble pendulum model of the golf swing. It serves to reduce the total torque applied to the club, allow-ing the club head path to maintain a larger radiusof curvature through the ball. As such, the nega-tive couple is a manifestation of the trade betweendistance and direction. • It remains the subject of future work to explain ex-actly how this negative force couple is generated.Given that it occurs over an imperceptibly shortperiod of time near to impact, and that nobodywas aware of it before MacKenzie’s experiments,this negative couple is possibly an involuntary fea-ture of the body when the hands/wrists are rotat-ing at very high speed. If so, it suggests golfershave learned to incorporate this natural negativecouple into their golf swings in a way which allowsthem to hit the ball straighter. Indeed, when train-ing golfers it may be better to simply focus on thepath of the club through the ball rather than tryingto measure the force couple at impact. • Golfers are going to ask how this information canbe used to improve their golf swing. This ques-tion is best addressed by professional golf instruc-tors. However, it is interesting to point out thatthe deceleration of the hands and the orientationof the force at impact highlighted in this paperis reminiscent of an approach to training the golfswing named the ‘Rotor Method’ that was pio-neered by Nichols in the 1970s [5] and recentlydemonstrated in a video by Malaska [6]. Quotingfrom [5], the downswing was characterized by ‘theexplosive movement of the ... right side against theresistance of the left’. This serves to enhance thedeceleration of the torso/arms/hands at impact. Atimpact Nichols stressed ‘ ... the weight of the clubhead must go down the line until just after impactand then upward’. Pulling the club upward just af-ter impact serves to help the golfer orient the forcesat impact towards the hub. When done correctly,this style of ‘swing produces a very shallow arc re-sulting in long, thin divots’. This is suggestive ofthe club head paths of Figs. 10 and 11. Perhapsthis training methodology from the 1970s can beadapted to the modern golf swing as a means oftraining the deceleration of the body and the hub-centric orientation of the applied force near to im-pact.
III. INTRODUCTION
This paper is motivated by the results of MacKenzie[1, 2, 7–9], Kwon [10] and Nesbit [11–13], who have used3-d motion analysis of the golf club to infer the forcesand torques necessary to move the club throughout theswing. A goal of this paper is to understand the role ofthe negative couple in the immediate vicinity of impact,as reported by MacKenzie [2].The golf swing has long been modeled as a double pen-dulum [14–16]. This paper makes use of this model inthe immediate vicinity of impact. There has been muchdiscussion about the general applicability of the doublependulum to the entire golf swing. For instance, it isknown the hub (i.e. the fixed pivot about which theproximal arm of the double pendulum rotates) is not rig-orously fixed throughout the entire swing [16], and thereare claims the length of the proximal arm can changesignificantly during the swing [12]. This paper is focusedon the dynamics in the immediate vicinity of impact. Anexplicit assumption of this paper is that near to impactthe hub is reasonably fixed and the proximal arm is ofconstant length. Under these conditions the double pen-dulum is a good approximation.The paper is divided into six sections. The first sectionintroduces a geometry particular to the double pendulumin which the club can access points along the target line.The length of the target line that is accessible dependson how far the golfer stands from the ball, but generallyextends from the middle of the stance out towards theforward foot.The second section uses this geometry to constrain thedynamics of the double pendulum so as to limit consid-eration to golf swings where the club head reaches max-imum speed as it moves down the target line at impacton a path with a specified radius of curvature.The third section begins with a derivation of the dou-ble pendulum Lagrangian, done in the coordinate systemused in this paper. The Lagrange equations of motion areused to calculate the external torques required to drivethe system at impact, given the constraints in the secondsection. It is in this section that the negative couple re-ported by MacKenzie is found to be a robust feature ofthe model.In the fourth section the equations of motion are usedto simulate the motion of the club in a region near to im-pact. The external applied torques are assumed constantthroughout this region, equal to the values required atimpact. Using inverse dynamics, similar to the approachof MacKenzie [2], Kwon [10] and Nesbit [11], the sim-ulated motion is used to recover the forces and torquesthat move the club.This fourth section provides the opportunity to look atthe problem from various different frames of reference,both inertial and non-inertial. This exercise serves toemphasize that the answer does not depend on the frameof reference in which the problem is solved. Hopefully,the discussions in this section can help to make clear someof the issues associated with working in different framesof reference [17].The fifth section of the paper performs a search overthe parameters of the model to find the best fit to theforces and torques at impact as reported by MacKenzie[2] for one particular golfer. It is demonstrated solutionsto the model can be found which agree quantitativelywith MacKenzie’s measurements to within a few percent.The final section of the paper speculates about various mechanisms by which the negative force couple can begenerated.
