A canonical Hamiltonian formulation of the Navier-Stokes problem
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John W. Sanders∗,a | A. C. DeVoriaa | Nathan J. Washutaa
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Gafar A. Elamina | Kevin L. Skenesa | Joel C. Berlinghierib

∗Corresponding Author<EMAIL_ADDRESS>
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aDepartment of Mechanical Engineering
The Citadel, the Military College of South Carolina
171 Moultrie St, Charleston, SC 29409
bDepartment of Physics
The Citadel, the Military College of South Carolina
171 Moultrie St, Charleston, SC 29409

## Abstract

This paper presents a novel Hamiltonian formulation of the isotropic Navier-
Stokes problem based on a minimum-action principle derived from the principle
of least squares. This formulation uses the velocities $u_{i}(x_{j},t)$ and
pressure $p(x_{j},t)$ as the field quantities to be varied, along with
canonically conjugate momenta deduced from the analysis. From these, a
conserved Hamiltonian functional $H^{*}$ satisfying Hamilton’s canonical
equations is constructed, and the associated Hamilton-Jacobi equation is
formulated for both compressible and incompressible flows. This Hamilton-
Jacobi equation reduces the problem of finding four separate field quantities
($u_{i}$,$p$) to that of finding a single scalar functional in those
fields—Hamilton’s functional $\text{S}^{*}[t,u_{i},p]$. Moreover, the
transformation theory of Hamilton and Jacobi now provides a prescribed recipe
for solving the Navier-Stokes problem: Find $\text{S}^{*}$. If an analytical
expression for $\text{S}^{*}$ can be obtained, it will lead via canonical
transformation to a new set of fields which are simply equal to their initial
values, giving analytical expressions for the original velocity and pressure
fields. Failing that, if one can only show that a complete solution to this
Hamilton-Jacobi equation does or does not exist, that will also resolve the
question of existence of solutions. The method employed here is not specific
to the Navier-Stokes problem or even to classical mechanics, and can be
applied to any traditionally non-Hamiltonian problem.

## 1 Introduction

Given the title of this paper, it is incumbent on the authors to assure the
reader that we do not claim to have done the impossible. A viscous fluid is,
after all, a non-Hamiltonian system. There is no action for which Hamilton’s
principle [1, 2, 3] yields the Navier-Stokes equations [4] in their usual
form, and we do not claim otherwise. Remarkably, however, a Hamiltonian
formulation can still be found by considering a _mathematically equivalent
higher-order problem_ , as we will now demonstrate via simple example.

### 1.1 A motivating example

Consider the first-order initial-value problem

$\dot{v}=-v,\quad v(0)=1,$ (1)

with unique solution $v(t)=e^{-t}$. Here $v(t)$ can be interpreted as the
velocity of a lumped mass moving in a viscous medium in one dimension with
linear damping. Like the traditional Navier-Stokes equations [4], this too is
an intrinsically non-Hamiltonian problem, in that there is no action
$\mathcal{S}$ for which Hamilton’s principle ($\delta\mathcal{S}=0$) yields
the governing equation $\dot{v}=-v$. And yet, if we simply differentiate both
sides of the equation ($\ddot{v}=-\dot{v}$), use the original equation to
write $\dot{v}=-v$, and apply the additional initial condition
$\dot{v}(0)=-v(0)=-1$, we arrive at the mathematically equivalent second-order
problem

$\ddot{v}=v,\quad v(0)=1,\quad\dot{v}(0)=-1,$ (2)

which has the same unique solution $v(t)=e^{-t}$ but which _is_
Hamiltonian—not in the sense that the total mechanical energy is conserved,
but in the sense that it has mathematically Hamiltonian structure.

As first observed by Sanders [5, 6, 7, 8, 9], the associated action can be
obtained by writing the original equation in standard form
($\mathcal{R}\equiv\dot{v}+v=0$), squaring the residual $\mathcal{R}$, and
integrating over time:

$\mathcal{S}^{*}[v]=\int\text{d}t\left(\frac{1}{2}\mathcal{R}^{2}\right)=\int\text{d}t\left[\frac{1}{2}\left(\dot{v}^{2}+2v\dot{v}+v^{2}\right)\right]\sim\int\text{d}t\left[\frac{1}{2}\left(\dot{v}^{2}+v^{2}\right)\right],$
(3)

where we have used the fact that $2v\dot{v}=\text{d}(v^{2})/\text{d}t$ is a
total time derivative and can therefore be excluded from the action without
changing the resulting Euler-Lagrange equation [10]. This is the so-called
“time-averaged principle of least squares” [5, 6, 7, 8, 9]: since
$\mathcal{R}=0$ is a local minimum of $\mathcal{R}^{2}$, it is also a local
minimum of $\int dt(\mathcal{R}^{2})$. Varying $v$, the first variation of
$\mathcal{S}^{*}$ is

$\delta\mathcal{S}^{*}=\int\text{d}t\biggl{[}\dot{v}\delta\dot{v}+v\delta
v\biggr{]}=\int\text{d}t\biggl{[}(-\ddot{v}+v)\delta
v\biggr{]}+\biggl{[}\dot{v}\delta v\biggr{]}_{t_{1}}^{t_{2}},$ (4)

yielding the second-order equation $\ddot{v}=v$ and revealing the canonically
conjugate “momentum” $\pi\equiv\dot{v}$. Here and in what follows, we will use
the symbol $\pi$ for canonically conjugate momenta, as is customary in
Hamiltonian field theory, in order to avoid later confusion with the pressure
field $p$. Since the mathematical constant $3.14159...$ does not appear in the
present work, there will be no ambiguity.

