BALLET IS A BLEND OF BEAUTY and the physics of motion. A man who is interested in both components is Kenneth Laws, professor of physics at Dickinson College and a student of ballet with the Central Pennsylvania Youth Ballet. Much of the following analysis of certain positions and movements of ballet is based on information that he provided me with.

A large part of the early training of a ballet student is aimed at teaching her (or, to be sure, him) to maintain her balance as she moves gracefully through the ballet forms. Balance is achieved when an area of support lies in an appropriate place below the dancer's body. If the support area is off to one side, gravity pulls or turns her toward the floor.

Gravity, of course, pulls constantly on every part of the body, but the concept of a center of mass of the body helps to simplify the picture The center of mass is a mathematical point whose position is determined by the distribution of mass within the body. The combined pull of gravity on all the parts of the body is said to operate through the center of mass.

When that center is not over a support area on the Boor, the pull of gravity creates a torque on the body. A torque is the product of a force (here gravity) and a distance called the lever arm. Lines representing gravity and the lever arm appear in the illustration on the left, which shows a tilting human figure. The torque arising from the tilt tends to rotate the body about the feet and toward the floor. The greater the tilt, the longer the lever arm and the stronger the torque. If the dancer stood upright, the length of the lever arm would be zero and gravity would create no torque; the position would be stable.

The routine for the beginning student seeks to develop a sense of balance gradually For example, she learns the position called the arabesque in a simple first version and is then advanced to the first arabesque penchée. In the simple version (arabesque à terre) she puts her right leg forward and her left leg back with its toes touching the floor, her weight is therefore on the right leg. The right arm reaches forward, the left arm slightly back.

Moving the left leg to the rear shifts the center of mass in that direction. Without a compensating movement the dancer would fall over backward. To shift the center of mass back over the support area of the right foot she must lean forward and extend her right arm. This motion serves two purposes: it is graceful and it enables her to maintain her balance.

In the first arabesque allongée the dancer continues the motion until her torso and right arm are almost horizontal and her rear leg is tilted upward. In the first arabesque penchée her torso and right arm are tilted downward and her rear leg is tilted upward by 45 degrees or more. In both forms of the position the mass moved rearward by the rear leg must be matched by mass shifted forward by means of leaning the torso and extending the arm. Only then will the center of mass remain over the support area so that the dancer is stable.

Some of the fascination of ballet stems from illusions in which physical laws seem to be momentarily suspended. An example is provided by the grand jeté, a forward leap. According to Laws, a properly executed grand jeté suggests that the gravitational pull on the dancer somehow weakens when she is near the top of the leap.

Two things contribute to the illusion. First, some slowing does take place near the top of even an ordinary jump because of the rules governing motion Although the dancer's horizontal speed remains constant throughout the jump, her vertical speed is zero at the peak. Just before and just after that point her vertical movement is slow. As a result for about half of the time required for the entire leap she is within a fourth of the leap's maximum height.

In a grand jeté another element is added to enhance the illusion that the dancer is hovering near the peak. Because of a

certain shift of her arms and legs while she is in the air her path appears to be flattened at the top. The illusion depends on a shift of her center of mass as she moves her arms and legs. She launches herself with her arms downward. As she approaches the top of the leap she lifts and spreads her legs and arms. Her center of mass is then at a higher position in her body. Since the center of mass follows a fixed trajectory, the shift means the head and torso do not rise as far above the floor as they would have risen. In descending the dancer lowers her legs and arms, restoring the center of mass to its usual position.

In jeté en tournant, a turning leap, the dancer jumps into the air with no apparent spin around her vertical axis, yet near the top of the leap she begins to rotate. Impossible. One of the firm rules of physics is that the angular momentum of an object remains constant unless a torque acts on the object. If the dancer was not spinning when she left the floor, she cannot begin to spin in midair.

The explanation of the illusion is that the dancer does have a small spin at the beginning of the leap because of a torque she receives from the floor as she launches herself into the air. The spin is too slight for an observer to notice it. As the dancer rises, however, she pulls in her arms and brings her legs together. The effect is to decrease her moment of inertia. Since her angular momentum is fixed once she is in the air, the decrease in her moment of inertia increases her spin rate. One sees the same thing when a spinning figure skater pulls in her arms and so spins faster.

The dancer begins a jeté en tournant with her feet in what is called the fifth position. Her feet are parallel and pointing to the side in opposite directions with the left foot in front. Each heel is near the toes of the other foot. The dancer slides the left foot to the left and quickly comes into an arabesque. She bends her left knee in what is called the demi-plié position.

Now she quickly brings her right leg downward and forward, simultaneously launching herself with her left leg. In flight she not only rises above the floor but also travels horizontally. At the beginning of the leap her arms and one leg are extended, providing the basis for her subsequent movements to increase her rate of spin.

The dancer lands on her right leg and in demi-plié. She should be facing the audience. With experience she learns to control the leap and her moment of inertia to achieve this orientation.