IV. GEOMETRY OF THE DOUBLEPENDULUM NEAR TO IMPACT
The coordinate system is shown in Fig. 1. The x-axisis perpendicular to the target line, while the y-axis isoriented parallel to the target line. The double pendulumconsists of two arms, a proximal arm of length R anda distal arm of length R . The angles θ and φ describethe angle of the proximal and distal arms relative to thex-axis. The stationary end of the proximal arm (i.e. thehub) is attached to the origin, but is free to rotate aboutthe origin. The proximal arm is an approximation to theshoulders/arms/hands. The hinge between the proximalarm and the distal arm is where the hands attach to thehandle of the club. The distal arm is the golf club.The position ( x , y ) of the far end of the proximal arm(i.e. the hinge between the hands an the club handle) is x = R cos θ (1a) y = R sin θ (1b)Similarly, the position ( x , y ) of the end of the distalarm (i.e. the club head) is x = R cos θ + R cos φ (2a) y = R sin θ + R sin φ (2b)Assume the club is constrained to move straight downthe target line, parallel to the y-axis, a distance x = R + R − δ from the origin, where δ >
0. This is shownon the left side of Fig. 2. That x < R + R allowsthe club head to access a family of points straight downtarget line. This family is defined by the constraint R + R − δ = R cos θ + R cos φ (3)An additional constraint is that the hands should beslightly cocked ( θ − φ ) >
0. Note that Jorgensen [16]refers to the angle θ − φ as β . Subsequently, Nesbit [11]popularized the use of the Euler angle naming conven-tion α , β , and γ to describe rotation in the swing plane,out of the swing plane, and about the axis of the shaft,respectively. This convention has become popular in golfteaching circles, and thus the convention α = θ − φ isadopted in this paper.The final constraint is that the hands uncock as theclub moves towards impact δαδy < x, y ) = ( x , y min ) through ( x, y ) = y (meters) x ( m e t e r s ) θϕ FIG. 1. Geometry of the double pendulum, defining the an-gles θ and φ . The hub of the proximal arm is attached atthe origin, indicated by the green circle, and is free to rotateabout the origin. The proximal arm is meant to approximatethe shoulders/arms/hands of the golfer. The distal arm isthe golf club. The hands attach to the golf club at the hingeindicated by the red circle. The blue circle at the far end ofthe distal arm is the club head. ( x , y max ), where y min = 0 and y max = √ R δ − δ ≈√ R δ . Note that at y max , φ = 0. For all other pointsalong the line, φ <
0. Similarly, θ > θ and φ at all points where the clubcan access the target line subject to these constraints, asfollows. The parameters x and y describe the positionof the club head on the target line, x = R cos θ + R cos φ (5a) y = R sin θ + R sin φ (5b)Eliminate φ from these coupled equations by squaringand adding together, (cid:0) R cos φ (cid:1) = (cid:0) x − R cos θ (cid:1) (6a) (cid:0) R sin φ (cid:1) = (cid:0) y − R sin θ (cid:1) (6b)yielding2 x R cos θ + 2 y R sin θ = x + y + R − R (7)Simplify by defining the parameters A = x + y + R − R , B = 2 x R , and C = 2 y R . Reduce to terms onlyinvolving sin θ by again taking the square y (meters) x ( m e t e r s ) θϕ < 0R + R − δ ≈ √ δ FIG. 2. Geometry of the double pendulum, defining the an-gles θ and φ , near to impact. The club head, indicated bythe blue circle, is fixed to the target line, which is a distance R + R − δ from the y-axis. As is described in the text,the club head can access the target line from y = 0 thru y ≈ √ R δ . In practice, this spans club positions from themiddle of the stance out towards the forward foot. Note that φ ≤ (cid:0) B cos θ (cid:1) = B (cid:0) − sin θ (cid:1) = (cid:0) A − C sin θ (cid:1) (8)which yields a quadratic equation in sin θ (cid:0) B + C (cid:1) sin θ − AC sin θ + (cid:0) A − B (cid:1) = 0 (9)Changing parameters again, this time to a = B + C , b = − AC and c = A − B , and solving for sin θ ,sin θ = − b + √ b − ac a (10)Then use y = R sin θ + R sin φ to solve for sin φ .The resulting family of orientations of the double pen-dulum for which the club head can access the target lineis shown in Fig. 3. The length of the distal arm (i.e. theclub) R = 1 .
092 m, consistent with the value used in[12]. The length of the proximal arm R = 0 . R for noparticular reason other than the aspect ratio looks aboutcorrect. R + R = 1 .
856 m ≈
73 in. Finally, δ is chosen tobe 7 .
84 cm ≈ R + R − δ = 1 .
778 m =70 . √ R δ ≈ . y (meters) x ( m e t e r s ) R + R − δ ≈ √ δ FIG. 3. The family of orientations of the double pendulum forwhich the club head, shown as blue dots, can access pointsalong the target line. The hands, shown as red dots, arealways ahead of the club head. The hands have gone past thebottom of their arc, and as a result are traveling up and inrelative to the target line the forward foot (i.e. left foot for a right handed golfer).The complete set of model parameters used throughoutthis paper are provided in Appendix A.An important feature of Fig. 3 is the motion of thehands near to impact. The hands are always ahead ofthe club head. The hands have gone past the bottom oftheir arc, and as a result are traveling up and in relativeto the target line.
V. DYNAMICS OF THE DOUBLE PENDULUMNEAR TO IMPACT