The corresponding Hamiltonian is obtained via the Legendre transform

$H^{*}[v,\pi]=\pi\dot{v}-\frac{1}{2}\left(\dot{v}^{2}+v^{2}\right)=\frac{1}{2}\left(\pi^{2}-v^{2}\right).$
(5)

Notably, this Hamiltonian has nothing to do with the total mechanical energy
of the system, although it _is_ a conserved quantity. In fact, $H^{*}=0$ for
the actual motion satisfying $\pi\equiv\dot{v}=-v$. We note in passing that
Liouville’s theorem is satisfied, as the motion occurs along the line
$\pi=-v$, so that the phase-space volume, being always zero, is conserved.
Hamilton’s equations

$\dot{v}=\frac{\partial H^{*}}{\partial\pi},\quad\dot{\pi}=-\frac{\partial
H^{*}}{\partial v}$ (6)

are mathematically equivalent to the second-order problem $\ddot{v}=v$ and
therefore also mathematically equivalent to the original, first-order problem.

The associated Hamilton-Jacobi equation [1, 2, 3, 11, 12, 13, 10] is

$\frac{1}{2}\left(\frac{\partial\text{S}^{*}}{\partial
v}\right)^{2}-\frac{1}{2}v^{2}+\frac{\partial\text{S}^{*}}{\partial t}=0,$ (7)

where Hamilton’s principal function $\text{S}^{*}=\text{S}^{*}(v,t)$ serves as
the generating function for a canonical transformation to a new coordinate
$\phi$ which is constant and equal to its initial value. Although this is
almost identical to the Hamilton-Jacobi equation for the simple harmonic
oscillator—the only difference being the sign in front of $(1/2)v^{2}$—the
usual separable solution of the form $\text{S}^{*}(v,t)=W(v)+T(t)$ does not
work, as the reader may check.

Instead, let us use a trial solution of the form

$\text{S}^{*}(v,t)=F(t)v+\frac{1}{2}v^{2}+f(t),$ (8)

where $F(t)$ and $f(t)$ are as yet undetermined functions of $t$. This trial
solution was chosen to cancel the term $(1/2)v^{2}$ from the equation.
Substituting our trial solution into the Hamilton-Jacobi equation, we find
that

$\frac{1}{2}[F(t)]^{2}+[F(t)+F^{\prime}(t)]v+f^{\prime}(t)=0.$ (9)

In order for this equation to hold for all $v$, we must have the following:

$F(t)+F^{\prime}(t)=0\quad\Rightarrow\quad F(t)=\alpha e^{-t},$ (10)

where $\alpha$ is a constant of integration which will be used to transform to
the new coordinate, and

$\frac{1}{2}[F(t)]^{2}+f^{\prime}(t)=0\quad\Rightarrow\quad
f(t)=\frac{1}{4}\alpha^{2}e^{-2t}+\gamma,$ (11)

where $\gamma$ is another constant of integration which is simply additive and
can therefore be discarded.

In this way, we have that

$\text{S}^{*}(v,t;\alpha)=\alpha
e^{-t}v+\frac{1}{2}v^{2}+\frac{1}{4}\alpha^{2}e^{-2t}.$ (12)

With one constant of integration ($\alpha$) to match the single degree of
freedom ($v$), this is a complete solution to the Hamilton-Jacobi equation.
The new coordinate $\phi$ (which is constant and equal to its initial value)
is obtained via the canonical transformation

$\phi=\frac{\partial\text{S}^{*}}{\partial\alpha}=e^{-t}v+\frac{1}{2}\alpha
e^{-2t}.$ (13)

The numerical value of $\alpha$ is in turn obtained via the canonical relation

$\pi=\frac{\partial\text{S}^{*}}{\partial v}=\alpha e^{-t}+v,$ (14)

which, evaluated at $t=0$, gives $\alpha=-2$ (recall that $\pi=\dot{v}$, and
$\dot{v}(0)=-v(0)=-1$). Using the fact that the new coordinate $\phi$ is equal
to its initial value, we have that

$e^{-t}v-e^{-2t}=v(0)-1=0,$ (15)

giving the correct solution $v(t)=e^{-t}$.

In summary, by doubling the order of the governing equation and supplying
additional auxiliary conditions, we made a non-Hamiltonian problem into a
Hamiltonian one [5, 6, 7, 8, 9]. Furthermore, this simple example demonstrates
that the method correctly gives the solution to the original, non-Hamiltonian
problem. Indeed, _it would appear that every non-Hamiltonian problem belongs
to an equivalence class of problems with the same solution, and within that
equivalence class there are Hamiltonian variants_. The remainder of this paper
applies that concept to the isotropic Navier-Stokes problem [4].

