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Unit outline_

MATH3977: Lagrangian and Hamiltonian Dynamics (Adv)

Semester 2, 2023 [Normal day] - Camperdown/Darlington, Sydney

Lagrangian and Hamiltonian dynamics are reformulations of classical Newtonian mechanics into a mathematically sophisticated framework using arbitrary coordinate systems. This formulation of classical mechanics generalises elegantly to modern theories of relativity and quantum mechanics. The unit develops dynamics from the Principle of Least Action using the calculus of variations. Emphasis is placed on the relation between the symmetry and invariance properties of the Lagrangian and Hamiltonian functions and conservation laws. Coordinate and canonical transformations are introduced to simplify apparently complicated dynamical problems. Connections between geometry and different physical theories beyond classical mechanics are explored. Students will be expected to describe and solve mechanical systems of some complexity including planetary motion and to investigate stability. Hamilton-Jacobi theory will be used to solve problems ranging from geodesic motion (shortest path between two points) on curved surfaces to relativistic motion in the vicinity of black holes.

Unit details and rules

Academic unit Mathematics and Statistics Academic Operations
Credit points 6
Prerequisites
? 
A mark of 65 or greater in 12 credit points of MATH2XXX units of study
Corequisites
? 
None
Prohibitions
? 
MATH4077
Assumed knowledge
? 

None

Available to study abroad and exchange students

Yes

Teaching staff

Coordinator Holger Dullin, holger.dullin@sydney.edu.au
Type Description Weight Due Length
Supervised exam
? 
Final Exam
Final Exam
60% Formal exam period 2 hours
Outcomes assessed: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10
Small test Quiz I
Quiz I
13% Week 07
Due date: 12 Sep 2023 at 13:00

Closing date: 12 Sep 2023
50 minutes
Outcomes assessed: LO1 LO6 LO5 LO4 LO3 LO2
Small test Quiz II
Quiz II
13% Week 11
Due date: 19 Oct 2023 at 13:00

Closing date: 19 Oct 2023
50 minutes
Outcomes assessed: LO1 LO10 LO9 LO8 LO7
Assignment Assignment
Assignment
14% Week 13
Due date: 05 Nov 2023 at 23:00

Closing date: 12 Nov 2023
Submitted work
Outcomes assessed: LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8 LO9 LO10

Assessment summary

Detailed information for each assessment can be found on Canvas.

Final exam: If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as a viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original exam. The format of the alternative assessment will be determined by the unit coordinator.

Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

Representing complete or close to complete mastery of the material.

Distinction

75 - 84

Representing excellence, but substantially less than complete mastery.

Credit

65 - 74

Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.

Pass

50 - 64

Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.


.

For more information see guide to grades.

Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

  • Deduction of 5% of the maximum mark for each calendar day after the due date.
  • After ten calendar days late, a mark of zero will be awarded.

Academic integrity

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

Use of generative artificial intelligence (AI) and automated writing tools

You may only use generative AI and automated writing tools in assessment tasks if you are permitted to by your unit coordinator. If you do use these tools, you must acknowledge this in your work, either in a footnote or an acknowledgement section. The assessment instructions or unit outline will give guidance of the types of tools that are permitted and how the tools should be used.

Your final submitted work must be your own, original work. You must acknowledge any use of generative AI tools that have been used in the assessment, and any material that forms part of your submission must be appropriately referenced. For guidance on how to acknowledge the use of AI, please refer to the AI in Education Canvas site.

The unapproved use of these tools or unacknowledged use will be considered a breach of the Academic Integrity Policy and penalties may apply.

Studiosity is permitted unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission as detailed on the Learning Hub’s Canvas page.

Outside assessment tasks, generative AI tools may be used to support your learning. The AI in Education Canvas site contains a number of productive ways that students are using AI to improve their learning.

Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

WK Topic Learning activity Learning outcomes
Week 01 Calculus of Variations Lecture (3 hr)  
Week 02 Lagrangian Dynamics Lecture (3 hr)  
Week 03 Central Forces Lecture (3 hr)  
Week 04 Covariance of the Lagrangian Formalism Lecture (3 hr)  
Week 05 Incorporating Constraints Lecture (3 hr)  
Week 06 Hamiltonian Dynamics Lecture (3 hr)  
Week 07 Geometric Connections Lecture (3 hr)  
Week 08 Symmetry and Conservation Laws Lecture (3 hr)  
Week 09 Hamilton-Jacobi Theory Lecture (3 hr)  
Week 10 Completely Integrable Systems Lecture (3 hr)  
Week 11 Applications Lecture (3 hr)  
Week 12 Applications Lecture (3 hr)  
Week 13 Revision Lecture (3 hr)  

Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

  • LO1. Recall and explain fundamental definitions, equations and techniques of Lagrangian and Hamiltonian dynamics and the calculus of variations.
  • LO2. Predict essential properties of the motion in a central force field.
  • LO3. Create descriptions of new mechanical systems using Euler-Lagrange equations and analyse and describe the motion determined by these equations.
  • LO4. Explain the concept of a point transformation and apply these in a broad range of familiar contexts.
  • LO5. Design sets of coordinates that are adapted to describe a particular mechanical system.
  • LO6. Analyse systems with constraints using the Lagrangian approach.
  • LO7. Simplify dynamical problems by using familiar context-dependent approaches including applying the relationships between conservation laws and symmetries or separation of variables.
  • LO8. Solve separable dynamical systems with Hamilton-Jacobi theory.
  • LO9. Verify that a given transformation is canonical and produce examples of canonical transformations using generating functions. Apply the Poisson bracket.
  • LO10. Understand the concept of integrable Hamiltonian system and find action variables.

Graduate qualities

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

GQ1 Depth of disciplinary expertise

Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline.

GQ2 Critical thinking and problem solving

Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem.

GQ3 Oral and written communication

Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context.

GQ4 Information and digital literacy

Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies.

GQ5 Inventiveness

Generating novel ideas and solutions.

GQ6 Cultural competence

Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues.

GQ7 Interdisciplinary effectiveness

Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries.

GQ8 Integrated professional, ethical, and personal identity

An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context.

GQ9 Influence

Engaging others in a process, idea or vision.

Outcome map

Learning outcomes Graduate qualities
GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

This section outlines changes made to this unit following staff and student reviews.

No changes have been made since this unit was last offered

Disclaimer

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