Undergraduate Course: Lagrangian Dynamics (PHYS10015)
Course Outline
School | School of Physics and Astronomy |
College | College of Science and Engineering |
Credit level (Normal year taken) | SCQF Level 10 (Year 3 Undergraduate) |
Availability | Available to all students |
SCQF Credits | 10 |
ECTS Credits | 5 |
Summary | The principles of classical dynamics, in the Newtonian formulation, are expressed in terms of (vectorial) equations of motion. These principles are recapitulated and extended to cover systems of many particles. The laws of dynamics are then reformulated in the Lagrangian framework, in which a scalar quantity (the Lagrangian) takes centre stage. The equations of motion then follow by differentiation, and can be obtained directly in terms of whatever generalised coordinates suit the problem at hand. These ideas are encapsulated in Hamilton's principle, a statement that the motion of any classical system is such as to extremise the value of a certain integral. The laws of mechanics are then obtained by a method known as the calculus of variations. As a problem-solving tool, the Lagrangian approach is especially useful in dealing with constrained systems, including (for example) rotating rigid bodies, and one aim of the course is to gain proficiency in such methods. At the same time, we examine the conceptual content of the theory, which reveals the deep connection between symmetries and conservation laws in physics. Hamilton's formulation of classical dynamics (Hamiltonian Dynamics) is introduced, and some of its consequences and applications are explored. |
Course description |
- Revision of Newtonian Mechanics: Newton's laws; Dynamics of a Particle; Conservation Laws
- Dynamics of a system of particles; Momentum; Angular Momentum; Energy; Transformation Laws
- Use of centre of momentum; Noninertial rotating frames; Summary of Newton's scheme
- Constraints; Generalised coordinates and velocities
- Generalised forces; Derivation of the Lagrange equation
- Lagrangian; Examples
- Using Lagrangian Method. Examples: Atwood's Monkey; particle and wedge; simple pendulum; spherical pendulum
- Rotating frames; Calculus of Variations
- Applications of Variational Calculus; Hamilton's Principle
- Hamilton's Principle; Conservation Laws; Energy Function
- Energy Function; Conservation Laws and Symmetry
- Velocity-dependent Forces;
- Hamiltonian Dynamics; relationship to Quantum Mechanics
- Rigid Body Motion; Introduction; Euler's Equations
- The Symmetric Top - Precession; the Tennis Racquet Theorem
- Lagrangian for a Top; Equations of motion; Conservation Laws
- Symmetric Tops: Zones; Steady Precession; Nutation; Gyroscopes
- Small Oscillation Theory
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Information for Visiting Students
Pre-requisites | None |
High Demand Course? |
Yes |
Course Delivery Information
|
Academic year 2024/25, Available to all students (SV1)
|
Quota: None |
Course Start |
Semester 1 |
Timetable |
Timetable |
Learning and Teaching activities (Further Info) |
Total Hours:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 20,
Summative Assessment Hours 2,
Revision Session Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
52 )
|
Assessment (Further Info) |
Written Exam
100 %,
Coursework
0 %,
Practical Exam
0 %
|
Additional Information (Assessment) |
Degree Examination, 100% |
Feedback |
Optional hand-ins fortnightly |
Exam Information |
Exam Diet |
Paper Name |
Hours & Minutes |
|
Main Exam Diet S2 (April/May) | Lagrangian Dynamics May Exam | 2:120 | | Resit Exam Diet (August) | Lagrangian Dynamics Aug Exam | 2:120 | |
|
Academic year 2024/25, Part-year visiting students only (VV1)
|
Quota: None |
Course Start |
Semester 1 |
Timetable |
Timetable |
Learning and Teaching activities (Further Info) |
Total Hours:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 20,
Summative Assessment Hours 2,
Revision Session Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
52 )
|
Assessment (Further Info) |
Written Exam
100 %,
Coursework
0 %,
Practical Exam
0 %
|
Additional Information (Assessment) |
Degree Examination, 100% |
Feedback |
Optional hand-ins fortnightly |
Exam Information |
Exam Diet |
Paper Name |
Hours & Minutes |
|
Main Exam Diet S1 (December) | Lagrangian Dynamics Dec Exam | 2:120 | |
Learning Outcomes
Consolidation of the learning outcomes in the Entry Requirements in the context of more challenging classical dynamics problems, together with at least two of the following:
a. understanding of the Lagrangian formulation of classical dynamics and the ability to apply it to solve for the motion of point particles and simple bodies in terms of generalised coordinates;
b. understanding of the relationship between symmetries and conservation laws, and knowledge of the Hamiltonian formulation of classical dynamics and Poisson brackets;
c. ability to apply the calculus of variations to solve minimisation problems, and knowledge of the formulation of dynamics in terms of a variational principle;
d. ability to apply Lagrangian methods to solve for the motion of rigid bodies;
e. ability to solve for the small amplitude oscillations of coupled systems.
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Additional Information
Graduate Attributes and Skills |
Not entered |
Additional Class Delivery Information |
Workshop/tutorial sessions, as arranged. |
Keywords | LagD |
Contacts
Course organiser | Dr Alexander Morozov
Tel: (0131 6)50 5289
Email: alexander.morozov@ed.ac.uk |
Course secretary | Ms Alexis Heeren
Tel:
Email: Alexis.Heeren@ed.ac.uk |
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