Difference Between Eulerian and Lagrangian

Eulerian vs Lagrangian

“Eulerian” and “Lagrangian” are two adjectives that refer to two mathematicians, specifically to Leonhard Euler and Joseph Louis Lagrange. Both mathematicians contributed many great works not only in mathematics but also in other fields of study (which are also mathematically related) like physics, astronomy, and other disciplines.

Since both men are considered pioneers in the same fields and contributed greatly to these disciplines, concepts, techniques, and other disciplined-related items, these terms were named after them in recognition of their contributions. Some of the contributions were considered as a revolutionary or novel idea at the time of their conception or introduction. Another use of these adjectives is to have an easy reference and differentiation for a point of view when used in a discussion or as a comparative level.

Eulerian, as its name implies, is attributed to Leonhard Euler. Euler is a Swiss mathematician who is considered as the most prolific in the history of mathematics in terms of his contribution to the study and disciplines. Most of his contributions are considered revolutionary and created an impact on mathematics as a study and discipline. Among his contributions are: function notations, prime number theorem, and law of bioquadratic reciprocity in number theory (dealing with the relationship of numbers, their classifications, and groupings), topology (the qualification and classification of objects in a geometric sense), and various studies outside of mathematics. Other studies include his contributions in practical engineering (Euler-Bernoulli beam equation), and in astronomy (calculations of the planets’ motion). In physics he articulated Newtonian dynamics and has studied elasticity, acoustics, wave theory of light, and hydrometrics of ships.

On the other hand, Joseph Louis Lagrange is a contemporary mathematician of Euler. In the same case of Eulerian, Lagrangian is any concept that is attributted to Joseph Louis Lagrange in many fields. Though Lagrange is a great mathematician in his own right, his contributions are often mirrored by Euler’s work and contributions since the former introduced many of the mathematical concepts in the same time period.

Lagrange also has contributions of his own to mathematics among other studies. He introduced the first theory of functions of a real variable and made contributions in the study of dynamics, fluid mechanics, probability, and the foundations of calculus. Like Euler, Lagrange also worked on the number theory, and his input resulted in proving that every positive integer is the sum of four squares, and later he proved Wilson’s theorem.

Both mathematicians were familiar with each other as they both shared a position as Director of Mathematics at the Prussian Academy of Sciences in Berlin and corresponded with each other discussing mathematical concepts. Both men share in the conception of Euler–Lagrange equation, an equation which is used in calculus, specifically in the calculus of variations for the motions of fluids.

In the study of mathematics, concepts developed by both Euler and Lagrange are often studied and compared with each other. Since both mathematicians have different opinions about the same concepts, their observations and opinions are often pitted against each other on which is more effective in terms of application. In the course of study, there are also differences on how different the approach or theory of Euler is from Lagrange. These differences would often lead to discussions or even debates not just in theory but in practical use as well.

Summary:

1.“Eulerian” and “Lagrangian” are adjectives that pertain to Leonhard Euler and Joseph Louis Lagrange. Both Euler and 2.Lagrange are noted mathematicians that gave many contributions to the field of mathematics and other related fields of study.
3.Both Eulerian and Lagrangian theory perform a descriptive function in the field of mathematics. Both are very helpful in discussions or debates of concepts and viewpoints especially when comparing one concept from another part of their descriptive function which also acts as an immediate reference to a specific mathematician or concept being alluded to.

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