# ACM30010 Analytical Mechanics

This course introduces students to the principles of mechanics in a mathematical framework and establishes a bridge to the principles of modern physics. The module begins with an introduction to Lagrangian mechanics which enables Newtonian mechanics to be distilled in its purest form. The powerful theory of Calculus of Variations is introduced and several crucial examples are studied, such as the brachistochrone. The course continues by developing Hamiltonian mechanics, a re-formulation of classical mechanics that was invented by William Rowan Hamilton, Ireland's greatest Mathematician.

Course Outline (sample):

1. Introduction to Analytical Mechanics. Generalised Co-ordinates and Generalised Forces in the Mechanical Description of a Single Particle. Degrees of Freedom. Equations of motion. Lagrange equations. Special case of conservative forces: Lagrangian. Ignorable coordinates (e.g., central force problem). Example: the simple pendulum.

2. The Lagrange Equations for a system of particles. Special case of conservative forces: Lagrangian of the N-particle system. Examples: Double pendulum and small oscillations thereof: normal modes and normal frequencies of oscillation.

3. Small Oscillations. General treatment in the conservative-force case. Lagrangian for small oscillations. Normal modes and normal frequencies of oscillations. Diagonalisation of the Lagrangian. Examples: different configurations of systems of masses connected by springs.

4. Hamilton's Principle. Historical context. Minimisation of the action integral. Calculus of Variations. Examples: shortest-length paths (geodesics) on the plane; the brachistochrone; geodesics in spherical surfaces (routes of international airplanes). Extension to N-dimensional paths: the Euler-Lagrange equations. Ignorable coordinates. Hamilton's Principle for mechanical systems.

5. Rigid Body Motion and Euler-Lagrange equations. Degrees of freedom. The angular velocity. The equations of motion and the inertia tensor. Application: freely rotating bodies. Lagrangian formulation. Holonomic constraints. Kinetic energy. Euler angles.

6. The Hamilton Equations. Motivation as the basis for statistical mechanics and quantum mechanics. The Hamiltonian. Conjugate variables. Examples for particles constrained to move on given surfaces.

7* (Extra Material). Classical relativistic particles immersed in a given electromagnetic field: Hamiltonian and Lagrangian approaches. Introduction to the quantum theory through Feynman's path-integral formulation. Example: the double-slit experiment for electron scattering/diffraction.

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Curricular information is subject to change

Learning Outcomes:

Calculate Lagrangians for simple and complex systems and determine equations of motion from a Lagrangian. Model extremum problems using the Calculus of Variations. Explain the behaviour of a system under small perturbations from equilibrium. Understand and model the motion of a rigid body under appropriate constraints and external forces. Calculate Hamiltonians for simple and complex systems and determine equations of motion from a Hamiltonian. Extra Material: Understand the first principles of quantum mechanics in terms of the sum over the classical paths in the path-integral formulation of Feynman.

Applications include: resonant systems, action-at-a-distance interacting particles, systems of constrained particles: rigid body, particles moving on a prescribed surface, coupled pendula, etc.; oscillations near the equilibrium positions, Fermat's principle of light propagation, the brachistochrone, the catenary, minimising/maximising surfaces and curves (geodesics), motion of a relativistic particle in a prescribed electromagnetic field.

Student Effort Hours:
Student Effort Type Hours
Lectures

30

Tutorial

12

Specified Learning Activities

24

Autonomous Student Learning

40

Total

106

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning.
Requirements, Exclusions and Recommendations
Learning Recommendations:

It is recommended that students know vector integral and differential calculus (ACM20150 or equivalent).

Module Requisites and Incompatibles
Not applicable to this module.

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Continuous Assessment: Take-home assignments and in-class exams Varies over the Trimester n/a Standard conversion grade scale 40% No

40

Examination: End of semester examination 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

60

Carry forward of passed components
No

Resit In Terminal Exam
Spring Yes - 2 Hour
Feedback Strategy/Strategies

• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Not yet recorded.

Name Role
Constantinos Menelaou Tutor
Timetabling information is displayed only for guidance purposes, relates to the current Academic Year only and is subject to change.

Autumn

Lecture Offering 1 Week(s) - Autumn: All Weeks Fri 11:00 - 11:50
Tutorial Offering 1 Week(s) - 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 Mon 13:00 - 13:50
Lecture Offering 1 Week(s) - Autumn: All Weeks Tues 11:00 - 11:50
Lecture Offering 1 Week(s) - Autumn: All Weeks Wed 10:00 - 10:50
Autumn