### 1.2 The Navier-Stokes problem

The incompressible Navier-Stokes equations [4] are given by

$\rho\dot{u}_{i}+\rho u_{i,j}u_{j}+p_{,i}-\mu u_{i,jj}-\rho b_{i}=0,$ (16)
$u_{i,i}=0,$ (17)

where $\rho$ is the constant and uniform density, $u_{i}=u_{i}(x_{j},t)$ is
the velocity field, $p=p(x_{j},t)$ is the pressure field,
$b_{i}=b_{i}(x_{j},t)$ is the body force field, subscript Roman indices label
Euclidean tensor components ($i,j=1,2,3$), the $x_{j}$ are Eulerian spatial
coordinates, $t$ is time, $\mu$ is the viscosity, a superscribed dot denotes a
_partial_ time derivative ($\dot{u}_{i}=\partial{u}_{i}/\partial t$), a comma
in a subscript indicates a spatial gradient ($p_{,i}=\partial p/\partial
x_{i}$), and we employ the Einstein summation convention on repeated subscript
indices. In the case of a uniform gravitational field, $b_{i}=g_{i}$ coincides
with the local acceleration due to gravity; however, in what follows, we make
no assumptions about the functional form of $b_{i}(x_{j},t)$: it is completely
arbitrary. There are four unknown field quantities: $u_{i}(x_{j},t)$ and
$p(x_{j},t)$.

We seek, ultimately, a functional

$H^{*}=H^{*}[t,u_{i},p,\pi_{i},\pi_{4}],$ (18)

where ($\pi_{i},\pi_{4}$) are suitable “momenta” conjugate to the field
quantities ($u_{i},p$), such that Hamilton’s canonical equations

$\displaystyle\dot{u}_{i}$ $\displaystyle=\frac{\delta H^{*}}{\delta\pi_{i}},$
$\displaystyle\dot{p}$ $\displaystyle=\frac{\delta H^{*}}{\delta\pi_{4}},$
(19) $\displaystyle\dot{\pi}_{i}$ $\displaystyle=-\frac{\delta H^{*}}{\delta
u_{i}},$ $\displaystyle\dot{\pi}_{4}$ $\displaystyle=-\frac{\delta
H^{*}}{\delta p},$ (20)

constitute a mathematically equivalent _second-order_ formulation of the
problem, where $\delta H^{*}/\delta u_{i}$, $\delta H^{*}/\delta p$, $\delta
H^{*}/\delta\pi_{i}$, and $\delta H^{*}/\delta\pi_{4}$ are the functional
derivatives of $H^{*}$ with respect to the field quantities and the conjugate
momenta. We will find that this is generally possible for a compressible
fluid. For an incompressible fluid, the equation $\dot{p}=\delta
H^{*}/\delta\pi_{4}$ will need to be replaced by the incompressibility
condition $u_{i,i}=0$, consistent with the well known result that the pressure
usually serves as a Lagrange multiplier for the incompressibility constraint
[10].

The remainder of this paper is organized as follows. Section 2 gives a
comprehensive overview of the relevant literature to date. Section 3 contains
the main results of the present work: a conserved Hamiltonian functional
$H^{*}$ satisfying Hamilton’s equations (19) and (20) for the mathematically
equivalent second-order problem, along with the accompanying Hamilton-Jacobi
equation [1, 2, 3, 11, 12, 13, 10]. Section 4 contains a discussion of the
physical interpretation of the second-order formulation. Section 5 presents a
brief case study in the form of one-dimensional flow over an infinite, flat
plate. Finally, Section 6 concludes the paper with a few closing remarks and
an outline of how the present formulation can aid in resolving the question of
existence and uniqueness of solutions to the Navier-Stokes problem.

By the end of the paper, we will have achieved precisely what the title
promises: a canonical Hamiltonian formulation of the problem, opening new
avenues toward resolution of one of the most famous unsolved problems in
mathematics.

## 2 Literature review

The field of analytical mechanics, with foundations planted in Hamilton’s
principle of stationary action [1, 2, 3] or d’Alembert’s principle of virtual
work [14], has been vital to the development of both classical and quantum
physics since the eighteenth century. This approach is versatile and helpful
to the physical understanding of the problem in question, and the foundation,
structure, and utility of Hamiltonian formalism is well-documented [15, 16,
17, 18, 19, 20]. The supporting mathematics of the calculus of variations as
well as symplectic and differential geometry can also be found in many
excellent sources [21, 22, 23, 24, 25, 26, 27]. It is therefore no surprise
that researchers have been applying analytical formalism to classical fluids
dating back to the time of Lagrange [28, 29, 30, 31, 32, 33].

The task of obtaining solutions to the governing equations of fluid flow
represents one of the most challenging problems in science and engineering. In
most cases, the mathematical formulation is expressed as an initial-boundary-
value problem: a set of coupled, nonlinear second order partial differential
equations, which are to be solved subject to various initial- and boundary
conditions. The degree of complication of the governing equations depends on
the type of the fluid. For a viscous fluid where the transport phenomena of
friction and thermal conduction are included, the governing equations are
called the Navier-Stokes equations [4]. The Navier-Stokes equations are
derived by applying fundamental physical principles—conservation of mass,
conservation of momentum, and conservation of energy—to a viscous fluid, and
the derivation can be found in any fluid mechanics textbook [34, 35, 36]. As
far as the present authors are aware, to date there is still no firm answer to
the question of whether or not there always exist unique, smooth, nonsingular
solutions to the three-dimensional Navier-Stokes equations [37], and this
constitutes one of the most famous unsolved problems in mathematics.

