Course Webpage for PHYS-510 (2025 Spring) at SIU--Carbondale

Course Webpage for
PHYS 510: Classical Mechanics (2025 Spring)
Department of Physics, Southern Illinois University--Carbondale
Last updated: May 14, 2025. (This page is no more maintained.)

General information:
- Instructor: K. V. Shajesh (Office: Neckers 491, Tel.: +1-618-453-5212)
- Class meetings: TR 9:35-10:50 AM. in Neckers 410.
- Office hours for instructor: MWF 9:30-11:00 AM; MF 3:00-4:00 PM.
- Syllabus for the course. The Provost Office has put together a Syllabus Attachment to assist the student. It lists and provides web links for many elements that are important for a student.
Resources:
- References:
  - Classical Mechanics, by Goldstein, Poole, and Safko. (Available in archive.org.)
  - Mechanics, by L.D. Landau and E.M. Lifshitz. (Available in archive.org.)
  - Mechanics, by Arnold Sommerfeld. (Available in archive.org.)
  - Mathematical Methods of Classical Mechanics, by V. I. Arnold. (Available in archive.org.)
  - Calculus of Variations, by I. M. Gelfand and S. V. Fomin. (Available in archive.org.)
- Past course webpages: [2024S, 2022S, 2021S, 2020S, 2019S, 2018S, 2017S]
- Lectures
  - Lecture recordings: [2024S, 2022S, 2021S, 2020S]
  - Informal Notes: This is not a substitute for a textbook.
- Quizzes and Homework
  - Quizzes: Due in class in the beginning of the next lecture.
  - Homework: Due in Mail Box before 4:30 PM on due date.
- Exams: Old Exams are available in past course webpages.
- Miscellaneous
  - Grades will be posted on SIU MyCourses (password protected).

COURSE CONTENT:

Conservative forces in Newtonian mechanics:
1. Conservative forces
2. Position dependent forces
3. Velocity dependent forces
Resources:
Homework No. 01
Keywords: Conservative forces

Examples:
- Harmonic oscillator, gravitational, and electromagnetic forces
- Drag forces
- Motion of a charge particle in a uniform magnetic field.
Variational principle:
1. Review delta function
2. Multivariable derivative
3. Functional derivative [20240201video]
4. Fermat's principle
5. Geodesics on surfaces
6. Brachistochrone on surfaces
Resources:
Homework No. 02
Homework No. 03
Keywords: Calculus of variations, geodesics, brachistochrone, Fermat's principle, cycloid, catenary, isochrone

Examples:
- Snell's law (Fermat's principle) for layered configurations; Mirage phenomena
- The curve of shortest distance connecting two points on a plane is a straight line, and on the surface of a sphere is a great circle.
- Brachistochrone curve
  Cycloid: A mass (perfectly) rolling on a surface in uniform gravity.
  Catenary: A rope hanging in a uniform gravity.
Stationary action principle in 1-dimension:
1. Conservative force, path independence, potential energy, closed path integral, curl of force
2. Newtonian mechanics for conservative forces
3. Lagrangian, Action
4. Euler-Lagrange equations of motion
5. Canonical momentum
6. Hamiltonian, Legendre transformation
7. Hamilton equations of motion
Resources:
Homework No. 04
Keywords: Action principle, Lagrangian, Hamiltonian

Examples:
- A mass in a uniform gravitational field
- A mass in gravitational field due to spherical mass
Stationary action principle in 3-dimension:
1. Lagrangian and Euler-Lagrange equations of motion
2. Hamiltonian and Hamilton equationsa of motion
3. Symmetries, cyclic coordinates, and conserved quantities
Resources:
Homework No. 05
Keywords: Action principle, Lagrangian, Hamiltonian, symmetries, cyclic coordinate, conserved quantites

Examples:
- A mass in a uniform gravitational field
- Two-body problem
(Preview of) Midterm Exam No. 01
Midterm Exam No. 01 (February 20, Thursday).
Motion with constraints and the Lagrangian:
1. Holonomic constraints
2. Lagrangian for holonomic constraints
Resources:
Homework No. 06
Keywords: Holonomic constraints, generalized coordinates

Examples:
- Atwood's machine
- Motion on an incline
Lagrange maultiplier
1. Using the gradient operator to model the force of constraint.
2. Lagrangian multiplier
Resources:
Homework No. 07
Keywords: Lagrangian multiplier, D'Alembert's principle

Examples:
- Sphercal pendulum
- Atwood's machine
- Perfectly rolling wheel
Small angle approximation
1. Small angle approximation
2. Fourier transform method to solve linear differential equations
3. Normal modes
Resources:
Homework No. 08
Keywords: Normal modes, small angle approximation

Examples:
- Normal modes of two masses connected to three springs in series.
- Normal modes of a double pendulum.
- Normal modes in LC circuits.
Stationary action principle (with variation at boundaries)
1. Generators of infinitismal transformations
2. Noether's theorem: Transformation, Symmetry, and Conservation principles.
Resources:
Homework No. 09
Keywords: Generators, Noether's theorem

Examples:
- Non-conservation principles: Virial theorems
- Boost operator
- Infinitesimal unitary transformations in quantum mechanics
Midterm Exam No. 02 (March 27, Thursday).
Poisson bracket: Variational calculus of dynamics
1. Poisson bracket
2. Generators of infinitesimal transformations
3. Statement of symmetry and transformation in terms of Poisson bracket.
Resources:
Homework No. 10
Keywords: Generators, Symmetry and transformations

Examples:
- Poisson bracket for simple and compound functions
- Poisson bracket between generators
Kepler problem
1. Orbit: Elliptic, parabolic, hyperbolic, circular
2. Derive Kepler's laws of planetary motion
3. Laplace-Runge-Lenz vector
4. Precession of perihelion
Resources:
Homework No. 11
Homework No. 12 (optional)
Keywords: Kepler's laws

Examples:
- Elliptic orbits versus hyperbolic orbits
Special relativity
1. Lorentz transformation
2. Geometry of Lorentz transformation
3. Relativistic kinematics
4. Relativistic dynamics
Resources:
Homework No. 13 (optional)
Homework No. 14 (optional)
Keywords: Special relativity

Examples:
- Relativistic motion of an electric charge in a uniform magnetic field.
- Relativistic motion of an electric charge in a uniform electric field.
Final Exam (May 8, Thursday, 8:00 AM-10:00 AM, in class)

More information:

Evaluation:

Quiz 10% and y (required to benefit from the curve)

Homework 30%

Midterm Exams (2x20%) 40%

Final Exam 20%

Exam schedule:

Midterm Exam 1 --- Fenruary 20 (Thursday)

Midterm Exam 2 --- March 27 (Thursday)

Final Exam --- May 8 (Thursday, 8:00 AM-10:00 AM, in class)

Last updated on May 14, 2025 (and is no more maintained) by K. V. Shajesh (kvshajesh@gmail.com).

Quiz	10% and y (required to benefit from the curve)
Homework	30%
Midterm Exams (2x20%)	40%
Final Exam	20%

Midterm Exam 1	---	Fenruary 20 (Thursday)
Midterm Exam 2	---	March 27 (Thursday)
Final Exam	---	May 8 (Thursday, 8:00 AM-10:00 AM, in class)