## MA222-5-SP-CO:

Analytical Mechanics

## Key module for

BSC G1F4 Mathematics with Physics (Including Placement Year),

BSC G1F5 Mathematics with Physics (Including Foundation Year),

BSC GCF3 Mathematics with Physics (Including Year Abroad)

## Module description

This module introduces general concepts and methods for the description and analysis of the motion and dynamics of particles, systems of particles, rigid bodies, and fields.

Assuming a basic knowledge of Newtonian mechanics, students will develop advanced techniques necessary to study more complicated, multi-particle systems and rigid bodies. The central part of the module is the Lagrangian and Hamiltonian formulation of Classical Mechanics, which allow for simplified treatments of many interesting problems and provide the foundation for the modern understanding of dynamics.

## Module aims

The aims of this module are:

- To provide a detailed introduction to the analytical foundations of Classical mechanics.
- To deal with the fundamentals of Lagrangian and Hamiltonian mechanics.
- To cover the range of applications from motion in central potentials, scattering and collisions, small oscillations, and stability, to motion and dynamics of rigid bodies.

## Module learning outcomes

By the end of this module, students will be expected to be able to:

- Demonstrate knowledge and understanding of the Lagrangian formalism of classical mechanics.
- Find and solve the Euler-Lagrange equations associated with simple problems.
- Understand the concept of symmetries and conserved quantities (integrals of motion). Recognise and make use of conserved quantities.
- Demonstrate knowledge and understanding of the Hamiltonian formalism of classical mechanics.
- Find the Hamilton equations associated with simple problems.
- Demonstrate ability to use a range of established techniques and a commensurate level of skills in solving problems relating to dynamics of particles and rigid bodies.

## Module information

*Indicative Syllabus*

Newton's laws of motion (single particles and systems of many particles).

- Kinematics and dynamics of particles and systems of particles; review of Newton's laws of motion; angular momentum; conservation laws; energy; momentum; examples.

The Lagrangian formalism

- The principle of least action; changing coordinate systems; constraints and generalised coordinates; Noether's theorem and symmetries; examples.

Small oscillations and stability

- Simple harmonic oscillations; stability; double pendulum.

The motion of rigid bodies

- Kinematics; inertia tensor; Euler's equations; free tops; Euler's angles; examples.

Hamiltonian formalism

- Hamilton's equations; the Legendre transform; conservation laws; Liouville's theorem; Poincaré's recurrence theorem; Poisson brackets; canonical transformations; examples.

## Learning and teaching methods

Teaching in the School will be delivered using a range of face-to-face lectures, classes, and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.

## Bibliography

## Assessment items, weightings and deadlines

Coursework / exam | Description | Deadline | Coursework weighting |
---|---|---|---|

Coursework | Test | ||

Exam | Main exam: In-Person, Open Book (Restricted), 120 minutes during Summer (Main Period) | ||

Exam | Reassessment Main exam: In-Person, Open Book (Restricted), 120 minutes during September (Reassessment Period) |

### Additional coursework information

Reassessment strategy: If a student fails the test and the exam they will only take the resit exam which will be worth 100%; If a student fails the exam but passes the test they will only take the resit exam, the mark for which will be re-aggregated with the test mark; If a student fails the test but passes the exam they will take the resit test, the mark for which will be re-aggregated with the exam mark.

### Exam format definitions

**Remote, open book:**Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.**In-person, open book:**Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.**In-person, open book (restricted):**The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.**In-person, closed book:**The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary, for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.

Your department will provide further guidance before your exams.

### Overall assessment

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

10% | 90% |

### Reassessment

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

10% | 90% |

## External examiner

**18**hours,

**17 (94.4%)**hours available to students:

**1**hours not recorded due to service coverage or fault;

**0**hours not recorded due to opt-out by lecturer(s), module, or event type.

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