Mécanique Lagrangienne from Coursera

Ces quelques leçons de mécanique lagrangienne font partie d'un cours de formation de base en mécanique Newtonienne présenté sous la forme d'un MOOC en quatre parties :

1. Lois de Newton

https://www.coursera.org/learn/mecanique-newton

2. Mécanique du point matériel

https://www.coursera.org/learn/mecanique-point-materiel

3. Mécanique du Solide Indéformable

https://www.coursera.org/learn/mecanique-solide

4. Mécanique Lagrangienne

Le formalisme de Lagrange permet une résolution efficace de problèmes complexes de mécanique. Il permet aussi d'apporter un éclairage plus fondamental sur les lois de conservation (théorème de Noether). A titre d'illustration de la méthode de Lagrange, on traitera le problème très important des oscillateurs harmoniques couplés, exprimé comme un problème de valeurs propres et de vecteurs propres. On termine avec un formalisme permettant d'analyser les résonances paramétriques, notion illustrée par l'expérience montrant la stabilité d'un pendule inversé forcé.

What's inside

Syllabus

Méthode de Lagrange

La méthode de Lagrange permet de résoudre de manière très efficace des problèmes d'une grande variété en utilisant des coordonnées généralisées. Ici, les équations de Lagrange sont démontrées pour des systèmes de points matériels soumis à des contraintes qui s'expriment sous la forme d'équations pour les coordonnées généralisées. Une généralisation sera vue plus loin.

Application du formalisme de Lagrange

La méthode de Lagrange permet d'obtenir les lois de conservations de la quantité de mouvement et du moment cinétique comme la conséquence de symétries fondamentales. On va voir aussi qu'on peut résumer toute la mécanique sous la forme d'un principe de minimisation d'une fonction qu'on appellera l'action. Il s'agit d'un principe dit "variationnel". Comme il s'agit de quelque chose de tout à fait nouveau, on va développer un sens physique de tels principes variationnels en considérant des expériences simples et quelques exercices.

Systèmes vibratoires discrets et pendules couplés

Les systèmes vibratoires couplés se retrouvent dans toutes sortes de contextes en physique et jouent un rôle très important en ingénierie. L'étude d'oscillateurs couplés permet de mettre en jeu les concepts de valeurs propres et de vecteurs propres qu'on devait avoir vu dans un cours d'algèbre linéaire.

Résonance paramétrique

Quoi de plus simple qu'un enfant qui fait osciller la balançoire sur laquelle il se tient en pliant les genoux au bon rythme ? Et pourtant, il s'agit là d'un problème de mécanique des systèmes non-linéaires dont l'analyse nécessite de nouvelles approches. Ces problèmes exprimés par les équations dites de "Hill" ou de "Mathieu" se retrouvent dans toutes sortes de contextes, notamment en physique de la matière condensée.

Principe de relativité (optionnel)

Partant des concepts vu à la leçon 15 sur les changements de référentiel, on rappelle le principe de relativité de Galilée. Avec une condition supplémentaire sur la vitesse de la lumière, on voit qu'on a un problème et qu'il faut changer la façon de relier les coordonnées liées à deux référentiels d'inertie en translation uniforme l'un par rapport à l'autre.

Cinématique relativiste (optionnel)

Dans cette introduction à la cinématique relativiste, on applique des principes de symétrie très généraux pour arriver aux célèbres transformations de Lorentz. Avec elles, on explique ce qu'on entend par contraction des longueurs et dilatation du temps.

Dynamique relativiste (optionnel)

On prend le point de vue d'induire par des arguments relativistes la relation qui doit exister entre la quantité de mouvement et la vitesse d'un point matériel. L'introduction d'un quadri-vecteur quantité de mouvement aboutit à la plus célèbre des équations de la physique : E = mc^2. On évoquera aussi la notion de photon.

Clôture

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Covers Lagrangian mechanics, which is an efficient method to solve complex mechanics problems

Taught by Jean-Philippe Ansermet, an expert in mechanics

Suitable for learners with a background in Newtonian mechanics

Requires understanding of linear algebra for the section on coupled oscillators

Includes an optional section on the principle of relativity

Reviews summary

Lagrangian mechanics course

The majority of students rate this course very well citing that the materials and lessons are valuable. Some students complain of a lack of support from the instructor, but do not fault the course materials.

The course is well-received.

"Cours d'un très bon niveau"

"Merci. Beau cours, très pédagogique"

Some students complain of lack of support from the instructor.

