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alec-glisman / Swimming-in-Potential-Flow

Study dynamics of inertial active matter in a potential fluid (irrotational, incompressible).
https://doi.org/10.1017/jfm.2022.946
GNU General Public License v3.0
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Check symmetries of the unit quaternion time derivatives #7

Closed alec-glisman closed 2 years ago

alec-glisman commented 2 years ago

It can be shown that reflecting the system across the z-axis (wall problem) causes the unit quaternion to have its components go from (w, x, y, z) to (w, -x, -y, z).

Does the same symmetry apply to its first and second time derivatives?

alec-glisman commented 2 years ago

If the quaternion symmetry must be obeyed for all configurations and times, then the symmetry must apply to its time derivatives as well. If the derivatives did not have this symmetry, their time integrals would not either.