Ann Varela about Joseph Louis Lagrange

Feb 10, 2018 by

An Interview with Ann Varela about Joseph Louis Lagrange

Michael F. Shaughnessy –

1) Apparently, Joseph Louis Lagrange was both a mathematician and astronomer. In your mind, how do these two fit together?

When Lagrange studied the three-body problem for the Earth, sun, and moon (celestial mechanics) and the movement of Jupiter’s satellites, measurement was required to make comparative analyses. Measurement is math. Without mathematics, no quantitative data may be collected or analyzed. Subsequently, nothing may be inferred for future experiments without accurate quantitative results.

Much of the mathematics required for understanding information attained through astronomical observation originates with physics, but sometimes mathematics itself is needed to better understand phenomena. Today, astronomers use mathematics every time they look through a telescope. Cameras used for celestial observation are equipped with detectors to convert (count) photons or electrons and record data relating to the amount of light emitted by the observed objects. Other obvious mathematical calculations would be the distance from one celestial object to another or the distance from the Earth to a celestial body.

2) He reportedly made several significant contributions to the fields of analysis, and number theory. What do you mean by analysis here—and what is the basis of number theory?

Mathematical analysis is the branch of mathematics concerned with limits and related theories based on continuous change, such as differentiation (finding derivatives), integration, measure (length, area, volume), infinite series (addition of infinitely many quantities), and analytic functions. When calculating spatial quantities, such as the length of a curved line or the area under a curve, analysis must be employed. Examples of this concept include determining how much sod will be needed to cover an irregularly shaped yard, calculating the total distance traveled by a fish under water, or the cooling of a cup of coffee in a cold room.

Number theory is the study of the properties of positive integers, also known as natural numbers. Number theory includes the study of prime numbers and number families made out of integers, such as rational numbers. Lagrange’s work with the sum of four squares is an example of number theory at work. He also developed a method of approximating the real roots of an equation through continued fractions.

Some other numerical topics that number theorists contemplate include sums of squares, sums of cubes, sums of higher powers, infinitude of primes, shapes of numbers (triangular or square), perfect numbers (numbers that are equal to the sum of their factors), and the Fibonacci sequence.

3)  What were his contributions to both classical and celestial mechanics? In addition, how does this relate to math?

Classical mechanics describes the motion of macroscopic objects and astronomical objects, while celestial mechanics focusses mostly on the motions of planets and stars. Lagrange was interested in solving the three-body problem. This particular problem concentrates on taking an initial set of data that specifies the masses, positions, and velocities of three bodies for some particular point in time and then determining the motions of the three bodies, in agreement with Newton’s laws of motion and of universal gravitation.

Lagrange analyzed the consistency of planetary orbits, and discovered the presence of the Lagrangian points, as seen in Figure 1. The first three points, L1, L2, and L3, connect the two large bodies. The other two points, L4 and L5, each form an equilateral triangle with the two large bodies. Since objects can orbit around them in a rotating coordinate system tied to the two large bodies, points L4 and L5 are considered stable.

Figure 1. Lagrangian Points.

https://upload.wikimedia.org/wikipedia/commons/5/5f/Lagrangian_points_equipotential.jpg

https://upload.wikimedia.org/wikipedia/commons/5/5f/Lagrangian_points_equipotential.jpg

Lagrange also reformulated the principles of classical mechanics, emphasizing energy more than force and developing a method to use a single polar coordinate equation to describe any orbit. This is important because it allows for calculating the behavior or movement of planets and comets. Nowadays, design engineers and trajectory analysts are concerned with other objects in space, such as the path of spacecraft. Figure 2 shows the projectile motion function with horizontal distance as a function of velocity and launch angle.

Figure 2. Projectile Motion.

https://upload.wikimedia.org/wikipedia/commons/2/25/Projectile-Motion.png

https://upload.wikimedia.org/wikipedia/commons/2/25/Projectile-Motion.png

4) Every natural number is the sum of four squares. Can you give us an example of this?

First of all, one must know the definition of a natural number. The natural numbers are positive integers, beginning with one. Figure 3 shows a few examples of the theorem Lagrange proved in 1770, which stems from number theory.

Figure 3. Sum of Four Squares.

5) “Calculus of variations” seems to be associated with Lagrange- but what exactly does this mean?

The calculus of variation is a generality of calculus that refines the solution of the function down to a path, curve, point, or surface, etc. relating to a fixed value, such as a minimum or maximum. In 1754, Lagrange developed this principle of variation throughout his work on the tautochrone; however, Euler later coined the official name, calculus of variation, in 1766. Figure 1 shows how the time taken by an object descending without resistance in constant gravity to its minima is not dependent on its starting point.

Figure 4. Tautochrone Curve.

https://upload.wikimedia.org/wikipedia/commons/b/bd/Tautochrone_curve.gif

https://upload.wikimedia.org/wikipedia/commons/b/bd/Tautochrone_curve.gif

6) What are Lagrange “multipliers”?

The method of Lagrange multipliers is a strategy for finding the related maxima and minima of a function subject to equality constraints. The main point here is to find those values of the function, which produce the largest and smallest output, either within a specified range (specific input) or on the entire domain (all input considered) of a function. Equality constraints refer to conditions of the problem that the solution must placate. Constraints may be in the form of an equality, inequality, or an integer. For example, if one wants to impose a constraint to only include x-values that are greater than one, the notation would be x > 1.

7)  Lagrange was involved in the making of standards of measurement. Why is this important in terms of math?

Lagrange’s efforts with standardizing weights and measures began in 1790 as a member of an appointed committee. Lagrange was also involved with the development of the metric system. His major role was setting up the unit system of meter and kilogram, along with their decimal parts. Having standard units of measurement makes for easier communication and understanding of another mathematician’s work. Comprehensively, a standard unit of measure is indeed helpful for most fields of study. One can see the importance of the scientific method and realize how a standard unit of measure may ensure consistent results when conducting research.

8) Allegedly, he was a great mathematician but not a great teacher or professor. Where did he actually teach and what was said about his teaching?

Lagrange was employed as an assistant professor of mathematics at the Royal Military Academy in 1755. The subjects he was charged with were calculus and mechanics. Though he was obviously talented and fluent in these areas himself, teaching them to students was another matter entirely. Lagrange apparently had a style of teaching that was difficult to understand because his own learning style was supposedly unique and vague. Patience with engineering applications was another character trait that caused him anguish.

9) Lagrange’s tomb is apparently in the Pantheon. It sounds like a fitting tribute to a great mathematician. Your thoughts about his contributions?

Lagrange’s involvement with the calculus of variation is considered one of his greatest contributions. His work in this area began with presenting a better way to solve an equation we now know as Euler’s equation. Other prominent mathematicians used Lagrange’s findings as a springboard for their own work in this area. Next, Lagrange studied geometric-based topics including volumes and surface areas, which led him to the general theory of a surface integral.

Polynomial equations intrigued Lagrange, especially those of the fifth degree and higher. Like Cardano, he attempted to simplify polynomial equations by considering permutations of the roots. This technique was not successful so he halted his pursuit.

Lagrange’s work on functions was not quite the same as what is taught today, but it eventually led to the fundamental theorem of calculus.

Finally, Lagrange’s efforts on the topic of planetary motions was quite outstanding, as it led to certain limits for an orbit’s stability.

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