"Appaling teaching! One cannot teach physics by reading slides"

Activities

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Review basic concepts of Newtonian mechanics

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Reacquaint yourself with the fundamental principles of Newtonian mechanics to strengthen your foundation for Lagrangian mechanics.

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Review Newton's laws of motion and their applications.
Recall concepts of kinematics, including displacement, velocity, and acceleration.
Brush up on work, energy, and power.

Solve practice problems on Lagrangian mechanics

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Deepen your understanding of Lagrangian mechanics by engaging in targeted practice problems.

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Find equations of motion using Lagrangian formulation.
Apply Lagrangian mechanics to solve problems involving constraints.
Use Lagrangian mechanics to analyze oscillatory and coupled systems.

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Career center

Learners who complete Mécanique Lagrangienne will develop knowledge and skills that may be useful to these careers:

Theoretical Physicist

Theoretical Physicists aim to develop, understand, and explain the fundamental physical laws that govern our universe and nature. Lagrange mechanics, a formalism that allows for complex physics problems to be solved using generalized coordinates, is foundational to this field. This course provides a rigorous foundation in Lagrange mechanics, preparing you to analyze and solve problems in theoretical physics.

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Mathematical Physicist

Mathematical Physicists use advanced mathematical techniques to solve problems in physics. Lagrange mechanics is a key tool in their toolkit, enabling them to model and analyze complex physical systems. This course will equip you with the mathematical expertise necessary to succeed in mathematical physics, particularly in areas that involve Lagrange mechanics.

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Aerospace Engineer

Aerospace Engineers design, develop, and test aircraft, spacecraft, and related systems. Lagrange mechanics is widely used in aerospace engineering to analyze and optimize the dynamics of aircraft and spacecraft. By taking this course, you will gain a strong foundation in Lagrange mechanics, which is essential for a successful career in aerospace engineering.

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Mechanical Engineer

Mechanical Engineers design, develop, and test mechanical systems, such as engines, robots, and machinery. Lagrange mechanics is a fundamental tool in mechanical engineering, as it allows engineers to analyze and optimize the dynamics of mechanical systems. This course will provide you with a strong foundation in Lagrange mechanics, which is essential for a successful career in mechanical engineering.

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Robotics Engineer

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Civil Engineer

Civil Engineers design, construct, and maintain infrastructure, such as bridges, buildings, and roads. Lagrange mechanics is used in civil engineering to analyze and optimize the dynamics of structures. This course will provide you with a strong foundation in Lagrange mechanics, which is essential for a successful career in civil engineering.

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Chemical Engineer

Chemical Engineers design, develop, and test processes for producing chemicals and other products. Lagrange mechanics is used in chemical engineering to analyze and optimize the dynamics of chemical processes. This course will provide you with a strong foundation in Lagrange mechanics, which may be useful for a career in chemical engineering.

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Geophysicist

Geophysicists study the Earth's physical properties and processes. Lagrange mechanics is used in geophysics to analyze and optimize the dynamics of the Earth's systems. This course will provide you with a strong foundation in Lagrange mechanics, which may be useful for a career in geophysics.

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Oceanographer

Oceanographers study the oceans and their interactions with the atmosphere and land. Lagrange mechanics is used in oceanography to analyze and optimize the dynamics of ocean currents and waves. This course will provide you with a strong foundation in Lagrange mechanics, which may be useful for a career in oceanography.

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Materials Scientist

Materials Scientists develop and improve materials for a variety of applications, such as electronics, energy, and medicine. Lagrange mechanics is used in materials science to analyze and optimize the dynamics of materials. This course will provide you with a strong foundation in Lagrange mechanics, which may be useful for a career in materials science.

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Biophysicist

Biophysicists study the physical principles underlying biological systems. Lagrange mechanics is used in biophysics to analyze and optimize the dynamics of biological systems, such as cells and molecules. This course may be useful for a career in biophysics, as it will provide you with a strong foundation in Lagrange mechanics.

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Systems Engineer

Systems Engineers design, develop, and test complex systems, such as airplanes, power plants, and communication networks. Lagrange mechanics is used in systems engineering to analyze and optimize the dynamics of complex systems. This course will provide you with a strong foundation in Lagrange mechanics, which may be useful for a career in systems engineering.

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Data Scientist

Data Scientists analyze and interpret data to extract meaningful insights. Lagrange mechanics is used in data science to analyze and optimize the dynamics of data-driven systems. This course may be useful for a career in data science, as it will provide you with a strong foundation in Lagrange mechanics.

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Software Engineer

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Financial Analyst

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