The next step is to use this geometry to put constraintson the first and second time derivatives of θ and φ .Start by considering the velocity of the club head as itmoves down the line. It is useful to organize the expres-sions for position as a matrix equation, (cid:20) xy (cid:21) = (cid:20) cos θ cos φ sin θ sin φ (cid:21) (cid:20) R R (cid:21) (11)Taking derivatives of the equations above, (cid:20) ˙ x ˙ y (cid:21) = (cid:20) − sin θ − sin φ cos θ cos φ (cid:21) (cid:20) R ˙ θR ˙ φ (cid:21) (12) The club is constrained to move straight down the lineat impact, so ˙ x = 0 and ˙ y = v . (cid:20) v (cid:21) = (cid:20) − sin θ − sin φ cos θ cos φ (cid:21) (cid:20) R ˙ θR ˙ φ (cid:21) (13)Solving for ˙ θ and ˙ φ requires inverting the matrix (cid:20) − sin θ − sin φ cos θ cos φ (cid:21) − = 1sin α (cid:20) − cos φ − sin φ cos θ sin θ (cid:21) (14)where α = θ − φ and sin α = sin θ cos φ − sin φ cos θ . Thus (cid:20) R ˙ θR ˙ φ (cid:21) = 1sin α (cid:20) − cos φ − sin φ cos θ sin θ (cid:21) (cid:20) v (cid:21) = v sin α (cid:20) − sin φ sin θ (cid:21) (15)Through this entire region φ <
0, and α >
0. Therefore,both ˙ θ > φ > y/y max a n g u l a r s p ee d ( r a d / s ) ̇ϕ̇θ FIG. 4. Angular speeds ˙ φ and ˙ θ for points along the targetline calculated for a club head speed of 44 . / s (i.e. 100mph). To keep the club on the target line, the club rotation(i.e. distal arm of the double pendulum) accelerates and theshoulder/arm/hand rotation (i.e. proximal arm of the doublependulum) decelerates. The complete set of model parame-ters used throughout this paper are provided in Appendix A. Fig 4 shows ˙ θ and ˙ φ for points along the target linefrom y = 0 thru y = y max . The abscissa is setup toalign with the image in Fig. 3. These angular speedsare calculated for the case of a club speed of 44 . / s(i.e. 100 mph). Note that at y = 0, ˙ θ = ˙ φ , and thusthe proximal and distal arms move together. Out near y = y max at the end of the accessible target line, ˙ θ = 0and all motion of the club head is related to ˙ φ .The acceleration is constrained such that the clubcomes to its maximum speed at impact, ¨ y = 0. Ad-ditionally, the club travels from inside the line to insidethe line, so at impact ¨ x <
0. In principle, the magni-tude | ¨ x | should be as small as possible so that the clubhead travels a reasonably straight path down the targetline. In practice, it requires larger forces and torques asthe golfer makes | ¨ x | smaller, and it becomes impracti-cal to get the club head to travel perfectly straight downthe target line for an extended distance at speed. But,as will be shown below, the resulting radius of curvatureof the club path can be sufficiently large that the clubpath is reasonably approximated as a straight line nearto impact.The radius of curvature of the club head path at im-pact is given by the expression R c = ˙ y / ¨ x [18]. As willbe shown, it is useful to parameterize R c in terms of thedistance of the hub from the target line, R + R − δ .In particular, define R c in terms of the parameter ξ suchthat ˙ y / | ¨ x | = ξ ( R + R − δ ). The condition ξ = 1 corre-sponds to the case when the path is approximated by theperimeter of a circle of radius ( R + R − δ ). Expressing¨ x in terms of ξ ,¨ x = − ˙ y ξ ( R + R − δ ) . (16)The second derivatives ¨ x and ¨ y are given by the ex-pression (cid:20) ¨ x ¨ y (cid:21) = (cid:20) − sin θ − sin φ cos θ cos φ (cid:21) (cid:20) R ¨ θR ¨ φ (cid:21) − (cid:20) cos θ cos φ sin θ sin φ (cid:21) (cid:20) R ˙ θ R ˙ φ (cid:21) (17)where we impose the condition. (cid:20) ¨ x ¨ y (cid:21) = " − ˙ y ξ ( R + R − δ ) (18)Solving for R ¨ θ and R ¨ φ using the same matrix inversionfrom above (cid:20) R ¨ θR ¨ φ (cid:21) = 1sin α (cid:20) − cos φ − sin φ cos θ sin θ (cid:21) (cid:20) ¨ x ¨ y (cid:21) + 1sin α (cid:20) − cos α −
11 cos α (cid:21) (cid:20) R ˙ θ R ˙ φ (cid:21) (19)With these equations, ( θ , ˙ θ , ¨ θ ), and ( φ , ˙ φ , ¨ φ ) are fullyspecified at impact.Shown in Fig. 5 are ¨ φ and ¨ θ as a function of the distancealong the target line for various values of ξ ≥
1. Again, y/y max ̈ ϕ ̈ ( r a d / s ) ξ = 1.4ξ = 1.3ξ = 1.2ξ = 1.1ξ = 1.00.00.20.40.60.81.0 y/y max −3000−2500−2000−1500−1000−5000 ̈ θ ̈ ( r a d / s ) ξ = 1.0ξ = 1.1ξ = 1.2ξ = 1.3ξ = 1.4 FIG. 5. Angular acceleration ¨ φ and ¨ θ for points along the tar-get line for values ξ ≥
1. There is a general trend that largerradius of curvature requires acceleration of larger magnitude. y/y max −1000−50005001000 ̈ ϕ ̈ ( r a d / s ) ξ = 1.0ξ = 0.9ξ = 0.8ξ = 0.7ξ = 0.60.00.20.40.60.81.0 y/y max −2000−1500−1000−500050010001500 ̈ θ ̈ ( r a d / s ) ξ = 0.6ξ = 0.7ξ = 0.8ξ = 0.9ξ = 1.0 FIG. 6. Angular acceleration ¨ φ and ¨ θ for points along thetarget line for values ξ ≤
1. The dashed lines occur wheneither ¨ φ < θ > the abscissa is setup to align with the image in Fig. 3.Note that at y/y max = 0 and for ξ = 1 .
0, ¨ θ = ¨ φ ≈
0. Forlarger values of y/y max the magnitude of the requiredangular acceleration increases, with φ accelerating and θ decelerating. The deceleration of θ and the accelerationof φ near to impact is consistent with previous studies ofthe kinematic sequence [3].Fig. 6 details ¨ φ and ¨ θ as a function of the distancealong the target line for various values of ξ ≤
1. Herethe curves have a dashed region and a solid region. Thedashed regions occur when either ¨ φ < θ > φ ≥ θ ≤
0, and are the solutions which havea better chance of matching what is realized in actualgolf swings. Note that for the case of the solid lines,the magnitude of the acceleration gets smaller at smallervalues of ξ , which corresponds to the club path throughimpact being more curved. VI. THE LAGRANGIAN OF THE DOUBLEPENDULUM
The the double pendulum was originally used as amodel for the golf swing by Cochran and Stobbs [14]. TheLagrangian of the double pendulum and its applicationto the dynamics of the golf swing was subsequently pio-neered by Jorgensen [15]. In this section, the Lagrangianis re-derived using the coordinate system of this paper.
