## 7.6

Simulate SAWs in three dimensions. Determine the vairation of with step number and find the value of , where this parameter is defined through the relation (7.9). Compare your results with those in Figure 7.6. You should find that decreases for successively higher dimensions. (It is 1 in one dimension and 3/4 in two dimensions.) Can you explain this trend qualitatively?

## 5.13

Calculate the value of by using numerical integration to estimate the area of a circle of unit radius. Observe how your estimate approaches the exact value (3.1415926…) as the grid size in the integration is reduced.

2013届的同学们：

## 4.19

Study the behavior of our model for Hyperion for different initial conditions. Estimate the Lyapunov exponent from calculation of , such as those shown in Figure 4.19. Examine how this exponent varies as a function of the eccentricity of the orbit.

## Problem 4.9

In this section we saw that orbits are unstable for any value of that is not precisely 2 in (4.12). A related question, which we did not address (until now), is how unstable an orbit might be. That is, how long will it take for an unstable orbit to become obvious. The answer to this question depends on the nature of the orbit. If the initial velocity is chosen so as to make the orbit precisely circular, then the value of in (4.12) will make absolutely no difference. Of course, in practice it is impossible to construct an orbit that is exactly circular, so the instabilities when will always be apparent given enough time. Even so, orbits that start out as nearly circular will remain almost stable for a longer period than those that are highly elliptical. Investigate this by studying orbits with the same value of (say, ) and comparing the hebavior with different values of the ellipticity of the orbit. You should find that the orientation of orbits that are more nearly circular will rotate more slowly than those that are highly elliptical. (尝试利用不是2的 值构造圆形轨道，并讨论各种不同轨道情况下的稳定性。)