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Unit of study_

MATH4077: Lagrangian and Hamiltonian Dynamics

Lagrangian and Hamiltonian dynamics are a reformulation of classical Newtonian mechanics into a mathematically sophisticated framework that can be applied in many different coordinate systems. This formulation 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 make apparently complicated dynamical problems appear simpler. In this unit you will also explore connections between geometry and different physical theories beyond classical mechanics. You will be expected to solve fully dynamical systems of some complexity including planetary motion and to investigate stability using perturbation analysis. You will use Hamilton-Jacobi theory to solve problems ranging from geodesic motion (shortest path between two points) on curved surfaces to relativistic motion in the vicinity of black holes. This unit is a useful preparation for units in dynamical systems and chaos, and complements units in differential equations, quantum theory and general relativity.

Details

Academic unit Mathematics and Statistics Academic Operations
Unit code MATH4077
Unit name Lagrangian and Hamiltonian Dynamics
Session, year
? 
Semester 2, 2021
Attendance mode Normal day
Location Camperdown/Darlington, Sydney
Credit points 6

Enrolment rules

Prohibitions
? 
MATH3977
Prerequisites
? 
(A mark of 65 or greater in 12cp of MATH2XXX units of study) or [12cp from (MATH3061 orMATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3961 or MATH3962 or MATH3963 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3978 or MATH3979)]
Corequisites
? 
None
Assumed knowledge
? 

6cp of 1000 level calculus units and 3cp of 1000 level linear algebra and (MATH2X21 or MATH2X61)

Available to study abroad and exchange students

Yes

Teaching staff and contact details

Coordinator Holger Dullin, holger.dullin@sydney.edu.au
Type Description Weight Due Length
Final exam (Take-home short release) Type D final exam Final exam
Final exam
70% Formal exam period 2 hours
Outcomes assessed: LO9 LO8 LO7 LO6 LO5 LO4 LO3 LO2 LO1 LO10
Online task Quiz I
Quiz I
10% Week 07
Due date: 21 Sep 2021 at 12:00
50 minutes
Outcomes assessed: LO5 LO6 LO1 LO2 LO3 LO4
Online task Quiz II
Quiz II
10% Week 11
Due date: 28 Oct 2021 at 11:00
50 minutes
Outcomes assessed: LO9 LO10 LO1 LO7 LO8
Assignment Assignment
Assignment
10% Week 13
Due date: 14 Nov 2021 at 23:00
Assignment
Outcomes assessed: LO9 LO10 LO1 LO2 LO3 LO4 LO5 LO6 LO7 LO8
Type D final exam = Type D final exam ?

Detailed information for each assessment can be found on Canvas.

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.

For more information see sydney.edu.au/students/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.

Special consideration

If you experience short-term circumstances beyond your control, such as illness, injury or misadventure or if you have essential commitments which impact your preparation or performance in an assessment, you may be eligible for special consideration or special arrangements.

Academic integrity

The Current Student website provides information on academic honesty, academic dishonesty, 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 dishonesty or plagiarism seriously.

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

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 novel 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 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
No changes have been made since this unit was last offered

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