An equally beautiful but more difficult leap is the grand jeté en tournant entrelacé (also called the grand jeté en tournant or simply the tour jeté). The dancer starts with her right leg in demi-plié and her left leg lifted to the side at an angle of about 45 degrees with the floor. Next she steps backward onto her left leg, bringing her right leg upward and around to the front so that it points to her left just as she turns her body to the left. She jumps from the left leg. Once she is in the air she rotates about an axis that tilts from the vertical. She holds her arms above her head and close to the rotational axis, also bringing her left leg up near her right leg so that they both rotate about the axis. She lands on her right leg and comes into the first arabesque.

The rotation of the dancer in midair is rapid because her moment of inertia is small. As soon as she leaps she brings her

arms and legs into line with the rotational axis. Once she has made her turn she effectively stops her rotation by spreading her arms and extending her right leg for her landing.

Classical dance has many techniques for whirling. One of them is embodied in the soutenu en tournant. Starting with her feet in the fifth position, left foot in front, the dancer goes into a demi-plié with her weight on her left leg and her right leg extended to the side. She brings her right foot back to the fifth position in front of the left foot and rises on her toes, twisting her feet to develop a torque that begins her twirl to the left.

A pirouette is a more ambitious twirl in which the propulsion also comes from torques on the feet. Consider first a quarter-turn pirouette. From the fifth position (right foot in front) the right foot is moved to the side while the arms are brought forward and then spread out to the sides. Next the right hand is moved to the front, the right foot to the rear. Now the dancer pushes against the floor with her right foot to propel herself around to the right. Simultaneously she rises on her left foot, which acts as a pivot.

The rotation brought about by the torque from the right foot is helped by the fact that the dancer draws in her left arm. The movement not only adds grace to the turn but also decreases the moment of inertia, enabling her to do the quarter turn rapidly. After dropping back into the fifth position she can immediately begin another quarter turn.

A complete pirouette differs mainly in the motion of the head and in the strength of the angular acceleration. A larger torque is required for a full turn. At the beginning of the rotation the dancer continues to face straight ahead. When her body has turned about 90 degrees, she suddenly whips her head in the direction of the turn, bringing it sharply to the front again when her body has turned about 270 degrees. Soon thereafter her body is also facing forward once more.

A grand pirouette calls for the dancer to turn with one leg and both arms held horizontally to the side. A mathematical analysis of this pirouette is difficult unless the shape of the human body is simplified. Laws has done so in a model that appears in the illustration at the right. The "body" consists of an upper section of mass M and length L and a two-part leg section. The "shank" and the "foot" have a combined mass of m/3 and a length of L'/2; the "thigh" has the same length but a mass of 2m/3. For a male dancer the ratio M: m is about 3.8 and L' is approximately equal to L.

The upper body is vertical throughout the maneuver. In the model of the pirouette one leg is held horizontally to the side. The supporting leg is rigid and makes an angle theta () with the vertical axis through the supporting foot.

Laws was interested in the value of the angle required for a stable spin around the vertical axis. He first calculated it for a stationary dancer. As I have mentioned, the requirement for stability is that the dancer's center of mass be over the point of support. Since one of her legs is held horizontally to the side, she must tilt her other leg in order to shift mass and so reposition the center of mass. Stability is achieved when the angle between the supporting leg and the vertical is about 4.4 degrees.

Is it the same when the dancer is rotating? I would have thought so, but Laws's calculations indicate that the angle is instead about 3.5 degrees. Rotation imposes additional constraints on stability. The upper body must be closer to the vertical axis and the extended leg must stretch farther from the axis. Hence the dancer's center of mass is slightly off the vertical axis, and yet she is stable.

A more complicated rotation is part of the fouetté turn. (In Act 111 of Swan Lake the Black Swan does 32 consecutive turns.) The turn is essentially a full pirouette except that the source of the torque on the dancer is hidden. She continues to turn as though she were a toy top propelled by magic.

Once the pirouette has been started the right leg is not returned to the floor until the end of the turn. During most of the pirouette the right foot is held near the knee of the supporting left leg. The dancer is on the toes of her left foot. Just as her body begins to face forward again she thrusts her right leg forward and opens her arms to the audience. The heel of the left foot is brought down to the floor and the left leg is bent. Then the right leg, still pointed outward, is brought around to the left side, continuing the rotation, while the rest of the body remains facing the audience. When the right leg has been thus rotated through about 90 degrees, the right foot is brought back to the left knee and she comes back en pointe (on her left toes). She then repeats the entire procedure.

The key question is how the dancer generates enough torque to continue the rotation. With each turn the friction between the supporting foot and the floor robs her of angular momentum and spin. The secret is in the movement of the right leg. When it is brought forward and rotated to the right, it takes up whatever angular momentum the dancer still has. Her rotation stops, except for the right leg, giving her a moment to put her left foot flat on the floor. In this moment she can push against the floor, providing the torque for another rotation. To assist it she draws her right foot toward her left knee to reduce her moment of inertia as she spins.

