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 (¡¦(»ý·«)
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
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