Slippery Stuff

The Physics of Skating

This is not a guide on how to skate, but rather an investigation into how skating works.

My goal in skating was only partially to learn how to skate.  I was also profoundly curious to learn why skating works at all, what physical laws serve as one's allies in the fight to stay upright, or nearly so, on a slippery surface.  What I would like to describe here is some of the simple yet subtle physics which underlies the elegance of skating well.

Like Newton, whose understanding of gravity is said to have been inspired by having an apple fall upon his head, I have also found that mistakes and accidents are often most useful in illuminating the whys of the world.  If you want to know how, just do as you are told.  If you want to know why, it is best to learn by trying to do otherwise.  Science, both etymologically, and in the popular imagination, is the business of knowing the truth.  In practice, however, it is more a process of demonstrating why and how one's current understanding is false.

And so, without further belaboring the introduction, here are my insights on why skating works.  The accumulation of these insights into the "why" of skating has not always been particularly helpful in learning the "how".  Nevertheless, I hope they will illuminate the internal beauty of skating, and inspire some moments of contemplation out on the ice, where it all comes together.  For me, at least, the question of why skating works has become an integral part of why I am obsessed with actually doing it.

Bipeds on Ice - some basic considerations

For a biped who has evolved through millions of years walking on terra firma, a smooth ice sheet offers definite challenges, but as any skater knows, certain rewards.  Boys soon learn the possibilities of reduced friction offered by a frozen puddle in the school yard.  But after a short run, an exhilarating slide, and a few pratfalls, recess is over, and the possibilities of unassisted locomotion on a frictionless surface pretty well exhausted.

The kind of walking we are used to on solid ground requires that a foot stay planted reassuringly just where it is put down.  One can count on each footfall providing a firm foundation, from which we can move on, over and beyond it, pushing off for the next step.  To turn a corner, one plants one's next step to the outside of the desired curve, and pushes off towards the inside.  One can even twist one's body about a single planted foot, and turn to face one's altered direction.  Normal footwear on ice violates such ground rules in a dramatic fashion: nothing at all is fixed in the foundation here, be warned.  Push off very delicately indeed.  Skates, however, offer fascinating possibilities.

The sharp steel edge of a skate cuts into a smooth ice surface sufficiently to provide ample resistance to lateral motion,  and yet slides easily along its track, almost without friction.  The physical mechanisms which make ice slippery have been discussed extensively in the literature without complete consensus.  There is general agreement that a thin film of liquid water is instrumental, although the extent to which this film is produced by frictional heating, or the inherent surface properties of the ice, is not entirely clear.  Two observations will suffice for almost all that follows:
The beautiful simplicity and contrast between these two ideal rules contribute greatly both to the potential elegance of skating itself, and to the corresponding elegance of its physical analysis.

From the practical point of view of a biped interested in negotiating a smooth sheet of ice, skates offer a wonderful opportunity.  Laterally they offer the customary traction of terra firma.  Longitudinally they offer a magic release from the bonds of friction.  A consequence is the possibility of achieving the sensation of flight in two dimensions.

Longitudinal Stability - the impossible lightness of being

Sudden liberation from the bonds of friction creates problems for the neophyte, until freedom becomes a habit and a coat worn lightly.  When walking on terra firma, the ultimate source of longitudinal stability resides in forward velocity control.  If we want to move forward, we simply start leaning in that direction, and then step forward to prevent ourselves from falling on our face.  Once moving forward, we essentially regain the vertical, but with each step we constantly make corrections.  Either we delay our pushoff from the hind foot for an instant, to accelerate and increase forward lean, or we come up sharper on our leading foot to lean back a bit and put on the brakes.

On the ice, the rules change, and longitudinal stability, even when we are in motion, takes on the character of the stability mechanisms we are used to exercise on solid ground while standing still.  When standing, if something shifts our center of gravity forward, gravity, acting at that point, creates a torque about our feet which then increases our forward lean.   As we feel ourselves leaning forward, we then increase the pressure on our toes.  This creates a torque about the balls of our feet which opposes the one created by gravity.