A. Kinetic Energy
The Lagrangian of a rigid body can be calculated asthe difference between the kinetic energy and potentialenergies [19]. Thus, the first step is to define the kineticenergy of the moving parts in the double pendulum.Start by considering the proximal arm. Let r denotethe distance along the arm. The velocity of a point alongthe proximal arm is ˙ x = − r ˙ θ sin θ (20a)˙ y = r ˙ θ cos θ (20b)The square of the velocity is v = ˙ x + ˙ y = r ˙ θ (21)The kinetic energy is calculated by integrating the localkinetic energy over the entire proximal arm. KE = 12 Z dm v (22) Defining the linear mass density ρ ( r ) such that dm = dr ρ ( r ), the integral becomes KE = 12 Z R dr ρ r ˙ θ (23)The integral of the linear mass density is just the mass, M = Z R dr ρ (24)As such, ρ /M is a probability density,1 = Z R dr ρ M (25)With this interpretation, the integral over r is the secondmoment, < R > = Z R dr ρ M r (26)The kinetic energy can then be parameterized as KE = 12 M < R > ˙ θ (27)Now consider the distal arm. Define r to be the dis-tance along the distal arm. The velocity of a point alongthe distal arm is˙ x = − R ˙ θ sin θ − r ˙ φ sin φ (28a)˙ y = R ˙ θ cos θ + r ˙ φ cos φ (28b)The square of the velocity is v = ˙ x + ˙ y (29) v = R ˙ θ + r ˙ φ +2 R r ˙ θ ˙ φ (cid:0) cos θ cos φ +sin θ sin φ (cid:1) (30)which simplifies to v = R ˙ θ + r ˙ φ + 2 R r ˙ θ ˙ φ cos ( θ − φ ) (31)Once again, define the kinetic energy of the distal armas an integral of the local kinetic energy over the entiredistal arm KE = 12 Z dm v (32)Defining the linear mass density of the distal arm, ρ ,and using the definitions of the first and second momentsof the distal arm, < R > = Z R dr ρ M r (33) < R > = Z R dr ρ M r (34)the following expression for the kinetic energy of the dis-tal arm is obtained, KE = 12 M (cid:0) R ˙ θ + < R > ˙ φ + 2 R < R > ˙ θ ˙ φ cos ( θ − φ ) (cid:1) (35)The kinetic energy of the entire double pendulum isthen KE = KE + KE , KE = 12 (cid:0) M < R > + M R (cid:1) ˙ θ + 12 M < R > ˙ φ + M R < R > ˙ θ ˙ φ cos ( θ − φ ) (36)It is useful to define the following parameters A = M < R > + M R (37a) B = M < R > (37b) C = M R < R > (37c) α = θ − φ (37d)With these definitions, the kinetic energy simplifies to KE = 12 A ˙ θ + 12 B ˙ φ + C ˙ θ ˙ φ cos α (38)The values of the parameters in A , B , and C which areused in subsequent calculations in this paper are listedin Appendix A. B. Potential Energy
We will want to apply some external torques to thesystem. These torques are better described as force cou-ples [20], where a force couple K can be thought of as thetorque obtained by two forces, equal in magnitude F butopposite in direction, acting at two different points sepa-rated by a distance d . The net force is zero, so the coupledoes not accelerate the center of mass. The net torqueis K = F d , and results in rotation about the center ofmass.Assume a couple of constant magnitude K θ is appliedat the hub and has the orientation such that it increasesthe angle θ . The potential energy for this couple is P E = − K θ θ (39) Similarly, another couple, K α is applied at the hinge be-tween the two arms. It is applied such that it increasesthe angle φ relative to θ , and thus decreases θ − φ . Thepotential energy associated with a constant version ofthis couple is then P E = K α ( θ − φ ) (40)As will be shown below, K α corresponds to the couplereported by MacKenzie [2].The total potential energy is P E = − K θ θ + K α ( θ − φ ) (41)The resulting Lagrangian L is L = 12 A ˙ θ + 12 B ˙ φ + C ˙ θ ˙ φ cos α + K θ θ − K α ( θ − φ ) (42) C. Equations of Motion
The Lagrangian is of the form L ( x i , ˙ x i ), where i rangesover the independent coordinates, in this case θ and φ .The associated equation of motion for each coordinate isgiven by [19], ddt δLδ ˙ x i − δLδx i = 0 (43)The equation of motion associated with θ is A ¨ θ + C ¨ φ cos ( θ − φ ) + C ˙ φ sin ( θ − φ ) = K θ − K α (44)Similarly, the equation of motion associated with φ is B ¨ φ + C ¨ θ cos ( θ − φ ) − C ˙ θ sin ( θ − φ ) = K α (45)These are the two equations of motion which govern themotion of the double pendulum subject to couples K θ and K α . Given initial conditions ( θ , ˙ θ ) and ( φ , ˙ φ ),and the couples K θ and K α , the equations of motion canbe solved for θ ( t ) and φ ( t ). D. Solving for Couples
Consider the situation at impact. The values ( θ , ˙ θ ,¨ θ ) and ( φ , ˙ φ , ¨ φ ) are known from the constraint thatat impact the club moves down the target line at peakspeed on a path with a specified radius of curvature. Inthis section the equations of motion are inverted to solvefor the values of K θ and K α that are consistent with thiscondition.Start with the equations of motion above, now writtenin matrix notation (cid:20) A C cos αC cos α B (cid:21) (cid:20) ¨ θ ¨ φ (cid:21) + (cid:20) C sin α − C sin α (cid:21) (cid:20) ˙ θ ˙ φ (cid:21) = (cid:20) −
10 1 (cid:21) (cid:20) K θ K α (cid:21) (46)Invert the matrix in front of K θ and K α , (cid:20) −
10 1 (cid:21) − = (cid:20) (cid:21) (47)Solve for K θ and K α , (cid:20) K θ K α (cid:21) = (cid:20) (cid:21) (cid:20) A C cos αC cos α B (cid:21) (cid:20) ¨ θ ¨ φ (cid:21) + (cid:20) (cid:21) (cid:20) C sin α − C sin α (cid:21) (cid:20) ˙ θ ˙ φ (cid:21) (48)Multiplying out the matrix equations, K θ = ( A + C cos α )¨ θ + ( B + C cos α ) ¨ φ + C sin α ( − ˙ θ + ˙ φ ) (49) K α = C cos α ¨ θ + B ¨ φ − C sin α ˙ θ (50)Thus, given ( θ , ˙ θ , ¨ θ ) and ( φ , ˙ φ , ¨ φ ), the couples K θ and K α are determined.This formalism has been used to calculate the re-quired couples at points along the target line for var-ious values of ξ . Shown in Fig. 7 are results for ξ =(1 . , . , . , . , . K α whilethe lower graphic shows K θ . Shown in Fig. 8 are resultsfor ξ = (0 . , . , . , . , . θ ≤ φ ≥
0. The dashed lines ex-tend beyond this range and are shown for completeness;however, it is unlikely one would want to implement asolution in these regions. K θ is the primary couple driving ¨ θ , which is deceler-ating into impact. Thus, it is not surprising K θ < K θ depends linearly on our choiceof the inertial moment of the proximal arm of the pen-dulum. In these numerical experiments, that value waschosen by fiat and is not based on a biomechanical model.Thus, the absolute scale of K θ is not meaningful.There is likely some surprise that K α is negative, as itwas discussed above that ¨ φ >
