# Nonlinear Dynamics from Linear Quantum Evolutions

## TL;DR

This paper explores how restricting linear quantum evolutions to specific submanifolds can induce nonlinear dynamics, providing methods to analyze and approximate such behaviors in quantum systems.

## Contribution

It introduces two procedures for deriving nonlinear dynamics from linear quantum evolutions, including invariant and non-invariant submanifolds, extending variational methods.

## Key findings

- Restriction to invariant submanifolds yields nonlinear dynamics.
- Lagrangian formalism allows approximation on non-invariant submanifolds.
- Framework generalizes existing variational approaches.

## Abstract

Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given family of states, either as a consequence of experimental constraints or inside an approximation scheme. In this work we investigate such issues in connection with a one parameter group $\phi_t$ of transformations on a Hilbert space, $\mathcal{H}$, defining the unitary evolutions of a chosen quantum system. Two procedures will be presented: the first one consists in the restriction of the vector field associated with the Schr\"{o}dinger equation to a submanifold invariant under the flow $\phi_t$. The second one makes use of the Lagrangian formalism and can be extended also to non-invariant submanifolds, even if in such a case the resulting dynamics is only an approximation of the flow $\phi_t$. Such a result, therefore, should be conceived as a generalization of the variational method already employed for stationary problems.