APP MTH 3022 - Optimal Functions and Nanomechanics III

North Terrace Campus - Semester 2 - 2014

Many problems in the sciences and engineering seek to find a shape or function that minimises or maximises some quantity. For example, an engineer may design a yacht's hull to minimise drag. And in nature, the shape that a complicated protein might adopt is determined in part by the lowest-energy state available to the protein during the folding process. The Calculus of Variations extends familiar calculus techniques to answer questions regarding optimal geometry or functions. The Calculus of Variations is applicable to almost all continuous physical systems, ranging through elasticity, solid and fluid mechanics, electro-magnetism, gravitation, quantum mechanics and string theory. In this course we will consider, in particular, problems from Nanoscience. Nanoscience is a multidisciplinary field at the nexus of physics, chemistry and engineering. Materials and systems that may be very well understood at the macroscale can often exhibit surprising phenomena at the nanoscale. Topic covered are: Classical Calculus of Variations problems such as the geodesic, catenary and brachistochrone; derivation and use of the Euler-Lagrange equations; multiple dependent variables (Hamilton's equations) and multiple independent variables (minimal surfaces); constrained problems, problems with variable end points and those with non-integral constraints; conservation laws and Noether's theorem; computational solutions using Euler's finite difference and Rayleigh-Ritz methods. Many of the examples considered will draw from continuum modelling of the intermolecular interaction potential utilizing special functions (such as gamma, beta, hypergeometric and generalized hypergeometric functions of two variables) and by application of Euler's elastica.

• General Course Information
Course Details
Course Code APP MTH 3022 Optimal Functions and Nanomechanics III Applied Mathematics Semester 2 Undergraduate North Terrace Campus 3 Up to 3 contact hours per week MATHS 1012 (Note: from 2015 the prerequisites for this course will be (MATHS 2101 and MATHS 2102) or (MATHS 2201 and MATHS 2202) . Please plan your 2014 enrolment accordingly). MATHS 2101 and MATHS 2102 or MATHS 2201 and MATHS 2202 Many problems in the sciences and engineering seek to find a shape or function that minimises or maximises some quantity. For example, an engineer may design a yacht's hull to minimise drag. And in nature, the shape that a complicated protein might adopt is determined in part by the lowest-energy state available to the protein during the folding process. The Calculus of Variations extends familiar calculus techniques to answer questions regarding optimal geometry or functions. The Calculus of Variations is applicable to almost all continuous physical systems, ranging through elasticity, solid and fluid mechanics, electro-magnetism, gravitation, quantum mechanics and string theory. In this course we will consider, in particular, problems from Nanoscience. Nanoscience is a multidisciplinary field at the nexus of physics, chemistry and engineering. Materials and systems that may be very well understood at the macroscale can often exhibit surprising phenomena at the nanoscale. Topic covered are: Classical Calculus of Variations problems such as the geodesic, catenary and brachistochrone; derivation and use of the Euler-Lagrange equations; multiple dependent variables (Hamilton's equations) and multiple independent variables (minimal surfaces); constrained problems, problems with variable end points and those with non-integral constraints; conservation laws and Noether's theorem; computational solutions using Euler's finite difference and Rayleigh-Ritz methods. Many of the examples considered will draw from continuum modelling of the intermolecular interaction potential utilizing special functions (such as gamma, beta, hypergeometric and generalized hypergeometric functions of two variables) and by application of Euler's elastica.
Course Staff

Course Coordinator: Dr Barry Cox

Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
Course Learning Outcomes
1. Apply the calculus of variations to find optimal solutions to problems.
2. Appreciate the derivation of many physical laws from variational principles.
3. Express interaction calculations using hypergeometric and other special functions.
4. Formulate models for nanoscale interactions.
5. Find optimal solutions to variational problems both analytically and numerically, as appropriate.
6. Explain of the role of applied mathematics in interdisciplinary research.

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)
Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. 1,2,3,4,5,6
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 1,3,4,5
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 4,5
Skills of a high order in interpersonal understanding, teamwork and communication. 2,6
A proficiency in the appropriate use of contemporary technologies. 3,5
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. 1,2,3,4,5,6
A commitment to the highest standards of professional endeavour and the ability to take a leadership role in the community. 6
An awareness of ethical, social and cultural issues within a global context and their importance in the exercise of professional skills and responsibilities. 6
• Learning Resources
None.
Recommended Resources

Students may wish to consult any of the following books, available in the Library.

• Calculus of variations / by L. E. Elsgolc
• Calculus of variations / by I. M. Gelfand and S. V. Fomin
• The calculus of variations / by Bruce van Brunt
• Calculus of variations / by J. W. Craggs
• Calculus of variations : with applications to physics and engineering / by Robert Weinstock
• Lectures on the calculus of variations / by Gilbert A. Bliss
• Problems and exercises in the calculus of variations / by M.L. Krasnov, G.I. Makarenko, A.I. Kiselev
Online Learning

Assignments, tutorial exercises, handouts, and course announcements will be posted on MyUni.

• Advanced Engineering Analysis : The Calculus of Variations and Functional Analysis with Applications in Mechanics / by Leonid P. Lebedev, Michael J Cloud and Victor A Eremeyev
is available as an e-book via the Library catalogue.
• Learning & Teaching Activities
Learning & Teaching Modes
The lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Students are expected to attend all lectures, but lectures will be recorded to help with occasional absences and for revision purposes. In fortnightly tutorials students present their solutions to assigned exercises and discuss them with the lecturer and each other. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

 Activity Quantity Workload Hours Lectures 30 90 Tutorials 6 18 Assignments 6 48 Total 156
Learning Activities Summary
 Week 1 Background Introduction - Extrema - Vector Calculus refresher Week 2 Fundamental Euler-Lagrange equations - Autonomous systems Week 3 Fundamental More autonomous - Geodesics - Invariance Week 4 Extension Higher order derivatives - Special functions Week 5 Extension / Nanomechanics Several dependent variables - Van der Waals forces - Interaction potentials Week 6 Nanomechanics Hypergeometric functions - Nanotube oscillators Week 7 Extension Several independent variables - Numerical solutions - Ritz method Week 8 Fundamental Lagrange multipliers - Isoperimetric problems Week 9 Fundamental Multiple integral constraints - Natural boundary conditions - Curvature and the elastica Week 10 Nanomechanics Join regions for carbon nanostructures - Free endpoints - AFM cantilever Week 11 Extension Traversals - Broken extremals - Hamiltonian formulation Week 12 Extension Conservation laws - Classifying extrema
Tutorials at end of Weeks 2, 4, 6, 8, 10 and 12 will cover the material of the previous five lectures.
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

Assessment Summary
 Task Type Due Weighting Learning Outcomes Examination Summative Examination Period 70 % All Assignments Formative and Summative Weeks 2, 4, 6, 8, 10 and 12 30 % All
Assessment Related Requirements
An aggregate score of 50% is required to pass the course.
Assessment Detail
 Task Set Due Weighting Assignment 1 Week 1 Week 2 5 % Assignment 2 Week 3 Week 4 5 % Assignment 3 Week 5 Week 6 5 % Assignment 4 Week 7 Week 8 5 % Assignment 5 Week 9 Week 10 5 % Assignment 6 Week 11 Week 12 5 %
Submission
Homework assignments must be submitted on time with a signed assessment cover sheet. Late assignments will not be accepted. Assignments will be returned within two weeks. Students may be excused from an assignment for medical or compassionate reasons. Documentation is required and the lecturer must be notified as soon as possible.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
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