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Regarding the issue, I believe it is most likely caused by a mismatch in FPS settings. If you have not changed the default settings, the FPS for the physics constraint should be set to 60 FPS, while the FPS for GIF export should be set to 20 FPS by default. This is likely why the exported GIF does not appear as smooth as what you see in the viewport.
Please try increasing the FPS value in the GIF export settings and exporting it again.