0. The reason K α < K α in these calculations should be close to what isobserved in experiment, as the inertial properties of thedistal arm of the double pendulum are based on thoseof a golf club. Note that in all realizable cases K α isnegative with magnitude of order tens of N m. This isconsistent with the experiments of MacKenzie [2], and isthe central point of this paper. y/y max −70−60−50−40−30−20−100 K α ( N - m ) ξ = 1.4ξ = 1.3ξ = 1.2ξ = 1.1ξ = 1.00.00.20.40.60.81.0 y/y max −2500−2000−1500−1000−5000 K θ ( N - m ) ξ = 1.0ξ = 1.1ξ = 1.2ξ = 1.3ξ = 1.4 FIG. 7. Couples K α and K θ for points along the target linewith ξ ≥ K α is robustly negative, of magnitude -50 N m.As the radius of curvature decreases, the magntude of negativeforce couple K α gets smaller. y/y max −70−60−50−40−30−20−100 K α ( N - m ) ξ = 1.0ξ = 0.9ξ = 0.8ξ = 0.7ξ = 0.60.00.20.40.60.81.0 y/y max −1500−1000−500050010001500 K θ ( N - m ) ξ = 0.6ξ = 0.7ξ = 0.8ξ = 0.9ξ = 1.0 FIG. 8. Couples K α and K θ for points along the target linewith ξ ≤
1. As the radius of curvature decreases, the magn-tude of negative force couple K α gets smaller. E. Whence art thou, K α < (I) One can get a sense for why the K α < B ¨ φ = − C ¨ θ cos α + C ˙ θ sin α + K α (51)This is the equation of motion for the rotation of the clubin the non-inertial frame of the handle of the club (i.e.at the hinge between the proximal and distal arms of thedouble pendulum). The parameter B is the moment ofinertia of the club about the handle. What follows onthe right hand side are the various torques which driverotation about the handle, in the frame of reference ofthe handle. Because ¨ φ >
0, the total torque on the clubis positive.The first two terms on the left hand side are fictitiousforces due to the fact the position of the handle definesthe origin of a non-inertial reference frame. The firstterm is the torque due to the Euler force associated withthe linear acceleration of the handle, acting through thecenter of mass of the golf club. The second term is thecentrifugal force associated with the rotation of the han-dle about the hub, acting through the center of mass ofthe golf club. The final term is the couple K α .The four terms in this equation of motion are shown inFig. 9 for the case ξ = 1, for points along the target line.The solid black line is B ¨ φ , the solid red line is − C ¨ θ cos α ,the solid green line is C ˙ θ sin α , the solid blue line is K α , and the black open circles are the sum of the terms C ¨ θ cos α , C ˙ θ sin α , and K α . Near to y = 0, the fictitiouscentrifugal torque dominates the release (i.e. the greencurve), while closer to y = y max the torque is dominatedby the fictitious Euler force (i.e. the red curve). In allcases the sum of the Euler and centrifugal torques arelarger than B ¨ φ (i.e. the black curve). Thus, to achievethe requisite motion of the club one must include thecouple K α < VII. CALCULATING THE CLUB PATH USINGLAGRANGIAN DYNAMICS
In this section the equations of motion for the doublependulum are used to calculate the motion of the dou-ble pendulum near to impact. The initial conditions areobtained from the considerations of the previous sectionswith ξ = 1 and y = 0 . y max . The couples K θ and K α are assumed constant over the range of motion, and setequal to the values required at impact from the consid-erations above. The value of K α is 43 . y/y max t o r q u e ( N - m ) B ̈ϕ-C ̈θcos(β)C ̇θ ṡn(β)K α -C ̈θcos(β) ̈ C ̇θ ṡn(β) ̈ K α FIG. 9. Detail of the terms in the equation of motion for therotation of the club in the frame of reference of the handleof the club, for the case ξ =1. The varous curves in graphiccorrespond to terms defined in Eq.(51). The black curve is thetorque required to keep the club head moving on the specifiedradius of curvature. The red and green curves indicate thetorques associated with the fictitious Euler and centrifugalforces due to the acceleration of the non-inertial referenceframe. The sum of these two torques is always larger thanthe that of the black curve. To moderate these two forces, anegative couple is applied. This is shown as the blue curve.The sum of the red, blue, and green curves is shown as theblack open circles, and is equal to the black line. the center of mass of the golf club. The hinge betweenthe proximal and distal arms is marked as small blackcircles. The point of impact is marked as a larger blackcircle on the target line.Fig. 11 zooms in on the target line so as to show thatthe distal end of the double pendulum does in fact travelreasonably straight down the target line at impact, on apath which has some curvature. It was verified the radiusof curvature equals R + R − δ , consistent with ξ = 1.The club head is moving at 100 mph (44.7 m/s) at im-pact. The calculation is done with step sizes of 0 . . . −1.00−0.75−0.50−0.250.000.250.500.751.00 y (meters) x ( m e t e r s ) FIG. 10. Calculated path of the double pendulum near toimpact for the parameters ξ = 1 and y = 0 . y max . Thecouples K θ and K α are assumed constant over the range ofmotion, and chosen to match the initial conditions. move the center of mass. This inverse dynamics calcu-lation is meant to enable comparision with the inversedynamics anaysis of golf club motion, as implemented byMacKenzie [2], Kwon [10] and Nesbit [11]. The forcesare depicted in Fig. 12, shown as arrows acting at thehandle. Note that they all point in the general directionof the hub, which is consistent with the measurements ofMacKenzie [2]. The scale is not indicated in the figure,but is of order 260 N.The forces obtained using inverse dynamics can becompared with theory. The double pendulum imposesconstraints on the motion of the proximal and distalarms, such as the fixed pivot around which the proxi-mal arm rotates and the hinged connection between twoarms. These constraints result in forces that constrainthe motion of the system, but are not explicit in the La-grangian. The implicit force due to constraints actingon the club can be described as the sum of four terms.The first two terms originate from the dynamics of theproximal arm. They look as if the center of mass of thedistal arm were located at the hinge, F ¨ θ = (cid:0) − sin θ ˆ x + cos θ ˆ y (cid:1) M R ¨ θ (52) F ˙ θ = (cid:0) − cos