# The multiplicative Hamiltonian and its hierarchy

## TL;DR

This paper introduces a hierarchy of multiplicative Hamiltonian flows for systems with one degree of freedom and discovers a new class of canonical transformations called the {}-extended class, expanding the theoretical framework of Hamiltonian mechanics.

## Contribution

It presents a novel hierarchy of multiplicative Hamiltonian flows and identifies a new class of canonical transformations, enriching the understanding of Hamiltonian dynamics.

## Key findings

- Defined the multiplicative Hamiltonian flow hierarchy.
- Discovered the {}-extended class of canonical transformations.
- Connected the hierarchy to new transformation properties.

## Abstract

The multiplicative Hamiltonian flow on the phase space for a system with 1 degree of freedom was constituted from infinite hierarchy Hamiltonian flows. A new type of canonical transformation associated with the multiplicative Hamiltonian was found and called the {\lambda}- extended class of the standard canonical transformations.