The fouetté is difficult for dancers. Laws points out that a novice is likely to botch the turn by thrusting the right leg directly to the side instead of positioning it so that it absorbs the angular momentum properly. The novice might also extend the leg only partly forward toward the audience. It must be fully extended to absorb all the angular momentum.

The moment of inertia of the extended leg is about 1.7 times more than the body's. Hence when the leg absorbs the body's angular momentum, it does not turn as rapidly as the body. If the normal pirouette is at the rate of two revolutions per second, the extended leg turns at only 1.2. If the leg absorbs all the dancer's angular momentum, she has about .3 second to push off for another turn.

In the grand pas de chat the application of rotational dynamics is needed to enable the dancer to leap while she maintains the orientation of her body. From the fifth position of the feet she does a demi-plié on the right foot, bringing the left leg behind the right one at an angle of about 45 degrees from the vertical. The jump is made from the supporting leg. Once the dancer is in the air she brings the right leg into line with the left leg.

This alignment of the legs requires a rotation of the left leg about the dancer's center of mass. Yet her angular momentum in midair is essentially zero. How does she manage a rotation while keeping her angular momentum constant and maintaining the orientation of her torso? She rotates her arms in the direction opposite to the movement of her left leg. The arms are moved just enough to make the combined angular momentum of the arms and the right leg zero. By means of these countering rotations the dancer can rotate part of her body in the air.

Laws has analyzed the subtleties of balance achieved in the promenade en attitude derriere. Here a female dancer is in the position derrière en pointe: raised on the toes of one foot with the other leg poised in the air. One of her hands is held by a male partner, whom she faces. The other hand is raised in a graceful arc. The sustained pose is difficult because of the balance required. If the female dancer shifts into this pose from another movement, she is likely to be off balance.

The dancer could correct her balance by stretching her body in order to shift the center of mass back over the supporting foot, but that would disturb-the grace of the movement. She could also push or pull her partner's hand, but that would be likely to generate a torque that would make her turn on her supporting foot, rotating her away from her partner when she should face him.

A better method is to achieve the proper application of forces between the touching hands. According to Laws, both dancers' hands should be horizontal and the elbows should be raised. Then the female dancer can apply forces to her partner's hand in such a way that a turning torque does not develop. The forces on her hand are indicated in the illustration at the right. Two oppositely directed forces come from the male dancer's hand, each one displaced from the center of the hand by a distance d. The center of the hand is at a distance D from the vertical axis running through the female dancer's supporting foot.

The forces on the hands generate torques on the female dancer, but they tend to turn her in opposite directions. By properly controlling the forces she can make the two torques cancel. She makes the nearer force slightly larger than the farther one. Since torque is the product of a lever arm and a force and the lever arm to the nearer force is smaller, the torques cancel. The forces on her hand therefore do not cause her to rotate about her pointed toes. By twisting her hand in just the right way, however, she can move toward or away from her partner's hand, thereby adjusting the position of her center of mass in such a way that the audience does not realize what she is doing.

Several of the male dancer's leaps call for his beating his calves together while he is in flight. One is entrechat quatre, which begins with the feet in the fifth position. The dancer descends into a demi-plié and then propels himself directly upward. In the air he opens his legs to the side, brings them briskly together, separates them and then brings them together again as he lands in the fifth position demi-plié. An experienced dancer with a powerful leap might be able to complete two beats or more.

Laws has investigated the difficulty this movement presents for a large dancer. In general such a dancer cannot beat his legs together at the rate and amplitude that can be achieved by a smaller dancer. Even though the larger dancer may be stronger, his legs are more massive and therefore more difficult to rotate. He also has more mass to lift and more difficulty leaping to the same fraction of his height.

I have only touched on the movements of classical dance. You could do a good deal more. Try analyzing a ballet movement. It would be helpful to make a series of photographs with stroboscopic flash. A motion picture that can be run slowly or stopped periodically would be even better. None of these techniques will be easy because the shape and motions of the human body are complex. You might simplify the shape of the body as Laws has done in some of his analyses.

A study of ballet is also complicated by the requirement of grace and style. If you want to determine the physical principles underlying certain movements, you must distinguish the components of the movements that are done only for style. Both Laws and I shall be interested in hearing about what you find out.

Bibliography

AN ANALYSIS OF TURNS IN DANCE. K. L. Laws in Dance Research Journal, No. 11-12, page 16; 1978-1979.

PHYSICS AND BALLET: A NEW PAS DE DEUX. Kenneth Laws in New Directions in Dance, edited by Diana Theodores Taplin. Pergamon Press, 1979.

PRECARIOUS AURORA-AN EXAMPLE OF PHYSICS IN PARTNERING. Kenneth Laws in Kinesiology for Dance, No. 12; August, 1980.

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