Math problems that are easier to solve using an exponential-exponential coordinate system

Question

Are there any math problems that are easier to solve on a exponential-exponential coordinate grid? (as opposed to a normal x-y cartesian coordinate grid). Some problems are easier to solve in cylindrical coordinates, some are easier to solve in polar coordinates, some in cartesian coordinates etc.

Looking for a list of mathematical problems that are easier to solve using an exponential-exponential coordinate system.

Thanks.

Answer 1

One of the most notable needs for an exponential coordinate system, is exponential coordinates for measuring rigid body rotation. Here is a summary from an excerpt (which I will include in full in the citations) on a study in robotics utilizing exponential coordinates.

A common motion encountered in robotics is the rotation of a body about a given axis by some amount. For example, we might wish to describe the rotation of the link of a robot about a fixed axis, as shown in the following figure which shows tip point trajectory generated by rotation about the $\omega$-axis.

On the other hand, a twist $\hat{\xi} \in se(3)$ is defined as the set of $4 \times 4$ matrices parameterized by exponential coordinates $\xi = (v,\omega)$ where $v \in \mathbb{R^3}$ and $\hat{\omega} \in so(3)$ such that the set $so(n) :=$ {$S \in \mathbb{R}^{n \times n}: S = -S$}. We can derive the matrix of $\hat{\xi}$ when considering rotations about revolute and prismatic joints, where $\omega$ is the axis of rotation, and $v$ is the vector describing the translation. Without the use of an exponential coordinate system, describing this type of motion in robotics would be extraordinarily difficult.

With that summary in mind, exponential coordinates are also heavily studied when considering the rotation group $SO(3)$ and the map $exp: so(3) \to SO(3)$. The exponential map effects a transformation from the axis-angle representation of rotations to rotation matrices. By using a Taylor expansion you can derive a closed-form relation between these two representations. Given a unit vector $\omega \in so(3) = \mathbb{R^3}$ representing the unit rotation axis, and an angle, $\theta \in \mathbb{R}$, an equivalent rotation matrix $R$ may be defined (which will be omitted here to remain on topic and can be seen defined on the wikipedia page on axis-angle representation). Due to the existence of the above-mentioned exponential map, the unit vector $\omega$ representing the rotation axis, and the angle $\theta$ are sometimes called the exponential coordinates of the rotation matrix $R$.

Axis-angle representation and using exponential coordinates is convenient when dealing with rigid body dynamics (as mentioned in the robotics summary). It is useful to both characterize rotations, and also for converting between different representations of rigid body motion, such as homogeneous transformations.

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citations

$\bullet$ http://www.cds.caltech.edu/~murray/books/MLS/pdf/mls94-complete.pdf

Answer 2

I hope that I understand you correctly. You can take any simple differential equation and transform it to a Log-Log plane. The transformed equation will be simply solvable by doing the inverse, exponential-exponential map.

For instance, the equation $du/dt=u+t^2+1$ with the transformation $x=log(t), y=log(x)$ (and using $\frac{du}{dt} = \frac{dy}{dx} \frac{du}{dy} \frac{dx}{dt} \ $) becomes $e^{y-x}dy/dx=e^y+e^{2x}+1$. An exponential-exponential map reduces it back to the original.