Maximal invariance group for constant acceleration?

[strangerep]

Recently, over in the relativity forum, Micromass contributed a post:
https://www.physicsforums.com/showpost.php?p=4168973&postcount=89
giving a proof that the most general coordinate transformation preserving the property of zero acceleration (i.e., maps straight lines to straight lines) is of fractional-linear form.

I'm now wondering whether there are generalizations of this sort of thing applicable to, e.g., trajectories of constant acceleration (i.e., nonzero, but constant acceleration), and maybe also trajectories of constant jerk?
[ http://en.wikipedia.org/wiki/Jerk_(physics) ]

1. What is the most general coordinate transformation (in 1+1D for starters) that maps curves of constant acceleration into curves of constant acceleration?

2. Is there already a well-developed body of math covering such cases, perhaps generalizing the theory of projective spaces that arise from an ambient linear space?

(P.S: I wasn't sure whether to ask this under Calculus, or Group Theory, or General Math. It kinda reaches into several categories.)

(P.P.S: Haven't tried using bold font in questions before. Thought I'd try it here since it seems to work for Tom Stoer... :-)

 

[jvicens]

Hi there,

Thank you for bringing up this interesting topic! I am always fascinated by the connections between different areas of physics and mathematics.

To answer your first question, the most general coordinate transformation that maps curves of constant acceleration into curves of constant acceleration is known as a Galilean transformation. This transformation preserves the form of the equations of motion for objects moving with constant acceleration, meaning that the acceleration remains constant in all frames of reference. This is in contrast to the fractional-linear transformation mentioned in the original post, which preserves the property of zero acceleration.

As for your second question, there is indeed a well-developed body of mathematics that covers such cases. This area of mathematics is known as Galilean geometry, which generalizes the concepts of projective geometry to include the effects of constant acceleration. It is a fascinating field that has applications in both physics and mathematics.

I hope this answers your questions and provides some insight into the connections between different areas of physics and mathematics. Thank you again for bringing up this topic and sparking a discussion!