Rotational group SO(3) in classical mechanics

  • Thread starter m_dronti
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  • #1
m_dronti
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Hi!
This is my first post here. I'm currently studying analytical/classical mechanics and have some problems understanding how the Lie algebra is formed in relation to the SO(3) group of rotations. My problem is this:

Given a matrix representation R of some rotation around a fixed axis, we can write this as

R=exp[A]

where A is some matrix. We can also parametrize this as

R(t)=exp[tA]

where R(0)=1, where 1 is the identity matrix. I understand how SO(3) is formed, how it is isomorphic to P^3 and that it should be a Lie group (but I have a very vague understanding of Lie groups). But I don't understand at all

1) what is A exactly? How is it related to the angle and rotation axis?
2) given that A can be decomposed as a linear combination of the infinitesimal rotation generators (and how should one understand them), what does that actually tell us in terms of what the Lie Algebra is?
3) when doing the parametrization, does this involve a fixed rotation axis or only a change in angle?

I have some more questions on this, but it might be best to start of there and see where it leads. If anyone can recommend homepages with more info (basic) on Lie groups in relation to this I would appreciate it.

Cheers!
 

Answers and Replies

  • #2
OrderOfThings
52
0
Hi m_dronti,

For a particle with position vector [itex]\mathbf{x}[/itex] rotating around origin with angular velocity vector [itex]\mathbf{\omega}[/itex], you have

[tex]\dot{\mathbf{x}} = \mathbf{\omega}\times\mathbf{x}[/tex]

This can also be written as a matrix equation

[tex]\dot{\mathbf{x}} = \left(\begin{array}{ccc}0&-\omega_z&\omega_y\\
\omega_z&0&-\omega_x\\-\omega_y&\omega_x&0\end{array}\right)\mathbf{x}[/tex]

which has the solution

[tex]\mathbf{x} = e^{At}\mathbf{x}\left( 0 \right)[/tex]

where [itex]A[/itex] is the matrix above. So you have an isomorphism between angular velocity vectors and real 3x3 antisymmetric matrices, which are the Lie algebra of SO(3).
 
Last edited:
  • #3
jostpuur
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