θ ˆ x − sin θ ˆ y (cid:1) M R ˙ θ (53)The second two terms involve the dynamics of the distalarm −0.10.00.10.20.30.40.5 y (meters) x ( m e t e r s ) FIG. 11. Zoomed in view of the calculated path of the doublependulum near to impact. The time between increments is0 . F ¨ φ = (cid:0) − sin φ ˆ x + cos φ ˆ y (cid:1) M < R > ¨ φ (54) F ˙ φ = (cid:0) − cos φ ˆ x − sin φ ˆ y (cid:1) M < R > ˙ φ (55)The sum of the x-components and y-components ofthese forces are shown in Fig. 13 in comparison with theforces obtained from inverse dynamics. The solid linesare calculated from the theoretical expressions, above.The open circles are obtained from the inverse dynam-ics. Indeed, the inverse dynamics recover the theoreticalanswer. A. Whence art thou, K α < (II) The results of the previous section allow us to calcu-late the torques on the club about the center of massof the club in the reference frame of the center of massof the club. While the position of the center of massdefines the origin of a non-inertial reference frame, thefictitious forces associated with the acceleration of thisreference frame act through the center of mass and thusyield no torque because the moment arm is zero. Thisis why the center of mass reference frame is always a2 −1.00−0.75−0.50−0.250.000.250.500.751.00 y (meters) x ( m e t e r s ) FIG. 12. The forces that move the center of mass of thegolf club. The forces are obtained from the inverse dynamicsanalysis. The forces are shown as arrows being applied at thehandle of the club (i.e at the hinge between the proximal anddistal arms of the double pendulum). As is shown, they are alloriented in the general direction of the hub. The magnitudeof the force at impact is 260 N particularly convenient frame of reference from which tocalculate torques [21].There are only two torques which are relevant. Thefirst is the torque generated by the linear force that movethe center of mass, detailed in Eqs. (52) - (55). MacKen-zie refers to this torque as the moment of force, and isindicated here as M α . The other torque is the couple K α . Combined, these two torques must equal the totaltorque which rotates the club, I cm ¨ φ = T α = M α + K α .These torques are shown in Fig. 14 as the club movesthrough impact, from the simulation above. The solidred line is M α . The blue line is K α = 43 . T α = I cm ¨ φ . The open black circles arecalculated as the sum M α + K α . This analysis confirms T α = M α + K α .Once again, we see that while the total torque on theclub T α >
0, the couple K α has to be negative becausethe other torque in the problem M α would otherwise pro-vide more torque than what is required to move the clubhead on the path defined by the radius of curvature.It is interesting to point out that the value for K α wasset by balancing torques in the non-inertial frame of ref-erence of the handle of the club. In this section the anal-ysis was done in the non-inertial frame of reference of thecenter of mass of the club. In both cases, the couple K α has the same value. This serves to emphasize that if you −0.50.00.51.01.5 y/y max (meters) −250−200−150−100−500 f o r c e ( N ) F y inv_dynF y theoryF x inv_dynF x theory FIG. 13. The components F x and F y of the forces movingthe center of mass of the golf club. The solid lines are calcu-lated from theory, as described in the text. The open circlesare obtained from the inverse dynamics analysis. The forcesobtained from the inverse dynamics analysis are shown to re-cover the forces calculated from theory. solve for the forces and torques which move a rigid bodyin multiple reference frames, even non-inertial referenceframes, you should always recover the same answer. VIII. SEEKING A MATCH TO MACKENZIE’SDATA
The majority of MacKenzie’s video ‘In-Plane Coupleand Moment of Force During the Golf Swing’ [2] high-lights the golf swing of a single golfer. For this golfer inthe last frame before impact, the club head is moving at116.5 mph and the measured values of force and torquesare force F = 456 N, moment of force M = 55 . K = − . δ , ξ , and y , in an attempt to find thebest fit to F , M and K . All other parameters in theproblem, such as the length of arms of the double pen-dulum and the inertial properties of the golf club, are asdefined in Appendix A. As such they are just approxima-tions to what may have been used in the experiments ofMacKenzie.The result of this search is summarized in the chartsof Fig. 15. The different panels correspond to differentvalues of δ , ranging from 2 to 5 inches. The abscissacorresponds to different values of y , ranging from 0 to3 −0.50.00.51.01.5 y/y max (meters) −40−200204060 t o r q u e ( N - m ) M α T α K α M α + K α FIG. 14. Accounting of the torques in the frame of referenceof the center of mass of the golf club. The solid red line is M α . The blue line is K α = -43 . T α = I cm ¨ φ . The open black circles are calculated as the sum M α + K α . This analysis confirms T α = M α + K α . This resultis the central point to the paper: M α by itself is larger thanthe requried torque T α . To compensate for this, the torque K α <
15 inches. The ordinate corresponds to different valuesof ξ in the range 0.6-1.4. The color scale encodes thedifference between MacKenzie’s data ( F , M , and K )and the result obtained from the model ( F , M , and K ).This difference is calculated as the sum-of-squares aver-age fractional error E , E = 13 (cid:0) ( F − F F ) + ( m − m m ) + ( K − K K ) (cid:1) . (56)This treats the parameters F, M , and K as if they wereindependent. To this end, m = M/F , and is thus onlysensitive to the angle between F and the shaft of theclub. To accommodate the dynamic range, the color scaleencodes log ( E ).The value of δ is given in the top left corner of eachpanel. The minimum value of E is indicated in the topright corner of each panel. The values F , M , and K atthe minimum are listed at the top of each panel. It is aprimary result of this paper that the double pendulummodel of the golf swing is able to obtain the force andtorques reported by MacKenzie to within a few percent.It is possible the scale of these differences are consistentwith the instrumental noise in MacKenzie’s experiments. These data show that one can use the same set of forcesand torques F , M , and K to hit the ball standing differ-ent distances from the ball, δ , and from different positionsin the stance, y . As the ball is moved further forward inthe stance, the golfer must stand closer to the ball andthe radius of curvature of the club head path becomessmaller. IX. SPECULATION ABOUT HOW K α ISGENERATED