The application of analytical mechanics to the field of fluid mechanics [38,
10, 21, 39, 25] has recently seen a resurgence in interest [40, 41, 30] after
a long history. Serrin [42], Benjamin [43], and Holm  _et al._ [44] have all
described variational and Hamiltonian formulations of incompressible, inviscid
fluid flow. Roberts [45] presented a Hamiltonian dynamic for weakly
interacting vortices. This research obtained the canonical equations of
Hamiltonian dynamics a set of two well-separated vortex rings by setting up a
Hamiltonian to define the set. Olver [46] showed that the Euler equations of
inviscid and incompressible fluid flow can be put into Hamiltonian form.
Benjamin and Olver [47] investigated the Hamiltonian structure of the water
waves problem. They examined the symmetry groups of this problem, finding that
Hamiltonian analysis enables the solution of conservative elements of the
problem. However, the study also acknowledged that further study is needed to
identify the physical significance of the mathematical results. Maddocks and
Pego [48] presented a novel Hamiltonian formulation of ideal fluid flow
expressed in material coordinates. Their Hamiltonian formulation arises from a
general approach for constrained systems that is not restricted to problems in
fluid mechanics. Rather, it is widely applicable for obtaining unconstrained
Hamiltonian dynamical systems from Lagrangian field equations that are subject
to pointwise constraints. More recently, Arnold [49] also studied the
Hamiltonian nature of the ideal Euler equations.

Viscous forces are non-conservative, which presents a fundamental challenge
when applying Hamilton’s principle to fluid mechanics [37]. The mathematical
study of the variational methods as applied to the Navier-Stokes equations is
an ongoing endeavor, though obtaining the Navier-Stokes equations from a
purely Hamiltonian formulation remains a relatively unexplored area of study
[50, 51, 52]. Solutions to the problem of non-conservative forces in an
inherently conservative formalism have been attempted by many [53, 54, 55, 56,
57] resulting in several procedures to analyze certain non-conservative cases.

Oseledets [58] attempted to express the Navier-Stokes equations using
Hamiltonian formalism. He was able to formalize the incompressible Euler
equation but stated that his result is not valid for a compressible fluid.
More recent attempts, such as Fukagawa and Fujitani [59] and Jones [60], have
enforced dissipation using a non-holonomic constraint on the entropy.
Hochgerner [61] attempted to obtain a Hamiltonian interacting particle system
that could accurately model fluid dynamics. His research separated the
dynamics into slow (deterministic) and fast (stochastic) components to capture
fine-scale effects. The study was able to derive the Navier-Stokes equation
from a stochastic Hamiltonian system but ignored the stress tensor, was unable
to separate configuration and momentum variables, and did not establish energy
conservation or dissipation.

Rashad _et al._ [62] modeled the incompressible Navier-Stokes equations in so-
called “port-Hamiltonian” framework rather than the standard Hamiltonian
framework. Their model used vector calculus instead of exterior calculus to
minimize the number of operators. While the main goal of this research was
increasing the interest of computational researchers in using vector calculus,
they also demonstrated that vector calculus can help in the formulation of
individual subsystems of Navier-Stokes equations and boundary ports of the
model.

Particularly relevant to the present work, Sanders [5, 6, 7, 9] has shown that
higher-order dynamics are “intrinsically variational,” in the sense that
higher-derivative versions of the classical equations of motion can be derived
from a minimum action principle even for dissipative systems, thus allowing
inherently non-Hamiltonian problems to be treated as though they are
Hamiltonian. This discovery has already led to two applications: the direct
modal analysis of damped dynamical systems [6] and subsequently a new and more
efficient algorithm for computing a damped system’s resonant frequencies [8].
Higher-derivative theories had been studied before in the realm of quantum
gravity physics [63, 64, 65, 66, 67, 68, 69, 70] but until now they have not
been applied to classical fluids. While the Navier-Stokes equations, in their
standard form, may be unsuited to Hamiltonian formalism [71, 51, 52, 37], it
will be shown here that higher-order dynamics can be used to restate the
problem in a form consistent with Hamiltonian and Hamilton-Jacobi formalism.

In conclusion, although the body of research surrounding the Navier-Stokes
equations is extensive, it would appear that no canonical Hamiltonian
formulation of the Navier-Stokes equations has been found to date. Indeed,
that is what the present work aims to achieve.

