# On the Lagrangian Structure of Reduced Dynamics Under Virtual Holonomic Constraints

## TL;DR

This paper characterizes when the reduced dynamics of certain Lagrangian control systems with virtual holonomic constraints have a Lagrangian structure, providing necessary and sufficient conditions and classifying solutions.

## Contribution

It offers a complete characterization of the conditions under which reduced dynamics retain a Lagrangian structure in systems with virtual holonomic constraints.

## Key findings

- Reduced dynamics can be Lagrangian under specific conditions.
- Necessary and sufficient criteria for Lagrangian structure are derived.
- Solutions satisfying virtual constraints are fully characterized.

## Abstract

This paper investigates a class of Lagrangian control systems with $n$ degrees-of-freedom (DOF) and n-1 actuators, assuming that $n-1$ virtual holonomic constraints have been enforced via feedback, and a basic regularity condition holds. The reduced dynamics of such systems are described by a second-order unforced differential equation. We present necessary and sufficient conditions under which the reduced dynamics are those of a mechanical system with one DOF and, more generally, under which they have a Lagrangian structure. In both cases, we show that typical solutions satisfying the virtual constraints lie in a restricted class which we completely characterize.