Composite Galilean transformation in 2 dimensions

In summary, the Galilean transformations for rotations, boosts, and translations in 2D can be combined in different orders to create different transformations. There is no "correct" order, and they are generally non-commutative. Both the equations provided in the conversation are correct, but they describe different transformations depending on the order in which they are combined. It is possible to use both geometric and algebraic approaches when modifying these transformations.
  • #1
Afterthought
29
2
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.
 
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  • #2
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.
 
  • #3
How would I go about modifying them? Do I have to analyze everything geometrically, or are there purely algebraic ways of doing it?
 
  • #4
You have just done it yourself, so how did you do it?
 
  • #5
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?
 
  • #6
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.
 
  • #7
Ah okay, I didn't understand 100% correctly at first. Thank you.
 

Related to Composite Galilean transformation in 2 dimensions

1. What is a Composite Galilean transformation in 2 dimensions?

A Composite Galilean transformation in 2 dimensions is a mathematical concept used in physics to describe the transformation of coordinates between two reference frames that are in uniform motion relative to each other.

2. How is a Composite Galilean transformation different from other types of transformations?

A Composite Galilean transformation differs from other types of transformations, such as Lorentz transformations, in that it is only applicable to reference frames that are in uniform motion, rather than those in relative motion with varying velocities.

3. What is the formula for a Composite Galilean transformation in 2 dimensions?

The formula for a Composite Galilean transformation in 2 dimensions is: x' = x - vt and y' = y, where x and y are the coordinates in the original reference frame, x' and y' are the coordinates in the transformed reference frame, and v is the velocity of the transformed reference frame relative to the original frame.

4. What is the purpose of using a Composite Galilean transformation?

The purpose of using a Composite Galilean transformation is to describe the relationship between two reference frames that are in uniform motion relative to each other, in order to understand the effects of this motion on physical phenomena.

5. How is a Composite Galilean transformation used in practical applications?

A Composite Galilean transformation is used in practical applications, such as in classical mechanics, to describe the motion of objects in different reference frames and to predict the behavior of physical systems under uniform motion.

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