In an intriguing subtlety, skate blades are usually slightly curved long their length.  The standard radius of curvature near the center of figure skate blades is seven feet.  Hockey skates, although shorter, have even larger radii.  Therefore without making any muscular correction, a forward lean causes us to rock forward on the curved blades.  Since their radius of curvature is larger than the distance from the ice to our center of gravity, the point of contact with the ice moves forward further than the initial center of gravity displacement which created the lean.  A natural stability is therefore assured.  As long as we keep our center of gravity within the length of the blade, we are simply going to stay upright like one of those tippy toy clowns with a weighted rounded base, which simply refuse to be knocked over.

The fact that beginning skaters can, and frequently do fall longitudinally, particularly onto their bottoms, is clear evidence that the longitudinal stability offered by normal skates is not absolute.  At least a modest muscular effort is needed to maintain the center of gravity within the length of the skate blade.  But perhaps the chief problem for beginners is that they attempt strategies they are used to applying when walking on firm ground.  As customarily observed in the classic bottom bruiser, any attempt to "put on the brakes" in order to correct a backwards lean is quite ineffective.  The marvelous freedom from longitudinal friction is almost perfect, absolute, and in this case, unforgiving.

Due to the small coefficient of friction, these considerations of longitudinal stability remain valid even at speed.  An experienced skater can build up momentum with a series of powerful strokes, and then simply stand nonchalantly while zooming down the ice without a care in the world.  The maneuver is particularly impressive when executed backwards, but in fact requires minimal skill, other than a certain level of sangfroid.  The small frictional force experienced by the skates as they slide along the ice does create a torque about the skater's center of gravity which must be countered by the earthwards attraction of gravity through a slight backwards lean.  But since the coefficient of friction is only a few percent, a few centimeters displacement of the point of contact with the ice is sufficient.

Initial Transverse Stability - the two-footed solution

A skater must maintain stability laterally, as well as longitudinally.  Initially this presents the less troublesome problem for the beginner.  Just standing on the ice with both feet side by side provides natural stability, since the center of gravity automatically ends up between the feet, and the bite of the skates' edges into the ice allow the same transverse balancing strategies used on dry land to be instantly adaptable.

Although an attempt to mimic the direct longitudinal push-off so useful in terra firma walking is not going to get a skater very far, the rudiments of forward motion on the ice can be achieved through adapting the "duck walk" strategy long favored by toddlers.  With the feet well apart, the toes angled outward, forward motion is possible basically by pushing off diagonally, alternating from one foot to the other.  With a bit of practice, the efficiency of the basic skating stroke improves, the increased velocity of forward motion narrows the opening angle of the diagonal V's, the skates glide further on each step, and the basic skating gait is acquired.

Throughout these initial attempts at locomotion, lateral stability continues to be ensured by always keeping the center of gravity well between the feet, and keeping both feet on the ice a good fraction of the time.  The lateral component of the thrust used in shifting from one foot to the other is kept modest enough that the center of gravity rocks over towards the new skating foot, but never with enough energy to go over the top, and pass beyond the zone of safety which exists when it lies between the two feet.  Each stride ends reassuringly, with a gentle fall back onto the initiating foot.

The Zen of One Foot Skating - achieving stability on a single blade

If this were as far as things could go, skating would remain a rather pedestrian activity, too similar to walking on solid ground.  On terra firma, a foot, once planted, stays just where is both laterally and longitudinally for the entire duration of the step.  But the key to the next stage of locomotion on ice is that a sliding skate, once placed on the ice, is free to move wherever the skate blade should guide it.  This near-magical ability is the key to taking lateral stability on the ice to the next level.  If it should happen that one's center of gravity moves beyond the zone of safety, to the outside of the skating foot, that foot can be turned outwards so that the forward motion of the skate then brings the center of gravity back between the two feet again.  In fact, the ability to steer the foot laterally allows stability to be maintained on a single blade, by actively steering it to remain under the center of gravity.

In this sense skating is like running along, balancing a vertical bamboo pole in the palm of your hand.  If one finds the pole falling to the right, one can steer the butt end back under the center of gravity and recover equilibrium.

