# J. Sivaloganathan's teaching page

This page will contain information relating to courses I am teaching, copies of the tutorial sheets and summaries of the material covered to date.

## MATH40049: Elasticity

In this course, we develop the general principles of the theory of elasticity. The classical theory of linear elasticity can then be obtained by expanding the constitutive law relating stress to strain in a neighbourhood of the undeformed state.

Course Diary:

Week 1: Motions, deformations and kinematics, homogeneous deformations. Recap of basic linear algebra results

Week 2: The Polar Decomposition Theorem, square root of a positive definite symmetric matrix, stretch tensors, principal stretches, local strain

Week 3: Geometric properties of deformations, stretching of line, area and volume elements, change of variables formulae for volume and surface integrals

Week 4: Balance laws in elasticity, balance of mass, balance of forces and balance of torque, Cauchy stress vector and Cauchy stress tensor, 1st Piola Kirchoff stress tensor. Material-frame indifference, equations of equilibrium in the reference and current configurations.

Week 5: Response functions for Cauchy and Piola-Kirchoff stress tensors, frame indifference in terms of response functions, material symmetries and isotropy, hyperelastic materials, examples of compressible stored energy functions

Week 6: Frame-indifference and material symmetries in terms of the stored energy function. The stored energy function as a function of the principal stretches, examples. Boundary-value problems of elasticity.

Week 7: The equations of linear elasticity, the linear elasticity tensor, elastic moduli.

Week 8: Solutions of simple problems, Uniqueness of solutions in linear elasticity (Clapeyron's Theorem), Variational formulation of nonlinear elasticity (hyperelasticity)

Relevant text books include:

"Mechanics of Continuous Media" by S.C. Hunter

"Mathematical Elasticity, Volume 1: Three-Dimensional Elasticity" by P.G. Ciarlet

"A First Course in Continuum Mechanics" by O. Gonzalez and A.M. Stuart (online version of book available from the university library)

"Nonlinear Elastic deformations" by R.W. Ogden

"Continuum Mechanics" by A.J.M. Spencer

Interesting practical aspects of materials science are contained in:

"The New Science of Strong Materials" by J.E. Gordon (first published 1968)

## MATH40048: Analytical and Geometric Theory of Differential Equations

This course has the intertwoven themes of ordinary differential equations, the Calculus of Variations, Lagrangian Systems and Hamiltonian Systems, and looks at some of the elegant and striking connections between them.

Hamilton’s Principle states that a system evolves in such a way that a given integral over the path, called the action integral, is stationary. Fermat’s Principle, that light rays travel through a material along paths that minimise the total travel time, is an example of Hamilton’s Principle. The condition that the given integral is minimised then leads to a differential equation, called the Euler-Lagrange equation(s), for the path itself. This approach is known generally as a  ‘variational principle’ and many familiar problems in physics, such as Newton’s laws of motion, can be recast in this way. The study of such variational problems constitutes the Calculus of Variations. In many cases, the Euler-Lagrange equations can in turn be recast as a first order autonomous system of differential equations, called a Hamiltonian System, which allows us to deduce general properties of solutions by studying the geometry of the flow induced by the differential equation(s) in the corresponding phase space.

The concepts and principles introduced in this course have led to many developments in modern pure mathematics (such as symplectic geometry and ergodic theory) and, besides applying to the equations of classical mechanics, they have motivated much of modern physics.

Relevant text books include:

"Introduction to Analytical Dynamics" by N.M.J. Woodhouse

"Calculus of Variations"  by Gelfand and Fomin

"Mechanics"  by Landau and Lifshitz

"Optimization-Theory and Applications" by L. Cesari

"Variational Calculus and Optimal Control"  by J.L. Troutman

" Principles of Mechanics" by J.L. Synge and B.A. Griffiths

"Mathematical Methods of Classical Mechanics" by V.I. Arnold

"The Feynman Lectures on Physics",  Volume 2,  by R. Feynman

"Introduction to Dynamics"  by Percival and Richards

"Analytical Dynamics of Particles" by Whittaker

"Mechanics"  by V. Kibble

"A Geometrical Theory of ODEs" by V.I. Arnold

"A Treatise on Analytical Dynamics" by Pars