I don’t know why it took me so long, but the motivation of Lie Algebra and Lie Theory in general is very similar to that of Laplace Transform.
Fundamentally, both methods transform the space of operation into something that is more easy to work with.
| Laplace Transform | Lie Algebra (exp/log maps) |
|---|---|
| Time domain (convolution is hard) | Lie group (multiplication/constraints are hard) |
| ↓ Laplace transform | ↓ Logarithm map |
| Frequency domain (multiplication is easy) | Lie algebra (addition/calculus is easy) |
| ↓ Solve algebraically | ↓ Do calculus/optimization |
| ↓ Inverse Laplace | ↓ Exponential map |
| Back to time domain with solution | Back to Lie group with solution |
| Lie Algebra is to stateEstimation and Laplace Transform is to Control Systems |
The other thing is that we can create a set of elements that follow explicit constraints, and, so long as that set and operation form a Lie Group, we can process it in a linear fashion with Lie Algebra.
EXAMPLE SE(3)