This mode of balancing laterally is shared by many rolling and sliding sports, such a riding a bicycle or scooter, skiing, snowboarding, or skateboarding.  As such it provides a universal attraction for a biped used to being trapped in the pedestrian dynamics of normal walking.  In ice skating, this coupling between forward motion and the lateral position of foot or wheel achieves perhaps its purest expression.  Lack of friction leads to an effortless maintenance of forward velocity, while the shortness of the skate's blades leads to a very agile steering system, which allows an exceptionally rapid lateral transfer of the skating foot.

The most basic use of this lateral control allows development of the basic mature skating stride, where the push-off on each stroke becomes strong enough to carry the center of gravity to the outside of the skating foot, which then makes an outward arc.  This arc first serves to regain balance, and then, carried a bit further, shifts the center of gravity back to the inside of the skating foot, allowing a strong push-off from the inside edge to commence the next stroke.  Other modes also become possible.  Simulation of a skier's wedel is particularly satisfying, as the skate traces a sinuous curve from side to side, as the body hangs almost stationary in the middle.

Life on the Edge - locking into synchrony

Unlike a skier, who must always face the reality of the fall line, a skater lives on an ice sheet of unbroken rotational symmetry.  On the ice, the possibility therefore arises to begin an arc and simply continue it in an indefinite orbit, round and round, limited only by an almost vanishingly small coefficient of friction.  But unlike a planet spinning through the vacuum of space, the orbit of a skater must follow the direction set by his blade.  Unless he wishes to end twisted up like a crueiller, he must rotate once about his own axis for each complete circle of his orbit.  Like the moon, locked in synchronous orbit around the earth, he must maintain one side pointing to the center.  His month must correspond to his day.

In practice, skaters rarely perform even single complete orbits in such equilibrium, let alone orbits of many revolutions, and their motion is usually some superposition of the skier's wedel, where the body maintains a fixed orientation as his skates weave a sinuous path beneath him, and the moon's ever center-facing swoon about the earth.

 This dicussion brings up the essential concept of "edge".  A figure skater is almost always "on edge", meaning that he is skating on a single blade which traces an arc on the ice.  His body leans to the inside of the curve so that his weight balances the centrifugal force generated by his trajectory. Although a bend at the ankle often modifies the blade angle slightly, his skate naturally follows the inclination of his body, so that a single edge, on the inside of the arc, cuts into the ice.  In equilibrium,  the curvature of this arc remains fixed.  centrifugal forceCalculating torques about the horizontal pivot axis which the length of the blade makes on the ice, gravity, acting downward at the center of mass, creates a torque which would increase the skater's lean, while centrifugal force, also acting at the skater's center of mass, creates an outward horizontal force which opposes and just balances the gravitational torque.

equilibrium edgeAs the skater proceeds to orbit in equlibrium around a circle, his body must also spin about its own axis once per revolution.  His skate then naturally remains tangential to the circle as it rotates around the circumference.  Although it takes a bit of skill to initiate an edge with such matching rates of orbital and spin rotations, it soon becomes second nature.  An experienced skater has the impression that once on edge, no active corrections are required, and he could easily read a newspaper as he continues round and round on his trajectory, forever.

A figure skate, with its curved blade, has a length of contact with the ice of only a few cm.  Its "track" along the ice is therefore not inherently determined, like that of a ski in snow, and can be easily diverted by a slight counter rotation of the body.  How is this inherent lability to be reconciled with the undeniable feeling which a skater has of being "in the groove" as he orbits around on his edge, reading his newspaper?

 The restoring force which creates this stability is created by the lateral displacement of the skater's center of gravity away from the vertical, towards the inside of the arc.  In equilibrium, a line drawn on the ice from the point of contact of the skate, perpendicular to the skate blade, passes directly under the skater's center of gravity.  The centrifugal force acts exactly along this line, and therefore creates no torque about a vertical axis passing through the point of skate-ice contact.  equilibrium edgeIf, however, some external influence were to disturb this situation, say by causing the skate to twist toward the outside of the arc, the line perpendicular to the skate blade would then move slightly ahead of the skater's center of mass.  The centrifugal force then does create a torque about the point of contact, and its direction acts to twist the skate blade outwards, opposing the displacement which created it.

