When we describe motion, we often resort to using some coordinate system. Typically this choice is the Cartesian (xyz) system we are introduced to sometime between 3rd grade and high school graduation. An issue with this system is it makes some problems very difficult to solve. This usually occurs when the problem has some symmetry that isn't easily expressed in the coordinates. For those situations we can change to a different set. One such set that some people are somewhat familiar with is polar coordinates.

We can use the properties of a coordinate system to extract information about conserved quantities. In Cartesian coordinates we can show that if the acceleration along some axis is zero, then linear momentum along that axis will not change. That is almost trivially true for anyone who has completed a year of introductory physics. In polar coordinates we can show that if the acceleration around the origin is zero, then angular momentum about the origin is conserved. The idea of angular momentum conservation may be kind of familiar, but the notion of it being embedded in the coordinates might be new to many people with a BS in physics. What we want to do is see if we can find conserved quantities in other coordinates.

What we need to do

1. Determine how to describe position, velocity, and acceleration in parabolic coordinates.
2. Set the acceleration for one of our coordinates to zero and try to determine what restrictions that puts on the motion.
3. Assuming we cannot do #2, we can try to deal with some limited cases (force this or that to be zero or constant).
4. We can also try numerical solutions to see if we can extract some useful information.

Claude has been working on the first part. This is a non-trivial exercise for someone taking differential equations. As my background is not in mathematical physics, we are approaching this via a "brute force" method.

1. Define the coordinate curves.
2. Find the unit vectors in this space.
3. Relate those to the Cartesian unit vectors.
4. Write position in the new system.
5. Start taking derivatives.

Progress

• 9 Nov 2024: Second go on acceleration matched the latest. Claude is working on simplifying her expression. Before trying to sort out if there is some conserved quantity or if the path is something interesting, we're going to solve the projectile motion problem. As this motion is parabolic, we expect to see a path along a single parabola, so one coordinate value will not change.
• 5 Nov 2024: I reworked the results without reference to my prior work. Position and velocity agree, but acceleration is different for me this time. A factor of 2 has become a factor of 1/2. I need to do the entire acceleration again.
• 1 Nov 2024: Yesterday Claude showed me her acceleration expression. She's working on simplifying things. This is somewhat difficult because we need to recognize when some group of terms might be factorable, when it might make sense to divide out some product of terms, or when it might help to multiply by 1 in some clever form. That's the rub. We must be clever enough to see the potential use of some manipulation, even if the next step or two looks messier. Claude's work on this is very encouraging.