Jacobi and Lagrange So far we have explored the Jacobi Integral and the Lagrange points. Now, we are ready to combine the two into one of the most useful concepts in the Circular restricted 3-Body Problem (CR3BP), forbidden regions. You’ve seen them in my visualizations before, but now we’ll finally go over what they are.…
Balancing a Marble Imagine you had a marble and you wanted to put it at the point between the Earth and the Moon where the force of gravity was equal from both. The two forces would cancel out, and the marble would remain in that location forever. If the sun and the earth were not…
Recap In the last two posts on what is the 3-body problem and the dynamics of the three body problem I’ve shown you the following GIF with a spacecrafts trajectory in the earth moon system. The trajectory is bouncing around what I’ve labeled a ‘forbidden region’. In this post we’ll go over the number behind…
Chaos I’ve recreated one of my favorite mathematical demonstrations below. Three double pendulums, all starting with near identical initial conditions, all rapidly diverging. I love this demonstration because it’s such a great example of chaotic dynamics. What is Chaos? Chaotic dynamics, in a nutshell, means that a system is extremely sensitive to initial conditions. That…
Recap In the last post we introduced the 3-body problem along with some constraints that turned it into the circular restricted 3-Body Problem (CR3BP). In the beginning of that post, I had a GIF that traced out a trajectory in the Earth-Moon system. In this post, we will derive the equations of motion for the Circular…
Making a Lab a Home Engineering labs, like most work spaces, begin life as utterly functional sterile rooms. Sometimes they have cubicles, other times offices, and in a few rare occasions an open office layout, but they all begin sterile. Over time, the people who work there begin to fix it up and imbue within…
What is Journal club As a new grad student, I need to read a lot of papers. A rule of thumb for new grad students is to read 5-7 papers every week. As I read through them, I highlight statements, leave comments in the margins, and write up a small note about what I thought…
A Whole New Ballgame In the past, this blog has covered the 2-Body problem which is solvable (Kinda) and the general n-body problem which isn’t. We have a lot of insight into how the 2-body dynamics work, but unfortunately, they break down as soon as we get away from any major body. If you want…
Good Enough Numerical Methods are all about getting an answer that is good enough. A whole class of numerical methods are iterative methods, which continue running until they are good enough. Newton’s method is the example of an iterative method. It keeps going, making the error smaller and smaller until it finds a zero. A good…
Good Enough Numerical methods are always concerned with the term “good enough”. Whether you are using a finite difference to approximate a derivative, or Newtons method to find the roots of a function, we rarely get an exact answer. Usually, our answer is different from the truth by some amount. We generally don’t know what our error…