This torque is in fact the only external torque which acts upon a skater when only a single skate is on the ice.  Without this mechanism he would simply be unable to change his body's rate of rotation about its own axis.   Just as it provides the essential mechanism of stability for staying on-edge, it also provides the only mechanism for initiating rotational change.  It is an essential element of skating.  Let us call it "edge torque".

An essential prediction of this model is that the edge torque grows stronger as the skater's lean, and hence the lever arm between his center of gravity and his skate, increases.  This corresponds well to the clear sense of greater stability felt by a skater on a deep edge.

Thinking ahead - active trajectory control

Although this "edge torque"mechanism couples spin and orbital angular momenta, it does nothing to couple angle of lean, velocity, and radius of trajectory.  In other words, we can be "in the groove", spinning in synchrony with our orbital rotation, but be moving too fast or too slowly along our arc, so that centrifugal force is out of balance with gravity, and our lean angle accelerates towards the horizontal.

In this respect, there is no inherent stability mechanism to aid us, and we are on our own to actively avoid an eventual lie-down on the ice.  The crucial variable which can prevent this uncomfortable end-state is controlling the curvature of the trajectory.  If one finds oneself falling over,  the cure is to twist the blade into a tighter curve, thus increasing the centrifugal force, countering gravitation, and recovering equilibrium.  The blade can be twisted by counter-rotation of the hips, upper body, arms, or a swing of the free leg.

The basic mechanism of edge stability is the only way a skater generates net torque while riding a single edge, but by counter-rotating parts of his body, a skater can steer his skate's trajectory out of equilibrium, and thus control the direction and magnitude of this edge torque.  The trick is that one cannot shift the edge torque instantaneously.  As in solving a Rubik's cube, the order of rotations is important.  One must think ahead, and follow an indirect path.

Suppose one is on a right inside edge, arcing to the left, in near equilibrium, and wishes to turn forcefully to the right, without change of skating leg?  The strategy is to first counter-rotate to the right (clock-wise), thereby twisting the skate to the left.  This decreases the radius of curvature of the leftward arc, bringing his center of gravity up over the skate trace, while simultaneously generating a clock-wise edge torque.  (Note that it is exactly the net clockwise torque which the skater needs to get his body rotating in the correct manner to be in equilibrium with his intended change from left-ward to right-ward orbit.)

If left to itself, this edge torque would tend to simply restore the skater's original trajectory.  But the skater, thinking ahead, does not leave things to themselves.  After allowing the edge torque to act for a while, he reverses the counter-rotation which commenced the manouver, at about the moment that his center of gravity comes over center.  Voila, change of edge.  He settles into a right outside edge, arcing to the right.

An inexperienced skater is invariably uncomfortable falling rapidly sideways, far from equlibrium.  It takes confidence to believe that one can still catch the fall with a subtle twist of the hip, confidence which can only be built by pushing the envelope, catching the fall again and again, gradually further and further from equilibrium.

In this connection it is useful to point out that one is rarely in perfect equilibrium.  One is usually either falling, or, having caught the fall, bouncing back to the vertical, often even passing the vertical, and falling to the opposite side.  In fact, just like in riding a bicycle, if one wants to turn to the left, one must first steer a bit to the right.  This initiates a fall to the left, and establishes the requisite lean into the desired leftward curve.

Side-winding on a Sinusoid

As mentioned above, it is possible to execute a wedeling motion on a single skate, steering it back and forth from one side to the other as the body hangs more or less stationary over the center of the sinuous track.  During such a manouver the usual skater's contraint applies, and the force on the skate is always normal to the trajectory, and essentially zero along it.  Although the direction of force is constrained, its magnitude has greater freedom, and can in fact be modulated out of phase with the wiggles of the trajectory.  To avoid wandering off to right or left, the integral of the lateral component must of course be zero.  But it can be arranged that the force is larger during the forward-sloping than it is during the rearward-sloping portions of the sinusoidal track.  This  results in net forward propulsion, using only a single skate.