Analytical Mechanics I

Fall 2015

Course summary:

This junior-level course aims to introduce the techniques of analytical mechanics for studying the dynamics of single-particle objects. In a nutshell, the Lagrangian and Hamiltonian (i.e. analytical) mechanics are alternatives to the Newtonian (i.e. vectorial) description of motion, where the equation of motion is obtained from scalers instead of vectors like forces.
Computer skills to do symbolic calculations and plotting curves (Maple, Mathematica, etc.) is highly desirable.

Contents:

  1. Vector Calculus
  2. Newtonian Mechanics of particles in 1D and 3D
  3. Oscillations
  4. Gravity
  5. Lagrangian Mechanics
  6. Central-force Motion

Evaluation:

Midterm exam #1: 25% (Chs 1-4) 11 Aban: (Grades from 5 and Solutions)
Midterm exam #2: 30% (Chs 3-7) 23 Azar: (Grades from 6 and Solutions)
Final exam: 35% (Chs 1-8, excluding 3.9, 4.3, 4.5, 4.6, 4.7, 4.8, 6, 7.12, 7.13, 8.8, 8.9) 7 Dey: (Grades from 7 and Solutions)


Exercises and quizzes: 10% (to be gained in the Problem-Solving Sessions)
Computer Project(s): +5% (this extra credit aims to motivate students to develop their skills in using computers in physics)


Final Evaluation (exercises and computer projects not yet included!) (Grades from 18)

Reference Textbook:

Additional References:

  1. Analytical Mechanics, Fowles and Cassiday
  2. Mechanics, Symon




Problem-Solving Sessions:

Instructor: Ms. Zare

Day & Time: Saturdays, 3:00 PM to 4:30 PM


Homework Assignments:
Your solutions may be handed in on paper to Ms. Zare, or sent by email to analyticalmechfall2015@gmail.com.
Exercises sheets Topic Due
Problem Set #1 Transformation Matrix - Vectors Tuesday, 31st Sharivar
Problem Set #2 Vector Calculus - Angular Velocity Tuesday, 7th Mehr
Problem Set #3 Newtonian Mechanics (Newton's laws) Tuesday, 14th Mehr
Problem Set #4 Newtonian Mechanics (Energy) Saturday, 25th Mehr
Problem Set #5 Oscillations Tuesday, 5th Aban
Problem Set #6 Gravitation Tuesday, 12th Aban





Computer Projects:

Project #1: Numerical solution to range of a projectile including linear air resistance

Write a computer program (in your favorite language) to do the following tasks:

a) Find numerically the root of Eq. 2.44, i.e. the traveling time t for which Eq. 2.44 vanishes. The range is then calculated simply by plugging the evaluated t in Eq. 2.43. Plot the calculated range as a function of k. (You should reproduce Figure 2.9 of the textbook; parameters, as given in the caption of Figure 2.8, are theta = 60 deg., v0=600 m/s).

b) For k=0.05 1/s and v0=600 m/s, plot the range as a function of throw angle. For which angle, the range is maximized?

Hint: There are a number of iterative root-finding algorithms suitable for this problem. If you implement the Newton's method or the bisection method, you may use as an initial guess the time given by Eq. 2.37.

Hint: Gnuplot is a powerful and user-friendly package to generate scientific plots. The image on the right (see also Figure 2.8 in the textbook) is generated by this gnuplot script.





Project #2: Diagrams of underdamped motion
Using computer programing and graphics, plot the following diagrams for an underdamped oscillator with this parameters set: A = 1 cm, m = 100 g, k = 10 N/cm. Take the initial phase at your convenience.

a) Position and velocity as a function of time
b) Mechanical energy as a function of time
c) Rate of mechanical energy loss as a function of time
d) Speed-position phase diagram

Generate plots for three values of damping parameter: beta = 0.1, 0.2 and 0.5 (1/s).








Project #3: Simple Pendulum.

The equation of a simple pendulum motion is given by Eq. 4.21 of the textbook. Assume L= 1 m. Write a computer program (in your favorite language) to solve numerically Eq. 4.21 to find the angle as a function of time. t. Then plot the following graphs:

a) If the initial angle is 90 degrees and the pendulum starts from rest, plot the calculated angle as a function of time. Also plot the angle obtained by using the analytic solution for the small-angle approximation on the same graph for comparison.

b) Repeat part (a) for three different values of initial angles between zero and pi, or for three different initial velocities from zero to 3 rad/s.

c) Plot the phase diagrams (angular velocity vs angle) for part (b). (Hint: Your plot should look like Fig. 4.11)





Project #4: Free fall
Computer Problem: C2.1, page 81, Fowles' book.

Project #5: Spherical pendulum
Computer Problem: C10.1, page 464, Fowles' book.



Note:
Students who wish to obtain the score of the solving computer problems, should send the solutions (plots and the program source) to four of the projects not later than 19 Dey.
Email address: ali.sadeqiATgmail,
Subject of the email: Computer Problem #?