The scale of K α is of order 50 N-m. What can generatea couple of this magnitude? This section explores threepossibilities.It is important to remember this particular torque isa force couple. It can be thought of as being generatedby two linear force vectors, equal in magnitude F K butopposite in direction, separated through a distance d .Because the linear sum of the forces is zero, there is nonet force on the center of mass of the object due to thetwo force vectors. However, because they are separatedthrough the distance d , they yield a torque of magnitude d F K perpendicular to the plane defined by the two forcevectors, and thus generate rotation. A. The Hands
Suppose this couple is generated by forces applied bythe hands. This could be either because the hands areactively applying force, or because the hands can not keepup with the linear and/or rotational speeds at impact.For a right handed golfer, imagine the left hand ap-plying a force in the direction of motion of the club, andthe right hand applying a force of equal magnitude inthe opposite direction (i.e. opposing the motion of theclub). Suppose the distance from the pinky finger of theleft hand to the forefinger of the right hand when a righthanded golfer grips the club is 1/6 meter (i.e. 6-7 inches)and is the distance through which the couple is applied.Then to generate a couple applying 50 N m of torque,each hand would have to be applying 300 N of force inopposite directions. This is in addition to the hundredsof Newtons of linear force already discussed above, whichis presumably split between the two hands. 300 N of forceamounts to 70 lbs force. That seems like a lot of force foreach hand to be applying. For this reason, it would seemthat this explanation alone is insufficient to provide allof K α .However, it is important to note that this negativecouple is applied only 10-20 ms before impact. Thus,the resulting impulse (i.e. torque multiplied by time) isnot particularly large. If this torque were due to thefact the hands can not keep up with the release of theclub, it might be difficult for the golfer to perceive thisapplied torque. It would be quite spectacular if golfershave learned to harness this natural drag to help themto hit the ball straighter.4 R c / ( R + R − d ) δ = 2.0 in E.min = 3.9%@min, F=435.9N, M=50.8Nm, K=-60.2Nm δ = 3.0 in E.min = 5.4%@min, F=424.3N, M=54.4Nm, K=-61.5Nm R c / ( R + R − d ) δ = 4.0 in E.min = 5.0%@min, F=440.4N, M=49.7Nm, K=-60.3Nm δ = 5.0 in E.min = 3.4%@min, F=432.4N, M=53.8Nm, K=-60.4Nm −1 E FIG. 15. The results of a search over a grid of parameters in an attempt to match to force F and torques, M and K , reportedby MacKenzie. The the color scale encodes log ( E ), where E is the average fractional error, as described in the text. Thevalue of δ is given in the top left corner of each panel. The minimum error is indicated in the top right corner of each panel.The values F , M , and K at the minimum are listed at the top of each panel. The values to which they are being fit are F =456 N, M = 55 . K = − . B. Aerodynamic Drag of the Club head
Another possible source of negative couple is the aero-dynamic drag on the club head as it approaches impact.Imagine the size of that force is F d in the direction op-posing the motion of the club head. Now imagine thatthe hands apply a force of equal magnitude but in theopposite direction, counter acting this drag. The separa-tion between these two forces is the length of the club, ǫ = R . Henrikson reports [22] the scale of the drag forceto be 4.5 - 7.5 N. If we use 10 N as an upper limit, andassume a club of length 1 m, then this can yield a coupleof order 10 N m. Again, this is too small to give valuesas large as 50 N m. C. Inertia of the squaring of the club face
Missing from the model of the double pendulum is thefact that the club face goes from open to square to closedas the club moves through impact. This requires rotationof the club around the long axis of the shaft. It also re- quires the rotation of the arms and hands, which supportthe club. This motion is related to the β -torques and γ -torques described in Nesbit’s 2005 paper [11], whichinvolve motion out of the swing plane and about the axisof the shaft, respectively.For our purposes, consider that the motion caused bythe β -torque and γ -torque is coupled to the release ofthe hands, defined in this paper as the angle α = θ − φ .It is certainly the case that the club face is open when α ≈ π/
2, it is square near to impact where α ≈
0, andclosed after impact, when α ends at − π/ β -torques and γ -torques causes mo-tion that affects the moment of inertia relevant to themotion in the plane of the golf swing. This could involvethe relative positions of the arms and hands, the rota-tion of the club around its axis, motion of mass aboveand below the swing plane, etc.Further make the generalization that the kinetic energyassociated with the squaring of the club manifests itselfin the swing plane as KE s and that this can be parame-terized in terms of the angular speed ˙ α and a moment ofinertia I s ,5 KE s = 12 I s ˙ α (57)As long as we are only solving the double pendulum inthe vicinity of impact, this additional term can then beincluded in the Lagrangian of this paper (i.e. not makinggeneralizations beyond the immediate vicinity of impact).With this addition, the equations of motion become A ¨ θ + C ¨ φ cos ( θ − φ )+ C ˙ φ sin ( θ − φ ) = K θ − K α + I s ( ¨ φ − ¨ θ )(58) B ¨ φ + C ¨ θ cos ( θ − φ ) − C ˙ θ sin ( θ − φ ) = K α − I s ( ¨ φ − ¨ θ )(59)As has been shown above, ¨ φ > θ <
0, so the term − I s ( ¨ φ − ¨ θ ) functions as a negative torque.In the exercises above, K α was assumed to providethe full negative couple required to keep the club movingstraight down the line. For arguments sake, lets assumehere that all of the negative couple comes from I s . Evalu-ating the example above at impact, ( ¨ φ − ¨ θ ) ≈
500 rad s − which suggests I s ≈ . . We can compare this withthe value of the moment of inertial of the golf club aboutits handle, I R = 0 .
