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About this paper
Jake Welde, Riley Link, Pieter van Goor
The authors define weak invariance: symmetry-breaking dynamics must be generated by the symmetry group. This produces group-affine cascade dynamics. In an aerial-vehicle model under gravity, a nine-dimensional weak symmetry reduces the joint error system from 24 to 15 dimensions, versus 20 using strong symmetry; this is structural, and the error result requires an autonomous residual.
Key insights
- Weak invariance sits between strong invariance and partial symmetry: the residual may be nonzero, but its specific group-action structure enables stronger conclusions than tangency to group orbits alone.
- Weak symmetry produces quotient dynamics that evolve independently of the group state, together with group-affine dynamics; strong invariance is recovered when the group-linear component vanishes.
- Group-affine systems on a Lie group are exactly weakly invariant under left translation, connecting this established class to systems on more general manifolds.
Summary by alphaXiv.
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