There is much more to the arabesque, grand jete, and fouette than just hours of practice and strong toes. Dancers do not often understand the physics and mathematics describing their movements, but they know how to execute them with grace. If ballerinas were not able to feel physics, they would fall over. Legs, arms and head are oscillating as they turn– moving back and forth in seemingly separate patterns. For dancers, everything must move in separate patterns, yet at the same time, move at the same rhythm. This presentation introduces the history, mathematics, and physics behind ballet.
Ballet originated in the Italian Renaissance courts during the 15th century. Noblemen and women attended special events, such as wedding celebrations, where dancing and music were considered an elaborate spectacle. Dancing masters taught the steps to the nobility and the court participated in the performances. In the 16th century, Catherine de Medici, an Italian noblewoman and the wife of King Henry II of France, funded ballet within the French Court. A century later, King Louis XIV helped to popularize and standardize ballet. In 1661, the first dance academy opened in Paris and ballet moved from the courts to the stage.
The French opera created the opera-ballet tradition in France. During the mid-1700s, a French ballet master, Georges Noverre, rebelled against the opera-ballet because he believed that ballet could stand as its own art form. This led to the establishment of a dramatic style of ballet that conveys a narrative story. In the late 18th century, the popularity of ballet increased in Russia, where classical ballet was represented in its most classical form: Marius Petipa’s The Nutcracker, Swan Lake, and Sleeping Beauty, were composed.
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The main purpose of classical ballet was to display technique to its fullest— pointe work, high extensions, precision of movement, and turn-out (the outward rotation of the legs from the hip) are included. Demanding leaps, steps, and turns were choreographed into the story. During the Romantic Movement in the start of the 19th century, ballet was influenced by spirits, magic, and fragility; henceforth, the romantic ballets were established. During the Romantic Era, en pointe, dancing on the tips of toes, became the norm for ballerinas, and the “tutu,” a skirt made of tulle, was introduced.
Later on in the 19th century, New York City Ballet founder, George Balanchine, a Russian who immigrated to America, introduced what is now known as the neo-classical ballet. This is a form that expands the classical form. Balanchine is considered to be one of the greatest innovators of the “plot-less” contemporary ballet: without a definite story line, its purpose is to use movement to express the music, and to illuminate human emotion and endeavor. Today, ballet is multi-faceted: classical forms, traditional stories, and contemporary innovations intertwine to produce the character of a modern ballet.
Physics and mathematics are found in every aspect of dance. First, it is important to understand that the net force (Fnet) involves the addition of applied forces. The net force causes a change in an object’s momentum. For example, the net force on a dancer is the force of gravity acting down, the force from the support of the floor acting upon, and the sideways force of the friction from the floor. A torque in physics is an off-center force that causes something to spin. Also, the torque can change angular momentum of an object.
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In dance, it should be understood that the greater the force, the more quickly a dancer can spin. Torques are further generated from the spin-axis, which causes the dancer to spin more quickly. The longer a torque is activated, the greater the total change in angular momentum. This is expressed through the equation: ? L=?? t. If a torque is not applied to a rigid object, or a solid body of finite size, it spins at a constant rate. Its spin-axis does not wobble. Non-rigid objects, like people or dancers, constantly change shape and are more complicated.
A dancer’s rotational inertia (I) changes with time (t).
Static balance occurs when both the net force and net torque equal zero. Static balance is the point of center of gravity at which an object, like a dancer, has the ability to balance. All of the forces acting upon a balanced dancer sum to equal zero. A dancer’s center of gravity must remain directly above the area of contact with the floor; otherwise, he or she will fall over. Smaller floor contact area, like en pointe, for example, makes balance more difficult.
Maintaining such balance involves constant shifts and adjustments of the body. The net effect of gravity on a dancer is the same as if it were only acting upon the center of gravity, and not on all parts of the body. Forces on a balanced dancer like gravity (pulls down, acts upon center of gravity) and the support from floor (pushes up, acts upon feet) equal zero. Both forces act through the center of gravity, and so, the dancer’s torque also equals zero. Not only must a dancer balance, but a dancer must also balance while turning.
Not all statically balanced dance positions can be held while turning. Centrifugal forces exert a torque, which throws the dancer off balance, or off of his or her center of gravity. A shift in the center of gravity must occur in order to maintain dynamic, yet not static, balance. If a torque is not applied, then the angular momentum (L) is constant. If the rotational inertia (I) is decreased, then the angular velocity (? ) must increase, or spin faster. If the rotational inertia (I) is increased, then the angular velocity must decrease, or spin slower.
Rotational inertia is an object’s resistance to its rotation. Through calculations, it is proven that the greater the mass of an object, the greater the rotational inertia. When mass is distributed farther from its rotational axis, there is a larger rotational inertia. The equation, I=mr2, represents how increasing the rotational inertia will decrease the rate at which a dancer spins. The Conservation of Angular Momentum explains how a dancer is able to change the speed of a turn when en pointe by extending or retracting his or her leg.
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Friction between the floor and a dancer’s foot can generate a torque. Dancers turn and spin on stage; however, dancers must also perform turning movements in the air. In certain movements, the body parts which carry the initial angular momentum change during the course of the turn. For example, during a tour jete, the angular momentum associated with the raising of the left leg (1) is taken up by the trunk and arms (2), then the left leg (3), then both legs (4).
Refer to the picture below. Jumps are essential movements in the world of ballet.
Gravitational force depends on mass. The greater the mass, the greater the gravitational pull. Also, momentum depends on mass. The greater the mass, the harder it is to stop moving at a particular speed (p=mv).
Doubling an object’s mass will double its momentum and double the strength of its gravitational force. This is proved through the equation, ? p = m? v = F? t. On the other hand, gravitational acceleration is independent of mass. The relationship between the position of a dancer’s center of gravity and the time during a jump is expressed.
The force of gravity influences the vertical component of motion, not the horizontal: Additionally, the relationship between the height of a jump (h) and time (t) is expressed through a parabolic function. The relationship between the horizontal distance (d) and time (t) is represented through a linear, straight line: The dancer’s trajectory in a jump is also a parabola: By raising his or her legs, which changes the distance between the center of gravity and the dancer’s head, he or she can create the illusion of floating.
The vertical motion of the head is less than the motion of the center of gravity. Mathematics Physics is a study of science that uses math. From simple geometry to advanced calculus, mathematics is present in the physics of ballet. Kinematics, a branch of mechanics in relation to motion of objects, is present in almost every aspect of ballet. A derivative (dy/dx) is a rate of change with respect to x and y. For objects in motion, like dancers, specific movement is related to the essential topic of calculus, which is the derivative.
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A dancer’s velocity (v) is the first derivative (? ) of his or her position (s), with respect to time (t).
A dancer’s acceleration (a) is the second derivative (? ) of his or her position (s) with respect to time (t) or the first derivative (? ) of his or her velocity (v).
Conclusion The elements of physics and mathematics are in use whether a dancer is balancing, spinning, jumping, or just standing still. Force, velocity, momentum, rotational inertia, static balance, and torques are specific variables that are utilized in the every aspect of ballet.
