## PHYSICS 3006: Advanced Dynamics and Relativity

**Level:**III

**Semester:**2

**School:**Physics

**Units:**3

**Prerequisites:**Classical Physics II, Electromagnetism II, Multivariable and Complex Calculus II, Differential Equations II

**Corequisites:**None

**Assumed Knowledge:**None

**Recommended Courses:**Physics III, Quantum Mechanics III

**Courses that have this as a prerequisite:**Differential Geometry and General Relativity (Honours), Quantum Field Theory (Honours), Relativistic Quantum Mechanics (Honours), Gauge Field Theory (Honours)

**Requirement for Majors:**Theoretical Physics, Experimental and Theoretical Physics

**Lecturer(s) in 2014:**Professor Anthony Williams (CSSM Website, Staff Directory, Office: Physics 126)

**Syllabus:**The course can broadly be divided into three sections. One half of the course is analytical mechanics. One third of the course covers an introduction to Special Relativity and relativistic kinematics. The remaining sixth of the course is an introduction into Electrodynamics.

Special Relativity (SR) is an important part of any physicist's knowledge set. It was pioneered by Einstein in the early 20th century and is fundamental to the understanding of Einstein's later General Relativity and is critical to the development of Quantum Field Theories. Many of the concepts introduced are also valuable in later courses in Honours Physics.

SR begins with an introduction of the formalism of inertial frames, fundamental to the course. From there, the two Einstein axioms are introduced, from which the remainder of SR is derived. From these key principles, the Lorentz Transformations are derived for boosts and rotations. Sometimes the mathematics can be overwhelming and may require further study out of class hours. A number of key consequences can then be derived, including time dilation, length contraction, velocity addition and the Doppler Effect. A few lectures will focus on how to manage position, velocity, force and momentum in four dimensions, with some key results and ideas about four-vectors. Four-momentum conservation is then used to calculate key quantities from particle collisions.

Analytical Mechanics is a reformation of Newtonian Mechanics into a form which is easier to solve and more applicable for Quantum Mechanics. The concept of forces is largely replaced with that of the Action, from which the Lagrangian and Hamiltonians are introduced. Each of these has equations of motion which can be solved for systems more complex than have been dealt with before, yet only a small number of solutions are analytical. Further analytical results can be obtained when it is assumed that oscillations are small. Some of the concepts taught here like Lagrangians and Noether's Theorem might seem pointless at the time, but play a very important role in the Standard Model.

The Electrodynamics section builds briefly upon Maxwell's equations, developed at the end of the Electromagnetism component of Physics III. Maxwell's equations are extended to four dimensions, leading to the field-strength tensor. The concepts here are difficult and confusing, but will be returned to in more detail in the study of Quantum Field Theories during Honours.

**Textbook(s):**The textbook for the Special Relativity section is Rindler, W.,

*Introduction to Special Relativity*, 2nd ed, OUP 1991 (Library (2 Copies),

**Unibooks, Amazon, Online PDF). Although it is a good reference text, the content is covered sufficiently well in lectures and the class notes, so there is little need to purchase the textbook outright. However, the section about covariant and contravariant vectors and tensors can be a little confusing, so reading up is necessary.**

The textbook for the Lagrangian Mechanics section is Goldstein, H., Poole, C., and Savko., J

*Classical Mechanics*3rd ed., Addison-Wesley, 2002 (Library (Several copies), Unibooks, Amazon, Online PDF). The Goldstein text has been the gospel of Analytical Mechanics since the 1950s. In a field that doesn't change much these days, it is an excellent text which explains the detail of Lagrangian and Hamiltonian Mechanics to a depth much deeper than is covered in lectures. However, unless you plan additional reading into Hamilton-Jacobi dynamics, the book is not a necessary purchase. Although that topic is interesting, it is not required for further understanding of Quantum Mechanics. Nevertheless, a bookshelf of Physics books would not be complete without Goldstein.

There is no textbook for the Electrodynamics section.

**Class Notes:**As of 2014, the lecture notes used were a combination of handwritten and printed notes written by Rod Crewther when he last taught the course. Those notes were added to by Professor Williams. However, in 2015, a set of fresh, printed notes might be available. It is also worth noting that Professor Williams is very ready to adapt his teaching style on request of the class, though sit near the front as his whiteboard markers are often nearly empty.

**Assessment in 2014:**Exam (60% to 70%), One major assignment (Lagrangian Mechanics) (15% to 20%), One mid-semester test (Special Relativity)

**Exams:**Past exams are available on MyUni, back to 2006. Each exam is typically 3 hours in length, and divided into three sections. There is typically a section on both Lagrangian Mechanics and Special Relativity, though the questions were mixed in the 2014 exam. Questions for Lagrangian Mechanics typically involve definitions and deriving the equations of motion for different mechanical systems. Questions for Special Relativity are typically derivations done in class. The questions presented in section C are typically very mathematical and require more than a page of working out. However the number of questions is limited, and past exam questions tend to repeat over time.

**Tutorials:**Tutorials are conducted weekly by a PhD student. There is one tutorial stream for the entire class. Tutorials are not compulsory, nor is attendance or effort marked. However, completing the tutorials is critical to the understanding of the course content, and should be completed weekly before the tutorial in order to ask questions.

**Advice:**ADR is one of the most conceptually demanding courses in Level III physics. However, the content is critical for any future in theoretical or quantum physics, and those students interested in these fields should find this course enjoyable. Additionally students studying Astrophysics will find this course useful for future studies of general relativity. Nevertheless, good marks are definitely achievable if you focus well.

**Links:**Advanced Dynamics and Relativity in Course Planner.