Classical Mechanics

The details

Requisites for this module

BSC G100 Mathematics,

BSC G102 Mathematics (Including Year Abroad),

BSC G103 Mathematics (Including Placement Year),

BSC G104 Mathematics (Including Foundation Year),

BSC G1F3 Mathematics with Physics,

BSC G1F4 Mathematics with Physics (Including Placement Year),

BSC GCF3 Mathematics with Physics (Including Year Abroad)

BSC G102 Mathematics (Including Year Abroad),

BSC G103 Mathematics (Including Placement Year),

BSC G104 Mathematics (Including Foundation Year),

BSC G1F3 Mathematics with Physics,

BSC G1F4 Mathematics with Physics (Including Placement Year),

BSC GCF3 Mathematics with Physics (Including Year Abroad)

This course concerns the general description and analysis of the motion of systems of particles acted on by forces. Assuming a basic familiarity with Newton's laws of motion and their application in simple situations, the students will develop the advanced techniques necessary to study more complicated systems. The students will also consider the beautiful extensions of Newton's equations due to Lagrange and Hamilton, which allow for simplified treatments of many interesting problems and provide the foundation for the modern understanding of dynamics. The module also includes an introduction to the calculus of variations, which allows the solution of an important class of problems involving the optimisation of integral quantities.

Main Topics:

1. Kinematics and dynamics of particles and systems of particles.

2. Calculus of variations (Equations of Lagrange - first and second kind. Least action principle).

3. Lagrangian formalism: D'Alembert principle vs variational principle.

4. Oscillations.

5. Hamiltonian formalism. Canonical transformations. Noether theorem and symmetries.

Main Topics:

1. Kinematics and dynamics of particles and systems of particles.

2. Calculus of variations (Equations of Lagrange - first and second kind. Least action principle).

3. Lagrangian formalism: D'Alembert principle vs variational principle.

4. Oscillations.

5. Hamiltonian formalism. Canonical transformations. Noether theorem and symmetries.

The aim of the course is to develop an understanding of how Newton's laws of motion can be used to describe the motion of systems of particles and solid bodies, and how the Lagrangian and Hamiltonian approaches allow use of more general coordinates systems, and how the calculus of variations can be used to solve simple continuous optimization problems.

On completion of the course, students should be able to:

* model simple mechanical systems, using Lagrange's equations;

* analyse the dynamics of systems near equilibrium; find the normal modes of oscillation;

* relate the Hamiltonian and Lagrangian approaches;

* recognise and make use of conserved quantities;

* find the Euler-Lagrange equation associated with simple variational problems.

* model simple mechanical systems, using Lagrange's equations;

* analyse the dynamics of systems near equilibrium; find the normal modes of oscillation;

* relate the Hamiltonian and Lagrangian approaches;

* recognise and make use of conserved quantities;

* find the Euler-Lagrange equation associated with simple variational problems.

'A' level Maths or equivalent is required.

Syllabus:

1. Kinematics and dynamics of particles and systems of particles

Review of Newton's laws; centre of mass; basic kinematic quantities: momentum, angular momentum and kinetic energy; circular motion; 2-body problem; conservation laws; reduction to centre of mass frame.

2. Calculus of variations

Examples of variational problems; derivation of Euler-Lagrange equations; natural boundary conditions; constrained systems; examples.

3. Lagrangian formulation of mechanics

Lagrange's equations and their equivalence to Newton's equations, generalised coordinates; constraints; Kepler's law.

4. Oscillations;

Simple harmonic oscillations; coupled harmonic oscillators; normal modes.

5. Hamiltonian formulation

Hamilton's equations, equivalence with Lagrangian formulation; conserved quantities;Poisson brackets, canonical transformations, Liouville's theorems.

Syllabus:

1. Kinematics and dynamics of particles and systems of particles

Review of Newton's laws; centre of mass; basic kinematic quantities: momentum, angular momentum and kinetic energy; circular motion; 2-body problem; conservation laws; reduction to centre of mass frame.

2. Calculus of variations

Examples of variational problems; derivation of Euler-Lagrange equations; natural boundary conditions; constrained systems; examples.

3. Lagrangian formulation of mechanics

Lagrange's equations and their equivalence to Newton's equations, generalised coordinates; constraints; Kepler's law.

4. Oscillations;

Simple harmonic oscillations; coupled harmonic oscillators; normal modes.

5. Hamiltonian formulation

Hamilton's equations, equivalence with Lagrangian formulation; conserved quantities;Poisson brackets, canonical transformations, Liouville's theorems.

This module runs at 3 hours per week. There are 5 lectures and 1 class every fortnight. In the Summer term, 3 revision lectures are given.

- R. Douglas Gregory. (2006)
*Classical Mechanics*, Cambridge: Cambridge University Press.

The above list is indicative of the essential reading for the course. The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students. Further reading can be obtained from this module's reading list.

Coursework / exam | Description | Deadline | Weighting |
---|---|---|---|

Coursework | Test 1 | ||

Coursework | Test 2 | ||

Exam | 120 minutes during Summer (Main Period) (Main) |

Coursework | Exam |
---|---|

20% | 80% |

Coursework | Exam |
---|---|

0% | 100% |

Module supervisor and teaching staff

Availability

No external examiner information available for this module.

Resources

Further information

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