## Classical Mechanics

 Unit code: MATH20512 Credit Rating: 10 Unit level: Level 2 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

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#### Aims

The course unit aims 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, how the Lagrangian and Hamiltonian approaches allow use of more general coordinate systems, and how the calculus of variations can be used to solve simple continuous optimization problems.

#### Overview

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, we shall develop the advanced techniques necessary for the study of more complicated systems. We shall also consider the beautiful extensions of Newton’s equations due to Lagrange and Hamilton, which allow for simplified treatments of many interesting problems and which 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 maximization or minimization of integral quantities. The course is a useful primer to third and fourth level course units in physical applied mathematics.

#### Learning outcomes

On completion of this unit successful students should be able to:

• Calculate elementary mechanical properties of a continuum body or a system of interacting particles including total mass, moment of inertia, kinetic energy, and centre of mass and prove simple identities involving these quantities.
• Construct and solve practical optimization problems using the calculus of variations, and prove that the Euler-Lagrange equations provide the optimal solution.
• Construct Lagrangians and Hamiltonians for simple mechanical systems comprising continuum bodies, particles, and springs, and derive Lagrange's and Hamilton's equations respectively.
• Calculate the mass-inertia matrix associated with a mechanical system, and determine whether the Lagrangian is regular.
• Recall and apply Noether's theorem, and identify conserved quantities in Lagrangian and Hamiltonian formulations of a mechanical system.
• Find the equilibrium configurations of one-dimensional anharmonic mechanical systems, determine their stability, and find the Hills region for a given total energy.
• Linearize a two-dimensional mechanical system about a stable equilibrium, and obtain the general solution to the linearized solution by identifying and interpreting the normal modes and characteristic frequencies.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Coursework (worth 20%) set around the middle of the semester
• End of semester examination (worth 80%).

#### Syllabus

1. Newtonian Mechanics of 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. [5]

2. Calculus of variations
Examples of variational problems; derivation of Euler–Langrange equations; natural boundary conditions; constrained systems; examples. [5]

3. Lagrangian formulation of mechanics
Lagrange’s equations and their equivalence to Newton’s equations, generalized coordinates; constraints; cyclic variables; examples. [5]

4. Potential wells and oscillations
Particle in a potential well; coupled harmonic oscillators; normal modes. [4]

5.Hamiltonian formulation
Hamilton’s equations, equivalence with Lagrangian formulation; equilibria; conserved quantities. [3]

• Classical Mechanics, by R.D. Gregory, CUP.
• Classical Mechanics, by T.W.B. Kibble. F.H. Berkshire, Addison Wesley

#### Feedback methods

Feedback tutorials will provide an opportunity for students’ work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer’s office hour.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

Gareth Wyn Jones - Unit coordinator