# Geometry of the free-sliding Bernoulli beam

## TL;DR

This paper reviews the geometric formulation of free boundary variational problems and applies it to analyze the free-sliding Bernoulli beam, highlighting the role of boundary conditions emerging from the variational process.

## Contribution

It introduces a rigorous geometric framework for free boundary variational problems and applies it specifically to the free-sliding Bernoulli beam case.

## Key findings

- Geometric formulation of free boundary problems on manifolds
- Application to the free-sliding Bernoulli beam
- Insights into boundary conditions from variational principles

## Abstract

If a variational problem comes with no boundary conditions prescribed beforehand, and yet these arise as a consequence of the variation process itself, we speak of a free boundary values variational problem. Such is, for instance, the problem of finding the shortest curve whose endpoints can slide along two prescribed curves. There exists a rigorous geometric way to formulate this sort of problems on smooth manifolds with boundary, which we review here in a friendly self-contained way. As an application, we study a particular free boundary values variational problem, the free-sliding Bernoulli beam.