Analytical mechanics [SPHYB211]

Learning outcomes

To master the different formalisms of classical mechanics (Newton, Lagrange, Hamilton) and use them to tackle selected fundamental mechanical problems.

Content

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.

Table of contents

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

Exercises description

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.