Master the various formalisms of classical mechanics (Newtonian, Lagrangian and Hamiltonian) and use them to understand some essential applications of mechanics. Use them to study and understand important models such as the pendulum, solid rotation, the two-body problem and non-inertial motion. Model simple dynamic problems, obtain the equations of motion and deduce their eventual solution, or at least describe the main properties of motion. Notions of harmonic oscillator, principle of least action, gauge invariance, phase and configuration space, Euler angles, central forces, inertia tensor.
The main objective of the course is to introduce the main formalisms of classical mechanics (Newton, Lagrange, Hamilton, Hamilton-Jacobi) in generalized coordinate systems, as well as conservation laws through Noether's theorem. The course also aims to introduce essential physics concepts such as the principle of least action, the harmonic oscillator, gauge invariance, phase space, motion in a non-inertial frame of reference, Euler angles, the two-body problem, central forces, the pendulum, and the rotation of solids.
The course studies first the movement of point masses or solids subjected to forces. The motion and conservation laws of Newtonian mechanics are deduced, as well as the relations between speeds and accelerations observed in different moving reference frames (including that of the centre of mass). One chapter deals with the general study of the one-dimensional motion of a conservative system. Then, the concepts of generalized coordinate, constraint, virtual work and variational problem are introduced to tackle Lagrange's formalism et Hamilton's least action principle. The main characteristics of the formalism are studied (including the conservation laws and the case of electromagnetism). A few fundamental problems of mechanics are dealt with: small oscillations, the two-body problem in a central potential (Kepler's laws and Rutherford scattering), solid rotations. Finally, Hamilton's formalism is introduced and the properties of Hamilton's equations, phase space, canonical transformations, and action are analysed. Hamilton-Jacobi's equation is deduced and the method illustrated through Kepler's problem.
1. Newtonian mechanics
- Point particle mechanics
- One-dimensional conservative systems
- Systems of point particles
- Non inertial reference frames
- The solid (1st part)
2. Lagrangian mechanics
- Bases of Lagrange's formalism
- Hamilton's variational principle
- Properties of the Lagrangian
- Lagrangian systems
- Two-body problem in a central potential
- The solid (2nd part)
3. Hamiltonian mechanics
- Hamilton's formalism
- Canonical transformations
- Hamilton-Jacobi's equation
The goal of the supervised exercises is to apply the theoretical concepts taught during the course to concrete applications. A first part deals with the rotation of solid bodies. In particular, we shall study the rotation of typical geometrical bodies (such as cylinders and cones), the gyroscope, and the motion of trailers and helices. After reviewing mathematical tools (such as ordinary differential equations and curvilinear coordinate systems) required to solve mechanical problems, we shall use the Lagrangian and Hamiltonian formalisms in a second part. In particular, we shall use these formalisms will to describe the motion of simple systems (coupled pendulum, two-dimensional systems of coupled oscillators...) and of more complex systems (Foucault pendulum, Penning trap...). The last part of the supervised exercises will include the equation of Hamilton-Jacobi in a synthesis of the major formalisms of analytical mechanics.
The course is mainly taught using the projection of powerpoint slides (rendered progressively to show the progression of the mathematics and of the reasoning), a version of which is available in pdf format on webcampus. The fundamental concepts studied during the course are illustrated and put into practice during the practical exercises.
The course content taught in Q1 is evaluated with an exam E1 of theory and exercises (exam oragnized in January and August).
The course content taught in Q2 is evaluated with an exam E2 of exercises and with an exam E3 of theory (exams oragnized in June and August).
If the grade of the exams E1, E2 et E3 are each better than or equal to 8/20, the global grade is given by the following weighted average: 1/2 E1 + 1/2 (2/3 E2 + 1/3 E3). Otherwise (that is, if one of the three grades at least is lower than 8), the exam is automatically considered failed (independently of the grade average).
The course does not follow a specific textbook. Nevertheless, here are useful references.
- Mécanique : de la formulation lagrangienne au chaos hamiltonien, Claude Gignoux et Bernard Silvestre-Brac, EDP Sciences
- Problèmes corrigés de mécanique et résumés de cours : de Lagrange à Hamilton, Claude Gignoux et Bernard Silvestre-Brac, EDP Sciences
- Mécanique (volumes 1 et 2), Philippe Spindel, Editions Scientifiques GB
- The variational principles of mechanics, Cornelius Lanczos, Dover
- Classical dynamics of particles and systems, Jerry Marion and Stephen Thornton, Harcourt
| Training | Study programme | Block | Credits | Mandatory |
|---|---|---|---|---|
| Bachelor in Physics | Standard | 0 | 7 | |
| Bachelor in Physics | Standard | 2 | 7 |