# Hamiltonian hydrodynamics of eccentric discs

## TL;DR

This paper develops a Hamiltonian-based theoretical framework for the ideal hydrodynamics of eccentric astrophysical discs, capturing their slow nonlinear evolution and eccentric modes through a variational principle.

## Contribution

It introduces a Hamiltonian formalism for eccentric disc hydrodynamics, extending from 2D to 3D cases, and identifies nonlinear eccentric modes as energy stationary solutions.

## Key findings

- Hamiltonian formalism derived from variational principle.
- Identification of nonlinear eccentric modes and their properties.
- Numerical examples of eccentric modes up to limiting amplitudes.

## Abstract

We show that the ideal hydrodynamics of an eccentric astrophysical disc can be derived from a variational principle. The nonlinear secular theory describes the slow evolution of a continuous set of nested elliptical orbits as a result of the pressure in a thin disc. In the artificial but widely considered case of a 2D disc, the hydrodynamic Hamiltonian is just the orbit-averaged internal energy of the disc, which can be determined from its eccentricity distribution using the geometry of the elliptical orbits. In the realistic case of a 3D disc, the Hamiltonian needs to be modified to take into account the dynamical vertical structure of the disc. The simplest solutions of the theory are uniformly precessing nonlinear eccentric modes, which make the energy stationary subject to the angular momentum being fixed. We present numerical examples of nonlinear eccentric modes up to their limiting amplitudes. Although it lacks dissipation, which is important in many astrophysical contexts, this formalism allows a simpler theoretical approach to the nonlinear dynamics of eccentric discs than that derived from stress integrals, and also connects better with established methods of celestial mechanics for cases in which the disc interacts gravitationally with one or more orbital companion.