# Generators of Lie Groups and Angular Velocity

• observer1
In summary, the conversation discusses the relationship between rotation matrices and angular velocity matrices. It explains how the derivative of the exponential map can be used to obtain the angular velocity matrix, which represents the rate of change of a rotation matrix over a finite amount of time. This relationship is important in understanding Lie Algebras and Lie Groups.
observer1
I envision the three fundamental rotation matrices: R (where I use R for Ryz, Rzx, Rxy)

I note that if I take (dR/dt * R-transpose) I get a skew-symmetric angular velocity matrix.
(I understand how I obtain this equation... that is not the issue.)

Now I am making the leap to learning about Lie Algebras and Lie Groups

And I understand that any Rotation matrix can be represented with the exponential map
And with the exponential map, the generator of the map happens to have a form that is skew symmetric and (aside from the coefficients) of the same form as the angular velocity matrix.

Of course. A rotatoin matrix is a change. The skew symmetric angular velocity matrix is a RATE of change.
How is it possible for these to related to each other through an exponential map that does not involve TIME.

Last edited by a moderator:
The answer is: the exponential map represents an infinitesimal change in R. The angular velocity matrix is a measure of the rate of change of R over a finite amount of time. By taking the derivative of the exponential map, we can obtain an expression for the angular velocity matrix. This is how the two are related.

## 1. What is a generator of a Lie group?

A generator of a Lie group is an element of the Lie algebra of the group that, when exponentiated, generates a one-parameter subgroup of the group. It can also be thought of as an infinitesimal transformation of a group element.

## 2. How is the generator of a Lie group related to its Lie algebra?

The generator of a Lie group is an element of its corresponding Lie algebra, which is a vector space that captures the algebraic structure of the group. The elements of the Lie algebra can be thought of as "infinitesimal generators" of the group, and the commutator operation on the Lie algebra corresponds to the group's multiplication operation.

## 3. Can generators of a Lie group be used to describe the group's structure?

Yes, generators of a Lie group can be used to describe the group's structure. They provide a way to understand the group's transformations and its algebraic properties, and can also be used to calculate group elements and their properties.

## 4. What is the relationship between generators of Lie groups and angular velocity?

Generators of Lie groups are closely related to angular velocity, as they both describe infinitesimal transformations. In fact, the generators of the special orthogonal group (SO(n)) correspond to the angular velocity of rotations in n-dimensional space.

## 5. Can generators of Lie groups be used to solve practical problems?

Yes, generators of Lie groups have many practical applications in fields such as physics, engineering, and computer graphics. They can be used to model and solve problems involving rotations, transformations, and symmetries, among others.

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