# Duality for systems of conservation laws

## TL;DR

This paper explores the geometric duality of certain systems of conservation laws, revealing their structure through ruled surfaces and Legendre submanifolds, and characterizes special classes like Hamiltonian and Temple class systems.

## Contribution

It introduces a geometric duality framework for systems with additional conservation laws, linking Hamiltonian systems to ruled quadrics and describing Temple class systems via web geometry.

## Key findings

- Hamiltonian systems are autodual and their ruled surfaces lie in quadrics.
- Temple class systems with three components are dual to systems with constant speeds.
- A complete geometric description of 3-component nondiagonalizable systems is provided.

## Abstract

For one-dimensional systems of conservation laws admitting two additional conservation laws we assign a ruled surface of codimension two in projective space. We call two such systems dual if the corresponding ruled surfaces are dual. We show that a Hamiltonian system is autodual, its ruled surface sits in some quadric, and the generators of this ruled surface form a Legendre submanifold for the contact structure on Fano variety of this quadric. We also give a complete geometric description of 3-component nondiagonalizable systems of Temple class: such systems admit two additional conservation laws, they are dual to systems with constant characteristic speeds, constructed via maximal rank 3-webs of curves in space.