[Paper Review] Navier-Stokes Hamiltonian for the Similarity Renormalization Group

Billy D. Jones|arXiv (Cornell University)|Jul 3, 2014

Quantum chaos and dynamical systems2 references3 citations

TL;DR

This paper derives the Navier-Stokes Hamiltonian from first principles for a compressible viscous fluid, formulating it using six scalar potentials as dynamical coordinates and incorporating nonholonomic entropy constraints. The key contribution is a consistent Hamiltonian framework with dissipative canonical Poisson brackets, enabling application of the similarity renormalization group to turbulence via a flow equation for the Hamiltonian and a closed-form galilean algebra for viscous fluids.

ABSTRACT

The Navier-Stokes Hamiltonian is derived from first principles. Its Hamilton equations are shown to be equivalent to the continuity, Navier-Stokes, and energy conservation equations of a compressible viscous fluid. The derivations of the Euler and Navier-Stokes Hamiltonians are compared, with the former having identical dynamics to the Euler equation with the viscosity terms of the Navier-Stokes equation dropped from the beginning. The two Hamiltonians have the same number of degrees of freedom in three spatial and one temporal dimension: six independent scalar potentials, but their dynamical fields are necessarily different due to a theory with dissipation not mapping smoothly onto one without. The dynamical coordinate field of a dissipative fluid is a vector field that stores the initial position of all of its fluid particles. Thus these appear to be natural coordinates for studying arbitrary separations of fluid particles over time. The classical similarity renormalization group is introduced and the first steps are carried out deriving a flow equation for the Navier-Stokes Hamiltonian. Finally, the symmetries of a nonrelativistic viscous fluid are discussed through its galilean algebra with dissipative canonical Poisson brackets.

Motivation & Objective

• To develop a Hamiltonian field theory framework for compressible viscous fluids that incorporates dissipation consistently.
• To derive the Navier-Stokes Hamiltonian from first principles using a nonholonomic constraint on entropy.
• To establish a dissipative canonical Poisson bracket structure compatible with energy and momentum conservation.
• To lay the foundation for applying the similarity renormalization group (SRG) to fluid dynamics and turbulence.
• To analyze the symmetries of a nonrelativistic viscous fluid through its galilean algebra with dissipative dynamics.

Proposed method

• Derives the Navier-Stokes Hamiltonian from the Lagrangian of a viscous fluid using variational principles and nonholonomic constraints on entropy.
• Introduces six independent scalar potentials as dynamical coordinates, with the velocity field encoded via the Jacobian of the mapping from initial to current particle positions.
• Constructs the Hamiltonian using canonical conjugate momenta and derives the Hamilton equations equivalent to the continuity, Navier-Stokes, and energy conservation equations.
• Establishes dissipative canonical Poisson brackets using a functional derivative approach that respects the nonholonomic entropy constraint.
• Applies the classical similarity renormalization group (SRG) to derive a flow equation for the Navier-Stokes Hamiltonian, enabling scale-by-scale integration of fluid dynamics.
• Verifies the galilean algebra by computing Poisson brackets of generators (mass, momentum, angular momentum, boost) with the Hamiltonian, confirming conservation laws under dissipative dynamics.

Experimental results

Research questions

• RQ1 Can the Navier-Stokes equation be derived as the Hamiltonian equations of motion for a viscous fluid using first principles?
• RQ2 How can dissipation be consistently incorporated into a Hamiltonian field theory framework with canonical Poisson brackets?
• RQ3 What is the structure of the dynamical degrees of freedom in a viscous fluid when formulated in terms of initial particle positions?
• RQ4 How does the similarity renormalization group (SRG) framework apply to the Navier-Stokes Hamiltonian for studying turbulence?
• RQ5 What are the symmetries of a nonrelativistic viscous fluid, and how do they close under the dissipative Poisson bracket structure?

Key findings

• The Navier-Stokes Hamiltonian is derived from first principles using a nonholonomic constraint on entropy, ensuring equivalence to the standard continuity, momentum, and energy equations.
• The dynamical coordinates are six scalar potentials encoding the initial particle positions, forming a natural basis for tracking fluid particle separation over time.
• The Hamiltonian equations reproduce the Navier-Stokes dynamics exactly, with the dissipative canonical Poisson bracket structure preserving energy and momentum conservation.
• A flow equation for the Navier-Stokes Hamiltonian is derived within the classical similarity renormalization group framework, enabling scale-by-scale analysis of fluid dynamics.
• The galilean algebra closes under the dissipative Poisson brackets, with time derivatives of generators like center-of-mass and momentum corresponding to physical symmetries such as free motion of the center of mass.
• The center-of-mass motion is shown to follow free particle dynamics: $\mathbf{R}_{cm} = \mathbf{R}_{cm}(0) + \mathbf{P}t/M$, derived directly from the Hamiltonian and Poisson bracket structure.