Composite Galilean transformation in 2 dimensions

[Afterthought]

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

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

Boosts:
x' = x - v_xt
y' = y - v_yt

Translations:
x' = x - d_x
y' = y - d_x

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 - d_x - v_xt)cosθ + (y - d_y - v_yt)sinθ
y' = -(x - d_x - v_xt)sinθ + (y - d_y - v_yt)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 - v_xt - d_x
y' = -xsinθ + ycosθ- v_yt - d_y

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.

 

[Orodruin]

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.

 

[Afterthought]

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

 

[Orodruin]

You have just done it yourself, so how did you do it?

 

[Afterthought]

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]

No, both are right. They just describe different galilean transformations. Just as you get different rotations by first rotating around the y-axis and then the z-axis as compared to doing it in the other order. One describes translate and boost first, then rotate. The other rotate first, then boost and translate.

 

[Afterthought]

Ah okay, I didn't understand 100% correctly at first. Thank you.