calculus of variations
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What is the calculus of variations?
The calculus of variations is a field of mathematics focused on determining the maximum or minimum value of certain integrals. These integrals contain functions, so the calculus of variations seeks to find the function that optimizes the integral.
The origin of the calculus of variations can be traced back to the isoperimetric problem, which was known to Greek mathematicians. Modern interest began in 1696 with Johann Bernoulli's brachistochrone problem, which was independently solved by mathematicians such as Isaac Newton and Gottfried Wilhelm Leibniz.
Variational principles are applicable in diverse fields including aerodynamics, electromagnetic theory, elasticity, quantum electrodynamics, and the theory of vibrations.
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How does the calculus of variations differ from regular calculus?
Calculus of variations is a field of mathematics that uses calculus methods to find the maxima and minima of functions, whose values depend on another function or curve.
In regular calculus, you find the maximum or minimum value of a function by finding critical points where the derivative is zero. The calculus of variations, however, seeks to find the function that optimizes a certain integral. This involves finding a function that makes the integral's value the largest or smallest possible, subject to certain constraints.
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What is a functional in calculus of variations?
In calculus of variations, a functional is a function of a function. It establishes a relationship between objects, which can be numbers, vectors, or functions. Differentiation is a functional because it links a function to its derivative, and integration is another example.
The calculus of variations uses functionals to find a function that maximizes or minimizes a certain integral. This involves finding a differential equation for the optimal function within a class of functions.
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What is the Euler–Lagrange equation and why is it important?
The calculus of variations is a field of mathematics focused on maximizing or minimizing the values of definite integrals.
In this field, variational principles describe scientific laws by asserting that a given integral is at a maximum or minimum. An example is the principle of least action, where minimizing an integral led to an explanation of Newton’s laws of motion.
The Euler–Lagrange equation is important because its solutions are the functions that minimize or maximize the integral. It has uses in elasticity, electromagnetic theory, aerodynamics, the theory of vibrations, and other areas in engineering and science.
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How is calculus of variations used in real-life problems, like physics or engineering?
Calculus of variations is a field of mathematics used to find the maximum or minimum value of integrals and has found many real-world applications.
Variational principles can formulate scientific laws. These principles assert that a given integral is either maximized or minimized. Examples include aerodynamics, where the goal is to find the shape of a solid with a given volume that will encounter minimal resistance as it travels through the atmosphere, and Pierre de Fermat's principle of least time.
Applications of variational principles can also be found in elasticity, electromagnetic theory, the theory of vibrations, and other areas of engineering and science.
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Who were some important mathematicians involved in the development of calculus of variations?
The calculus of variations emerged in the late 17th century, with mathematicians tackling problems of maximizing or minimizing integrals.
Key figures in its development include:
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Johann Bernoulli: In 1696, Bernoulli posed the brachistochrone problem, which spurred interest in the field.
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Jakob Bernoulli: Independently solved the brachistochrone problem. The brothers' keen rivalry pushed them to develop ideas that would later become the calculus of variations.
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Gottfried Wilhelm Leibniz: Also solved the brachistochrone problem.
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Isaac Newton: Solved the brachistochrone problem.
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Joseph-Louis Lagrange: Developed a new algorithm in the calculus of variations in 1755. Euler adopted Lagrange's idea, and the two systematically revised the subject using the new techniques.
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Leonhard Euler: Systematically separated calculus from geometry. In 1755, he had devoted an important treatise to the calculus of variations.
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calculus of variations, branch of mathematics concerned with the problem of finding a function for which the value of a certain integral is either the largest or the smallest possible. Many problems of this kind are easy to state, but their solutions commonly involve difficult procedures of the differential calculus and differential equations.
The isoperimetric problem—that of finding, among all plane figures of a given perimeter, the one enclosing the greatest area—was known to Greek mathematicians of the 2nd century bce. The term isoperimetric problem has been extended in the modern era to mean any problem in the calculus of variations in which a function is to be made a maximum or a minimum, subject to an auxiliary condition called the isoperimetric condition, although it may have nothing to do with perimeters. For example, the problem of finding a solid of given volume that has the least surface area is an isoperimetric problem, the given volume being the auxiliary, or isoperimetric, condition. An example of an isoperimetric problem from the field of aerodynamics is that of finding the shape of a solid having a given volume that will encounter minimum resistance as it travels through the atmosphere at a constant velocity.
Modern interest in the calculus of variations began in 1696 when Johann Bernoulli of Switzerland proposed a brachistochrone (“least-time”) problem as a challenge to his peers. Suppose that a thin wire in the shape of a curve joins two points at different elevations. Further suppose that a bead is placed on the wire at the higher point and allowed to slide under gravity, starting from rest and assuming no friction. The question is: What should be the shape of the curve so that the bead will reach the lower point in the least time?
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The problem was solved independently in 1696 by Johann Bernoulli, his brother Jakob Bernoulli, the German Gottfried Wilhelm Leibniz, the Frenchman Guillaume-François-Antoine, marquis de L’Hôpital, and the Englishman Isaac Newton. Their basic idea was to set up an integral for the total time of fall in terms of the unknown curve and then vary the curve so that a minimum time is obtained. This technique, typical of the calculus of variations, led to a differential equation whose solution is a curve called the cycloid.
It is possible to formulate various scientific laws in terms of general principles involving the calculus of variations. These are called variational principles and are usually expressed by stating that some given integral is a maximum or a minimum. One example is the French mathematician Pierre-Louis Moreau de Maupertuis’s principle of least action (c. 1744), which sought to explain all processes as driven by a demand that some property be economized or minimized. In particular, minimizing an integral, called an action integral, led several mathematicians (most notably the Italian-French Joseph-Louis Lagrange in the 18th century and the Irish William Rowan Hamilton in the 19th century) to a teleological explanation of Newton’s laws of motion. Nevertheless, a general appreciation of the principle of least resistance came only with its use in the 1940s as a foundation for quantum electrodynamics.
Applications of variational principles also occur in elasticity, electromagnetic theory, aerodynamics, the theory of vibrations, and other areas in engineering and science.