# Composite Galilean transformation in 2 dimensions

## Main Question or Discussion Point

The Galilean transforms for rotations, boosts and translations in 2D are the follows:

Rotations:
x' = xcosθ + ysinθ
y' = -xsinθ + ycosθ

Boosts:
x' = x - vxt
y' = y - vyt

Translations:
x' = x - dx
y' = y - dx

I wanted to combine these into a single pair of equations, so my first thought was to combine boosts and translations and plug into rotations. Doing that, you get:

x' = (x - dx - vxt)cosθ + (y - dy - vyt)sinθ
y' = -(x - dx - vxt)sinθ + (y - dy - vyt)cosθ

However, I realized that if you combined the equations differently, say by first plugging rotations into translations, and then plugging that into boosts, you get:

x' = xcosθ + ysinθ - vxt - dx
y' = -xsinθ + ycosθ- vyt - dy

Which is the correct order, if any, and why? It's also possible that I'm doing the composite wrong somehow, haven't really done that sort of thing since pre-calc.. I'm a junior now.

Related Special and General Relativity News on Phys.org
Orodruin
Staff Emeritus
Homework Helper
Gold Member
There is no "correct" order, you can do it either way, but you will have to use different boosts and displacements. The galilei transformations are generally non-commutative (it is not the same to make a particular boost firs and then a particular rotation as it is to do them in the opposite order). This is nothing strange, rotations in dimensions higher than two are not commutative either.

How would I go about modifying them? Do I have to analyze everything geometrically, or are there purely algebraic ways of doing it?

Orodruin
Staff Emeritus
Homework Helper
Gold Member
You have just done it yourself, so how did you do it?

I did translations and boosts geometrically, and used linear algebra for rotations. Although I probably could have done rotations geometrically. At any case I'd prefer an algebraic approach, although I don't know how possible it is to divorce the algebra from geometry.

Edit: Nearly forgot, but do you mean that neither of the two equations I put above are right *as is*, or that the first one is right, but the second needs modification?

Orodruin
Staff Emeritus