Background

Physics: Classical orbital dynamics are completely described by Newtonian Gravity and Newton's Third Law:

Math: In a multi-body system, these equations for a set of coupled differential equations. To solve the positions of the bodies as a function of time, one must employ numeric integration techniques. One such technique is Euler's Method:
Other algorithms exist which may provide increased accuracy and/or reduced computational burden in exchange for increased complexity. Use of a more advanced algorithm should improve your results (and increase your score!).

Computer Science: The previously mentioned physics and numerics are sufficient to compute the properties of a given system. The challenge is often to search for a system with certain properties or, more specifically, to find the set of initial conditions that result in such a system. This could lead you to employ a search algorithm. Humans are particularly good at searching, so beginning your search by hand could point you in the right direction. From there, you may want to automate the search. There are many automated search algorithms out there (various forms of interpolation/extrapolation); we recommend you start with a simple one.

Problem

Develop a "map-quest" for the solar system. Below is a list of some tasks a "map quest" might be able to accomplish, in order of difficulty (in our minds). These are simply suggestions; you are encouraged to implement alternative/additional functionality:

1. Trace the path of an object traveling in the vicinity of the Earth and Moon, subject to the gravitation of both.

2. Find the two stable orbits between the Earth and the moon.

3. Trace the paths of Voyager I and Voyager II.

4. Find fly-by routes for each of the outer planets.

5. Find the optimal route between any two points in space, where "optimal" could mean "fastest" given a fixed amount of energy or "most efficient" given a maximum trip duration.

Hints:
You might want to spend some time thinking about data/visualization. Though computing and printing certain numbers could be sufficient, finding those values could be difficult. Visualization is always helpful... To make your task easier, you probably want to make some assumptions. For example, you could assume that your spacecraft has constant mass. Remember to clearly state your assumptions.

Recommended Readings:
Euler's Method

Orbits

Newton's Law of Universal Gravitation

Voyager Program

NASA Horizons Web Interface