Proving Unique Decomposition of a Galilean Transformation

[rayohauno]

hi... I´m attending a course of advanced classical mechanics.
I´m working on statistical mechanics, so I´m not so familiar with some things on the course.

I must solve the follwing problem for homework:

show that every galilean transformation g on the (galilean space, using natural galilean structure) space RxR^3 (first R linked to time, R^3 linked to 3 spatial coordinates) can be written in a unique way as the composition of a uniform motion g_1, a translation g_2, and a rotation g_3:

g = g_1 . g_2 . g_3

(thus the dimension of the galilean group is 3 + 4 + 3 = 10)

it is easy to show that g_1, g_2 and g_3 are galilean transformation. but I don´t know how to show that
they can ¨represent¨ any galilean transformation g. and less I can show uniqueness of it.

best regards
rayo

 

[rayohauno]

I´m editing this

 

[tiny-tim]

Hi rayohauno!

Let's start at the beginning …

What definition of galilean transformation (on RxR^3) are you working with?

 

[rayohauno]

hi tiny tim, thanks for looking my questions :)

I´ve been trying to write something more specific in latex, but I had problems :P...

anyway, I will try in this way. first the book (Arnold V.I. Mathematical Methods of Classical Mechanics (2ed , Springer, 1989)) talks about a galilean structure:

an affine 4-dimensional space A^4 called universe, its elements (points) are called events. The parallel displacements of the universe A^4 constitutes a vectorial space R^4. for to this has any sense, along with A^4 a subtraction operation must been given between any pair of points in A^4, then for any a,b in A^4 then a-b is on R^4.

then there is a linear mapping t : R^4 -> R from the vector space of parallel displacements to the real "time" axis. the time interval from event a in A^4 to event b in A^4 is t(b-a). if t(b-a)=0 then a and b are called simultaneous events. the set of events simultaneous with a given (any of them) one, its called
a space of simultaneous events A^3

the distance between simultaneous events

d(a,b) = | b-a | = sqrt( (b-a,b-a) )

where (,) its a scalar product on the space R^3. this distance makes every space of simultaneous events into an euclidean space.

then the book says that a space A^4 equipped with a galilean space-time structure like this its called
galilean space.

the the galilean group, its the group of all the transformations of a galilean space which preserve its structure. the elements of this group are called galilean transformations.

then in the book, RxR^3 its just a particular case of A^4. and talks about the natural galilean
structure in RxR^3, so I assume that it speaks about R as time, and R^3 as the spatial coordinates,
the classical galilean time measure (the natural one), and the natural distance measure.

best regards
rayo

 

[tiny-tim]

… one step at a time … !

the distance between simultaneous events

d(a,b) = | b-a | = (b-a,b-a)​

where (,) its a scalar product on the space R^3. this distance makes every space of simultaneous events into an euclidean space.

then the book says that a space A^4 equipped with a galilean space-time structure like this its called galilean space.

… the the galilean group, its the group of all the transformations of a galilean space which preserve its structure. the elements of this group are called galilean transformations.

then in the book, RxR^3 its just a particular case of A^4. and talks about the natural galilean
structure in RxR^3, so I assume that it speaks about R as time, and R^3 as the spatial coordinates,
the classical galilean time measure (the natural one), and the natural distance measure.

Hi rayohauno!

ok, you approach a fundamental problem like this … converting one definition to another … one step at a time.

First step: what is the starting definition (the definition of a galilean transformation)?

Answer: one that preserves the galilean structure.

Second step: what does "preserves the structure" mean?

Answer … ?

 

[rayohauno]

hi again tiny-tim :)

ok... I forgotten to say that I´ve been solved a great part of the problem. there only remains to prove one thing.

why any transformation g that preserves the galilean structure must be linear ??
if this is the case, then it need some work to figure out, but it is not difficult to prove the remaining things (may be there could be some error, but nothing big deal). those thing are the one that I´m trying to write down here in latex.. but I´ve had some problems with it yesterday, so I will tray again soon.

another thing, obviously if any galilean transformation g, its linear, then t its linear also, so:
why it must be assumed in the definition of galilean structure ??

best regards
rayo

 

[tiny-tim]

… Zzz …

those thing are the one that I´m trying to write down here in latex.. but I´ve had some problems with it yesterday, so I will tray again soon

Hi rayohauno!

I have to go to bed now … it's well after midnight here in London … :zzz:

Meanwhile, maybe these will help with your LaTex problems:

http://www.physics.udel.edu/~dubois/lshort2e/node56.html#SECTION00850000000000000000

and

http://www.physics.udel.edu/~dubois/lshort2e/node51.html ​

 

[rayohauno]

look , the following text, its my first attempt to write a partial solution (it is not finished, I must resign), i have written [ tex] ... [ /tex] as a way to latex ignores those lines, but if I delete the spaces, the latex gives to me an error. I don´t know why, it seems to be right.

best regards
rayo
----------------------------------------------

ok, here goes my partial demonstration.

here $$A^n$$ denotes an affine space of dimension $$n$$ and $$R^n$$ denotes a vectorial space (over the real filed $$R$$) of dimension $$n$$.

$$(,)$$ denotes a scalar product.

preservation of the galilean structure

lets $$\left( A^4, - , t : R^4 \to R , (,) : A^4 A^4 \to R \right)$$ be a galilean structure.

lets $$G = \{ g \in G \, tq \, g:A^4 \to A^4 \}$$ be a group on $$A^4$$

then $$G$$ its a galilean group if for all $$g \in G$$ it happens that:

$$\left( A_g^4, - , t : R^4 \to R , (,) : A_g^4 A_g^4 \to R \right)$$ its also a galilean structure.

where:

[ tex] A_g^4 = \{ b=ga : a\in A^4 \}[ /tex] its the transformed space [ tex]A^4[ /tex] by [ tex]g[ /tex].

and if also holds that:

for any [ tex] a,b \in A^4 [ /tex] then

[ tex]t(a-b)=0[ /tex] iff [ tex]t(ga-gb)=0[ /tex]

and

[ tex] t(ga-gb)=0 [ /tex] implies [ tex] d(a-b)=d(ga-gb) [ /tex]

where [ tex]d : R^4 \to R[ /tex] and [ tex]d(a-b)=\sqrt{ (a-b,a-b) }[ /tex]

 

[rayohauno]

I think it is a problem of editor in php, cause I´m on linux. so the latex compiler doesn´t work properly.

anyway, I`ve had solved the remaining part of the problem. I will write it in latex, the I wil upload a .pdf file.

thanks for u time.

best regards
rayo

 

[rayohauno]

here it is all the solution to the problem... I must translate it to english. but the math it just there.

http://www.4shared.com/file/43397915/94ab4035/manuscrito_1.html

best regards
rayo