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PH4038   Lagrangian and Hamiltonian Dynamics

Academic year(s): 2019-2020

Key information

SCOTCAT credits : 15

ECTS credits : 7

Level : SCQF Level 10

Semester: 2

Availability restrictions: Not automatically available to General Degree students

Planned timetable: 10.00 am even Mon, 2.00 pm odd Fri, 10.00 am Tue & Thu

The module covers the foundations of classical mechanics as well as a number of applications in various areas. Starting from the principle of least action, the Lagrangian and Hamiltonian formulations of mechanics are introduced. The module explains the connection between symmetries and conservation laws and shows bridges between classical and quantum mechanics. Applications include the central force problem (orbits and scattering) and coupled oscillators.

Relationship to other modules

Pre-requisite(s): Before taking this module you must pass PH3081 or pass PH3082 or ( pass MT2506 and pass MT2507 ). In taking this module you will need a knowledge of vector calculus

Anti-requisite(s): You cannot take this module if you take MT4507

Learning and teaching methods and delivery

Weekly contact: 2 or 3 lectures and some tutorials

Scheduled learning hours: 32

Guided independent study hours: 118

Assessment pattern

As used by St Andrews: 2-hour Written Examination = 75%, Coursework = 25%

As defined by QAA
Written examinations : 75%
Practical examinations : 0%
Coursework: 25%

Re-assessment: Oral Re-assessment, capped at grade 7


Module coordinator: Dr B H Braunecker
Module teaching staff: Dr B Braunecker

Additional information from school

Aims & Objectives

To give students a solid grounding and sufficient training in Lagrangian and Hamiltonian techniques in classical mechanics and their applications, including


  • the Principle of Least Action as the starting point of Lagrangian mechanics
  • traditional applications of Lagrangian mechanics such as mechanical pendulums, planetary motion, collisions and some non-traditional ones
  • appreciating the problem-solving power, generality and elegance of Lagrangian and Hamiltonian techniques
  • understand the fundamental connection between symmetries and conservation laws (Noether theorem)


Learning Outcomes

By the end of the module, students will have a solid knowledge of the central concepts of Classical Mechanics and will have acquired and trained important problem-solving skills. They will be able to


  • establish the Lagrangian, and to derive and solve the equations of motions for many systems subject to the Principle of Least Action
  • calculate conserved quantities from symmetries
  • calculate the Hamiltonian and establish the Hamilton equations
  • be familiar with canonical transformations and Hamilton-Jacobi theory
  • understand the concept of phase space and the conservation of phase-space density (Liouville's theorem)
  • acquire a deep knowledge of the Hamiltonian formalism that is crucial for the formulation and understanding of quantum mechanics



Review of Newtonian mechanics.

Equations of Motion: The Principle of least action. The Euler-Lagrange equations. Free and interacting particles in non-relativistic mechanics. Introductory examples to illustrate the abstract concepts that follow in sections 2 and 3.

Conservation Laws: Energy, momentum, angular momentum, centre of mass. The Noether theorem.

Canonical formalism: Hamiltonian techniques. Canonical transformations. Liouville theorem (Hamilton-Jacobi theory).

Applications: Two-body problem. Kepler problem (planetary motion). Collisions.


Additional information on continuous assessment etc.

Please note that the definitive comments on continuous assessment will be communicated within the module. This section is intended to give an indication of the likely breakdown and timing of the continuous assessment.


This module is typically taken in JH by theoretical physicists, and in SH by those doing an MPhys in other degree programmes in the School. It is sufficiently core to the programmes that it is included in the summary of deadlines etc on the School’s Students and Staff web pages. Five tutorial sheets will be issued over the semester in two week intervals. They contain questions that will deepen the understanding of the current topics in the lectures, and they are required to be handed in for marking. This accounts for 25% of the module mark. Tutorials take the form of “whole class” tutorials where the solutions and underlying physics and problem-solving strategies can be discussed.


Accreditation Matters

This module may not contain material that is part of the IOP “Core of Physics”, but does contribute to the wider and deeper learning expected in an accredited degree programme. The skills developed in this module, and others, contribute towards the requirements of the IOP “Graduate Skill Base”.


Recommended Books

Please view University online record:


General Information

Please also read the general information in the School's honours handbook that is available via