## 3 Analysis

Although we are primarily interested in the incompressible form of the
equations given by (16) and (17), for reasons that will become clear shortly,
here we will begin with the compressible form of the equations, with the
understanding that we will eventually take the incompressible limit. For the
compressible case, the linear momentum balance and continuity equations are
given by

$\mathcal{R}_{i}[t,x_{j},u_{j},p,\rho]\equiv\rho\dot{u}_{i}+\rho
u_{i,j}u_{j}+p_{,i}-\mu u_{i,jj}-(\mu+\lambda)u_{j,ji}-\rho b_{i}=0,$ (21)
$\mathcal{R}_{4}[u_{j},\rho]\equiv\dot{\rho}+\rho_{,i}u_{i}+\rho u_{i,i}=0,$
(22)

where $\rho=\rho(x_{j},t)$ is the density field (now one of the unknown field
quantities along with $u_{i}$ and $p$), and $\lambda$ is an additional
viscosity coefficient which, under Stokes’s [4] hypothesis, is related to
$\mu$ as $\lambda=-2\mu/3$, ensuring that the mechanical pressure agrees with
the thermodynamic pressure. Henceforth we will assume that all quantities have
been suitably non-dimensionalized. The non-dimensional (constant) viscosities
in (21) and (22) may be regarded as inverse Reynolds numbers, and the non-
dimensional pressure may be considered to be normalized by the inertial scale
$\rho_{0}U^{2}$, with $\rho_{0}$ and $U$ appropriate density and velocity
scales.

In general, (21) and (22) would be appended with the energy equation, which
introduces additional thermodynamic variables, such as temperature and
enthalpy or entropy. Two of the thermodynamic variables are designated as
“primary,” and equations of state are required to relate the remaining
variables to the primary variables. Typically, pressure and temperature are
chosen as the primary variables, and the equation of state for the density,
for example, is expressed as $\rho=\rho(p,T)$. The conservation equations
along with the equation of state constitute six equations for the six unknowns
fields $(u_{i},p,T,\rho)$. Henceforth in the present work, we will take the
temperature to be constant, though we intend to consider variations in
temperature in future work.

An incompressible flow is one for which the material derivative of the density
vanishes, _i.e._ , $\text{d}\rho/\text{d}t=\dot{\rho}+\rho_{,i}u_{i}=0$, and
this condition serves as an equation of state. It is usually also assumed, for
the sake of simplicity, that the density is both constant and uniform, further
reducing the equation of state $\rho=\rho(p,T)$ to specification of
$\rho=\rho_{0}$ as a system parameter. Consequently, (22) reduces to $\rho
u_{i,i}=0$ and the energy equation is decoupled from the system. Accordingly,
there are now only four unknown field quantities $(u_{i},p)$ and the momentum
balance and continuity equations suffice for the governing field equations.

We pause here to remark that all four field equations (21), (22) are _first-
order_ in time with respect to the field quantities $u_{i}$ and $\rho$. This
will be important shortly, when we double the order of the equations. It
should also be noted, as mentioned previously, that the first-order problem
described above is inherently _non-Hamiltonian_ , in that there is no action
$\mathcal{S}$ for which Hamilton’s principle ($\delta\mathcal{S}=0$) yields
the first-order field equations. Finally, we note that in the incompressible
limit, $\mathcal{R}_{4}$ becomes independent of $\dot{\rho}$ and is no longer
first-order in time.

### 3.1 Second-order formulation

Although the first-order formulation of the problem is intrinsically non-
Hamiltonian, nevertheless a Hamiltonian for the system may be found by
considering a second-order formulation. Following Sanders [9], we observe that
the actual motion of the fluid corresponds to the particular fields
$(u_{i},p,\rho)$ for which the following action achieves a local minimum:

$\mathcal{S}^{*}[u_{j},p,\rho]=\int\text{d}^{4}x\left(\frac{1}{2}\mathcal{R}_{i}\mathcal{R}_{i}+\frac{1}{2}\mathcal{R}_{4}\mathcal{R}_{4}\right),$
(23)

where $\text{d}^{4}x=\text{d}x_{1}\text{d}x_{2}\text{d}x_{3}\text{d}t$, and
the integral is carried out over both the control volume $\mathcal{V}$
occupied by the fluid ($x_{j}\in\mathcal{V}$) and the time period of interest
($t\in[t_{1},t_{2}]$). It must be emphasized that this action contains no new
physics. Again, this is simply the principle of least squares averaged over
the spacetime occupied by the fluid. The entire physics of the problem are
already contained in the residuals ($\mathcal{R}_{i}$, $\mathcal{R}_{4}$).

Without an equation of state relating $\rho$ to $p$, the problem is
underconstrained with five unknown field quantities and only four dynamical
field equations. Anticipating the case of incompressible flow, where the
density is constant and the four field quantities are $u_{i}$ and $p$,
henceforth we will assume an equation of state of the form
$\rho=\hat{\rho}(p)$, with $\hat{\rho}$ a known function determined either
from first principles or empirically. In this way, the density field may be
eliminated in favor of the pressure field, and the field equations assume the
following form:

$\mathcal{R}_{i}[t,x_{j},u_{j},p]\equiv\hat{\rho}\dot{u}_{i}+\hat{\rho}u_{i,j}u_{j}+p_{,i}-\mu
u_{i,jj}-(\mu+\lambda)u_{j,ji}-\hat{\rho}b_{i}=0,$ (24)
$\mathcal{R}_{4}[u_{j},p]\equiv\hat{\rho}^{\prime}\dot{p}+\hat{\rho}^{\prime}p_{,i}u_{i}+\hat{\rho}u_{i,i}=0,$
(25)

where $\hat{\rho}^{\prime}=d\hat{\rho}/dp$. We note that, under equilibrium
conditions, $\hat{\rho}^{\prime}$ is related to the speed of sound $c$ and the
bulk modulus $K$ as $\hat{\rho}^{\prime}=1/c^{2}=\rho/K$ (for incompressible
fluids, $\hat{\rho}^{\prime}\equiv 0$ and the speed of sound and bulk modulus
are both infinite). Having specified $\hat{\rho}(p)$, $\mu$, $\lambda$, and
$b_{i}(x_{j},t)$, and having prescribed appropriate auxiliary conditions
(initial and boundary conditions), one seeks the four field quantities
$(u_{i},p)$ satisfying the governing field equations and the auxiliary
conditions. To recover the case of incompressible flow, we will eventually
take $\hat{\rho}^{\prime}\equiv 0$.

