카이스트 고전역학 기말 족보.docx

2015 Classical Mechanics I Final-term Exam
2015. 6. 19 20:00 ~

1. (To be scored by Prof.)

2. (a) Show that the superposition principle does not hold for a nonlinear differential equation such as . Here is a constant. (4 points)

Let and are possible solutions of the differential equation. If we apply a trial superposition solution to the equation, then . Therefore, the superposition principle does not hold for such a nonlinear differential equation.
(b) Consider the system represented as
Here, is a small, positive constant. Show that the system has a simple limit cycle, and describe the motions of the variables with brief sketches. (6 points)
Change the variables as and . Then .
In similar, .
Let , then .
Finally, where ,
For large , in which , and . Thus this system has a simple limit cycle with and . Or and , and and . (4 points)
A brief sketch can be drawn as (…(생략)

3. Consider an isolated two-particle system consisting of and . Gravitational force is the only interaction between the particles. Solve this problem using Lagrange’s or Hamilton’s methods.

(a) Find the Lagrangian for the system. (2 points)

(b) Derive the equation of the motion for the position of the center-of-mass (CM), , and describe the motion. (3 points)

(c) Find the Lagrangian function and derive Lagrange equation for the system in the CM reference frame, which is defined by the condition, . (5 points)

4. (a) Consider a simple plane pendulum consisting of a mass attached to a string of length . After the pendulum is s