Differential invariants of the motion group actions

TL;DR

This paper explores how differential invariants under the motion group change when restricted to invariant submanifolds, providing detailed analysis for the motion group acting on jet spaces and invariant equations.

Contribution

It offers a detailed description of the behavior of differential invariants under the motion group action on jet spaces and invariant equations, extending the Lie-Tresse theorem.

Findings

Differential invariants can significantly change when restricted to invariant submanifolds.

The algebra of invariants is governed by the Lie-Tresse theorem but varies with constraints.

Detailed analysis provided for the motion group acting on jet spaces and invariant equations.

Abstract

Differential invariants of a (pseudo)group action can vary when restricted to invariant submanifolds (differential equations). The algebra is still governed by the Lie-Tresse theorem, but may change a lot. We describe in details the case of the motion group $O(n)\ltimes\R^n$ acting on the full (unconstraint) jet-space as well as on some invariant equations.