We pause here to note that, even though the residuals ($\mathcal{R}_{i}$,
$\mathcal{R}_{4}$) vanish for the actual motion, they are _not_ trivially
zero. That is, the residuals _only_ vanish for the _particular_ fields
$(u_{i},p)$ which satisfy the first-order field equations (24) and (25); they
do not vanish for _every conceivable_ $(u_{i},p)$. Thus it is not appropriate
to take $\mathcal{R}_{i}\equiv 0$, $\mathcal{R}_{4}\equiv 0$. We will return
to this point later when we discuss the Hamiltonian formulation of the
problem.

For now, we note that the action $\mathcal{S}^{*}=\mathcal{S}^{*}[u_{i},p]$
defines a Lagrangian

$L^{*}[t,u_{i},p]=\int\text{d}^{3}x(\mathcal{L}^{*}),$ (26)

where the integral is carried out over the volume $\mathcal{V}$ only
($\text{d}^{3}x=\text{d}x_{1}\text{d}x_{2}\text{d}x_{3}$), with Lagrangian
density

$\mathcal{L}^{*}[t,x_{j},u_{j},p]=\frac{1}{2}\mathcal{R}_{i}\mathcal{R}_{i}+\frac{1}{2}\mathcal{R}_{4}\mathcal{R}_{4}.$
(27)

Because the residuals ($\mathcal{R}_{i}$, $\mathcal{R}_{4}$) have been non-
dimensionalized, the Lagrangian density is also dimensionless. Once again,
even though the Lagrangian vanishes for the actual motion, it is not trivially
zero, and it is not appropriate to take $L^{*}\equiv 0$.

As noted above, the actual motion of the fluid corresponds to the particular
fields $(u_{i},p)$ for which $\mathcal{S}^{*}$ achieves a local minimum. To
obtain the Euler-Lagrange equations, the conjugate momenta, and the natural
auxiliary conditions, we insist that $\mathcal{S}^{*}$ not vary to first order
($\delta\mathcal{S}^{*}=0$) under small variations in the fields $(\delta
u_{i},\delta p)$. Evaluating $\delta\mathcal{S}^{*}$, integrating by parts,
and collecting like terms, we find that

$\displaystyle\delta\mathcal{S}^{*}=\int\text{d}^{4}x$
$\displaystyle\biggl{\\{}\biggl{[}-\frac{\partial}{\partial
t}\left(\hat{\rho}\mathcal{R}_{i}\right)-\frac{\partial}{\partial
x_{j}}\left(\hat{\rho}\mathcal{R}_{i}u_{j}\right)+\hat{\rho}\mathcal{R}_{j}u_{j,i}-\mu\mathcal{R}_{i,jj}\biggr{.}\biggr{.}$
(28)
$\displaystyle\biggl{.}-(\mu+\lambda)\mathcal{R}_{j,ij}+\hat{\rho}^{\prime}\mathcal{R}_{4}p_{,i}-\frac{\partial}{\partial
x_{i}}\left(\hat{\rho}\mathcal{R}_{4}\right)\biggr{]}\delta{u}_{i}$ (29)
$\displaystyle+$
$\displaystyle\biggl{[}\hat{\rho}^{\prime}\mathcal{R}_{i}\dot{u}_{i}+\hat{\rho}^{\prime}\mathcal{R}_{i}u_{i,j}u_{j}-\mathcal{R}_{i,i}-\hat{\rho}^{\prime}\mathcal{R}_{i}b_{i}+\hat{\rho}^{\prime\prime}\mathcal{R}_{4}\dot{p}\biggr{.}$
(30) $\displaystyle\biggl{.}\biggl{.}-\frac{\partial}{\partial
t}\left(\hat{\rho}^{\prime}\mathcal{R}_{4}\right)+\hat{\rho}^{\prime\prime}\mathcal{R}_{4}p_{,i}u_{i}-\frac{\partial}{\partial
x_{i}}\left(\hat{\rho}^{\prime}\mathcal{R}_{4}u_{i}\right)+\hat{\rho}^{\prime}\mathcal{R}_{4}u_{i,i}\biggr{]}\delta
p\biggr{\\}}$ (31) $\displaystyle+\int$
$\displaystyle\text{d}^{3}x\biggl{[}\hat{\rho}\mathcal{R}_{i}\delta{u}_{i}+\hat{\rho}^{\prime}\mathcal{R}_{4}\delta{p}\biggr{]}_{t_{1}}^{t_{2}}$
(32) $\displaystyle+\int$
$\displaystyle\text{d}^{2}x\text{d}t\biggl{\\{}\biggl{[}\hat{\rho}\mathcal{R}_{i}u_{j}n_{j}+\mu\mathcal{R}_{i,j}n_{j}+(\mu+\lambda)\mathcal{R}_{j,i}n_{j}+\hat{\rho}\mathcal{R}_{4}n_{i}\biggr{]}\delta
u_{i}\biggr{.}$ (33)
$\displaystyle\biggl{.}\biggl{[}-\mu\mathcal{R}_{i}n_{j}-(\mu+\lambda)\mathcal{R}_{j}n_{i}\biggr{]}\delta
u_{i,j}+\biggl{[}\mathcal{R}_{i}n_{i}+\hat{\rho}^{\prime}\mathcal{R}_{4}u_{i}n_{i}\biggr{]}\delta
p\biggr{\\}}$ (34)