24 kg m used in this paper. Thus, I s needs to be of order 40% of the size of I R , which wouldbe a large perturbation. While this seems like a logicalavenue for the biomechanics community to explore, it ispossible it will not be large enough to explain all of K α . D. Speculation Summary
This section has explored three physical processes thatcould generate K α ≈ −
50 N m. Each one of them indi-vidually seems too small to provide a torque of sufficientmagnitude. Thus, instead of there being one clean sourceof K α , it seems likely the actual answer involves multipleterms, or phenomena not considered in this paper. X. SUMMARY
Motivated by MacKenzie’s observation of a negativecouple near to impact [1, 2], this paper has explored amodel for how the golf club moves near to impact. Itassumes the club is moving as the distal arm of a doublependulum and that at impact the club head is movingstraight down the target line, at its maximum speed, ona path of defined curvature. From this model, the forcesand torques required to move the club near to impact arecalculated.The results obtained from this model are shown to bequantitatively consistent with data reported by Macken-zie to within a few percent. Indeed, the negative couplenear to impact is found to be a robust feature of this model. It balances torques resulting from the forces thatdrive the center of mass of the golf club. These torquesreduce the radius of curvature of the path of the clubhead as it moves through impact. By applying a nega-tive couple the golfer is able to achieve a larger radiusof curvature. This reduces the difference between thepath of the club head and the target line as the clubhead moves near to impact. Because the negative couplecan also serve to reduce the rotational speed of the club,its presence in the golf swing manifests a trade betweendistance and direction.
Appendix A: Model Parameters
The properties of the golf club were taken from Nesbit[12], for consistency. They are: • R = 1 .
092 m, the length of the golf club in meters.Presumably measured from a place between the twohands to the middle of the club face. • M = 0 .
382 kg, the mass of the golf club. • < R > = 0 .
661 m, the first moment, which is thedistance from the hands to the center of mass ofthe club. • I ,CM = M < ( R − < R > ) > = 0 .
071 kg m ,the moment of inertia of the golf club measuredabout it’s center of mass.The properties of the proximal arm of the double pen-dulum were picked by fiat, and are not based on anybiomechanical model. • R = 0 . R , the length of the proximal arm of thedouble pendulum. This number is not based onany detailed measurement. It is meant to be a verycrude approximation. • M < ( R − < R > ) > = 3 ∗ I ,CM , the momentof inertia of the proximal arm of the double pen-dulum about the fixed hub. This number is just astab in the dark. Its only relevance is to scale themagnitude of K θ .The distance δ is taken to be 7 .
84 cm, which is justabout 3.1 inches. This was chosen so that R + R − δ =70 in. Again, there is no particular reason for this choiceother than it made the length of the accessible pointsalong the target line of order 12 in.6 [1] S. MacKenzie, M. McCourt, and L. Champoux, Inter-national Journal of Golf Science (2020).[2] S. MacKenzie, Vimeo (https://vimeo.com/158856998)(2016).[3] P. Cheethan, G. Rose, R. Hinrichs, R. Neal, R. Mottram,P. Hurrion, and P. Vint, in Science and Golf V: Proceed-ings of the World Scientific Congress of Golf , 520 (2005). [12] S. M. Nesbit and R. S. McGinnis, Journal of Sports Sci-ence and Medicine , 235 (2009).[13] S. M. Nesbit and R. S. McGinnis, Journal of Sports Sci-ence and Medicine , 859 (2014).[14] A. Cochran and J. Stobbs, Search for the Perfect Swing (The Golf Society of Great Britan, 1968).[15] T. Jorgensen, American Journal of Physics , 644(1970).[16] T. Jorgensen, The Physics of Golf
CalculusMultivariable , 9th ed. (John Wiley & Sons, Inc., Hobo-ken, NY 07030, 2009) Chap. 12.[19] H. Goldstein,
Classical Mechanics , 2nd ed. (Addison-Wesley, Reading, Ma., 1981).[20] K. Symon,
Mechanics , 3rd ed. (Addison-Wesley, Reading,Ma., 1971).[21] R. Feynman, R. Leighton, and M. Sands,
The FeynmanLectures on Physics , Vol. I (Addison-Wesley, Reading,Ma., 1977) Chap. 19.[22] E. Henrikson, P. Wood, and J. Hart, Procedia Engineer-ing72