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카이스트 고전역학 기말 족보.docx

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 (2 points)

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)

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

And, using , the Lagrangian of the isolated two-particle system thus becomes

, where .
Finally, for an isolated system, the CM canonical momentum is a constant of the motion as
(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)
The Lagrangian for an isolated two-particle system in the CM reference frame is

Hence, once the Euler-Lagrange equation for is given as

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


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