MMath Mathematics and Statistics

This course unit provides students with the methodology to study problems which arise in applied mathematics.  Often, the analytical solution to such a problem involves approximation in terms of a small parameter. We consider asymptotic expansions, such that the error made is controlled.

MATH35041 Pre-Requisites: MATH11412/22 and MATH24420 (or PHYS20171)

PHYS20171 is an acceptable alternative for those Maths-Physics students who took that unit instead of MATH24420

This course unit introduces students to important topics in applied mathematics, developing their understanding of asymptotic methods and calculus of variations. The syllabus is motivated by physical applications of historical importance.

• Apply asymptotic methods to obtain perturbation expansions of algebraic equations.
• Compute asymptotic expansions of integrals containing a parameter.
• Calculate asymptotic solutions to ODEs containing a parameter, matching the inner and outer solutions as appropriate.
• Interpret variational problems and derive and/or solve the corresponding Euler-Lagrange equations. 
• Calculate solutions to standard variational problems (e.g. the brachistochrone, Fermat’s principle, geodesics, minimal surface, the isoperimetric problem, the hanging chain).
• Formulate equations of motion in classical mechanics from a Lagrangian, via Hamilton’s Principle and describe solutions to mechanical problems using the Euler-Lagrange equations and/or perturbation theory.
• Estimate eigenvalues of Sturm-Liouville systems via the Rayleigh-Ritz method 

his course unit provides students with the methodology to study problems which arise in applied mathematics. Often, the analytical solution to such a problem involves approximation in terms of a small parameter. We consider asymptotic expansions, such that the error made is controlled.

Another common type of problem involves finding a shape or path which minimizes or maximizes a physical quantity, e.g. energy, area, time, etc. We study calculus of variations in order to solve such problems by deriving Euler-Lagrange equations.

Throughout, the focus will be on examples and on applying the mathematical methods in a physical context. The course is therefore most suitable for students with strengths and interests in applied mathematics. Fluency in calculus and ODEs will be beneficial.

Syllabus:

Asymptotic Methods:

Motivation, definitions. Perturbations to algebraic equations. Asymptotic expansions of integrals containing a parameter. Asymptotic solution of ODEs containing a parameter.

Calculus of Variations:

Motivational problems. The Euler-Lagrange equation. The Brachistochrone. Fermat’s Principle. Minimal surfaces (soap film). Extensions. Hamilton’s Principle (Least Action), Lagrangian mechanics, Physical applications.. Geodesics. . Constrained problems, the isoperimetric problem, shape of a hanging chain (catenary). Sturm-Liouville equations, the Rayleigh-Ritz method. 

Individual feedback via marking comments, and cohort feedback document circulated.
 

Hinch, E. J. (1991). Perturbation methods. Cambridge University Press. ISBN: 9781139172189

Bender, C.arl M. (2011). Advanced mathematical methods for scientists and engineers: asymptotic methods and perturbation theory. Springer. ISBN: 9781441931870

Riley, K. F. (2006). Mathematical methods for physics and engineering. Cambridge University Press. ISBN: 9780521861533

Freguglia, Paolo. (2016). The early period of the calculus of variations. Birkhäuser. ISBN: 9783319389455

Cline, Douglas. (2018). Variational principles in classical mechanics. University of Rochester River Campus Libraries University of Rochester. ISBN: 9780998837260