where the purely volumetric integral ($\text{d}^{3}x$) is carried out over
$\mathcal{V}$, and the surface integral ($\text{d}^{2}x$) is carried out over
the boundary $\partial\mathcal{V}$. Note that, because we are using Eulerian
coordinates $x_{j}$, the volume element $\text{d}^{3}x$ is not to be varied.

The Euler-Lagrange equations (which hold for all $x_{j}\in\mathcal{V}$) may be
read directly from the spacetime ($\text{d}^{4}x$) integral:

$\displaystyle\delta{u}_{i}:\quad$ $\displaystyle-\frac{\partial}{\partial
t}\left(\hat{\rho}\mathcal{R}_{i}\right)-\frac{\partial}{\partial
x_{j}}\left(\hat{\rho}\mathcal{R}_{i}u_{j}\right)+\hat{\rho}\mathcal{R}_{j}u_{j,i}-\mu\mathcal{R}_{i,jj}$
$\displaystyle-(\mu+\lambda)\mathcal{R}_{j,ij}+\hat{\rho}^{\prime}\mathcal{R}_{4}p_{,i}-\frac{\partial}{\partial
x_{i}}\left(\hat{\rho}\mathcal{R}_{4}\right)=0$ (35) $\displaystyle\delta
p:\quad$
$\displaystyle\biggl{.}\hat{\rho}^{\prime}\mathcal{R}_{i}\dot{u}_{i}+\hat{\rho}^{\prime}\mathcal{R}_{i}u_{i,j}u_{j}-\mathcal{R}_{i,i}-\hat{\rho}^{\prime}\mathcal{R}_{i}b_{i}+\hat{\rho}^{\prime\prime}\mathcal{R}_{4}\dot{p}\biggr{.}$
$\displaystyle-\frac{\partial}{\partial
t}\left(\hat{\rho}^{\prime}\mathcal{R}_{4}\right)+\hat{\rho}^{\prime\prime}\mathcal{R}_{4}p_{,i}u_{i}-\frac{\partial}{\partial
x_{i}}\left(\hat{\rho}^{\prime}\mathcal{R}_{4}u_{i}\right)+\hat{\rho}^{\prime}\mathcal{R}_{4}u_{i,i}=0$
(36)

It should be noted that all four Euler-Lagrange equations (35), (36) are
_second-order_ in time, as they involve time derivatives of the residuals. By
doubling the order of the equations, we have put the problem in Hamiltonian
form, consistent with the general result of Sanders [9]. We also note that all
four Euler-Lagrange equations of the second-order formulation are
automatically satisfied by the solution to the first-order formulation (_i.e._
, the actual motion), for which $\mathcal{R}_{i}=0$ and $\mathcal{R}_{4}=0$
everywhere and at all times.

Corresponding to each of the four field quantities is a canonically conjugate
“momentum” field, which can be read from (32). The momenta conjugate to the
velocities $u_{i}$ are

$\pi_{i}\equiv\hat{\rho}\mathcal{R}_{i},$ (37)

and the momentum conjugate to the pressure $p$ is

$\pi_{4}\equiv\hat{\rho}^{\prime}\mathcal{R}_{4}.$ (38)

The natural “temporal conditions” are that the conjugate momenta should vanish
at the endpoints $t_{1}$ and $t_{2}$. When enforced as initial conditions,
these guarantee that the solution to the second-order formulation coincides
with the solution to the first-order formulation (_i.e._ , the actual motion).
In other words, the actual motion _is_ the natural evolution of the second-
order formulation, and unlike the familiar form of Hamilton’s principle, here
the variations $(\delta u_{i},\delta p)$ are _not_ taken to vanish at the
endpoints. Although the conjugate momenta ($\pi_{i}$, $\pi_{4}$) do not
coincide with conventional linear or angular momenta, there is nonetheless a
curious mathematical connection between the conjugate momenta and the linear
momentum density $P_{i}=\rho u_{i}$, which we will see presently from the
natural boundary conditions.

