APP MTH 3022 - Optimal Functions and Nanomechanics III

North Terrace Campus - Semester 2 - 2019

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. Topics 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
    Course Optimal Functions and Nanomechanics III
    Coordinating Unit School of Mathematical Sciences
    Term Semester 2
    Level Undergraduate
    Location/s North Terrace Campus
    Units 3
    Contact Up to 3 contact hours per week
    Available for Study Abroad and Exchange Y
    Prerequisites (MATHS 2101 or MATHS 2202) and (MATHS 2102 or MATHS 2201)
    Assumed Knowledge Basic computer programming skills such as would be obtained from COMP SCI 1012, 1101, MECH ENG 1100, 1102, 1103, 1104, 1105, C&ENVENG 1012
    Course Description 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.

    Topics 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

    No information currently available.

    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.
    University Graduate Attributes

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

    University Graduate Attribute Course Learning Outcome(s)
    Deep discipline knowledge
    • informed and infused by cutting edge research, scaffolded throughout their program of studies
    • acquired from personal interaction with research active educators, from year 1
    • accredited or validated against national or international standards (for relevant programs)
    All
    Critical thinking and problem solving
    • steeped in research methods and rigor
    • based on empirical evidence and the scientific approach to knowledge development
    • demonstrated through appropriate and relevant assessment
    1-5
    Teamwork and communication skills
    • developed from, with, and via the SGDE
    • honed through assessment and practice throughout the program of studies
    • encouraged and valued in all aspects of learning
    3,6
    Career and leadership readiness
    • technology savvy
    • professional and, where relevant, fully accredited
    • forward thinking and well informed
    • tested and validated by work based experiences
    2,6
  • Learning Resources
    Required 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.
    Workload

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


    Workload
    Activity Quantity Workload Hours
    Lectures 30 90
    Tutorials 6 18
    Assignments 6 48
    Total 156
    Learning Activities Summary
    Schedule
    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
    Assessment
    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
    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.
    Course Grading

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

    M10 (Coursework Mark Scheme)
    Grade Mark Description
    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

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    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.

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