The natural boundary conditions are read directly from the surface
($\text{d}^{2}x$) integral:

$\displaystyle\biggl{.}\delta u_{i}:\quad$
$\displaystyle\hat{\rho}\mathcal{R}_{i}u_{j}n_{j}+\mu\mathcal{R}_{i,j}n_{j}+(\mu+\lambda)\mathcal{R}_{j,i}n_{j}+\hat{\rho}\mathcal{R}_{4}n_{i}=0\biggr{.}$
(39) $\displaystyle\biggl{.}\delta u_{i,j}:\quad$
$\displaystyle-\mu\mathcal{R}_{i}n_{j}-(\mu+\lambda)\mathcal{R}_{j}n_{i}=0\biggr{.}$
(40) $\displaystyle\biggl{.}\delta p:\quad$
$\displaystyle\mathcal{R}_{i}n_{i}+\hat{\rho}^{\prime}\mathcal{R}_{4}u_{i}n_{i}=0\biggr{.}$
(41)

This last condition, (41), establishes a connection between the new conjugate
momenta and the conventional linear momenta. Multiplying (41) by $\hat{\rho}$,
and noting that $\hat{\rho}u_{i}=\rho u_{i}=P_{i}$, we find that

$(\pi_{i}+\pi_{4}P_{i})n_{i}=0.$ (42)

Evidently, boundary condition (41) states that the flux of the vector
$\Pi_{i}\equiv\pi_{i}+\pi_{4}P_{i}$ through the boundary $\partial\mathcal{V}$
should vanish. It is interesting that this new vector $\Pi_{i}$ contains both
old and new momenta, with $\pi_{4}$ “carried” (_i.e._ , given direction) by
$P_{i}$. The actual physical meaning of these natural boundary conditions is
less clear and may require further investigation.

### 3.2 Equivalence of the first- and second-order formulations

The first- and second-order formulations are mathematically equivalent, in the
sense that imposing _identical auxiliary conditions_ on the two formulations
will yield identical solutions $(u_{i},p)$. In other words, with identical
auxiliary conditions, $(u_{i},p)$ is a solution to the first-order formulation
_if and only if_ the same $(u_{i},p)$ is a solution to the second-order
formulation.

The proof is straightforward. Consider the two formulations separately, and
impose on the second-order formulation identical auxiliary conditions to those
of the first-order formulation. In particular, just like the simple example
given in Section 1.1, the second-order formulation requires additional
auxiliary conditions over and above those applied to the first-order
formulation. These include initial conditions making
$\mathcal{R}_{i}(x_{j},0)=0$ and $\mathcal{R}_{4}(x_{j},0)=0$ for all
$x_{j}\in\mathcal{V}\cup\partial\mathcal{V}$, along with boundary conditions
making $\mathcal{R}_{i}(x_{j},t)=0$, $\mathcal{R}_{i,j}(x_{j},t)=0$, and
$\mathcal{R}_{4}(x_{j},t)=0$ for all $x_{j}\in\partial\mathcal{V}$ and all
times $t$. By supposition, the auxiliary conditions applied to the two
formulations are identical, so it suffices to show that $(u_{i},p)$ satisfies
the governing field equations (24), (25) of the first-order formulation
($\mathcal{R}_{i}=0$ and $\mathcal{R}_{4}=0$) everywhere in $\mathcal{V}$ and
at all times _if and only if_ $(u_{i},p)$ satisfies the Euler-Lagrange
equations (35), (36) of the second-order formulation everywhere in
$\mathcal{V}$ and at all times.

Suppose first that $(u_{i},p)$ satisfies the governing field equations (24),
(25) of the first-order formulation everywhere in $\mathcal{V}$ and at all
times. Then $\mathcal{R}_{i}=0$, $\mathcal{R}_{4}=0$, and $(u_{i},p)$ is a
trivial solution to the Euler-Lagrange equations (35), (36) of the second-
order formulation.

Conversely, suppose that $(u_{i},p)$ satisfies the Euler-Lagrange equations
(35), (36) of the second-order formulation everywhere in $\mathcal{V}$ and at
all times. We note that $(\mathcal{R}_{i}=0,\mathcal{R}_{4}=0)$ constitutes an
equilibrium solution of the Euler-Lagrange equations (35), (36). Thus, because
the initial conditions are chosen such that $\mathcal{R}_{i}(x_{j},0)=0$ and
$\mathcal{R}_{4}(x_{j},0)=0$ for all $x_{j}\in\mathcal{V}$, and because the
boundary conditions are chosen such that $\mathcal{R}_{i}(x_{j},t)=0$ and
$\mathcal{R}_{4}(x_{j},t)=0$ for all $x_{j}\in\partial\mathcal{V}$ and all
times $t$, then $\mathcal{R}_{i}$ and $\mathcal{R}_{4}$ will remain
identically zero everywhere in $\mathcal{V}$ for all future times. Thus,
$(u_{i},p)$ satisfies the governing field equations (24), (25) of the first-
order formulation everywhere in $\mathcal{V}$ and at all times. This completes
the proof, and we have established that the two formulations are equivalent.
$\square$

### 3.3 Hamilton’s equations

We are now ready to proceed with the Hamiltonian formulation of the problem.
The Lagrangian $L^{*}$ has a corresponding Hamiltonian

$H^{*}=\int\text{d}^{3}x(\mathcal{H}^{*}),$ (43)

with the Hamiltonian density $\mathcal{H}^{*}$ obtained from the Lagrangian
density $\mathcal{L}^{*}$ via the Legendre transform

$\mathcal{H}^{*}=\pi_{i}\dot{u}_{i}+\pi_{4}\dot{p}-\mathcal{L}^{*}=\pi_{i}\dot{u}_{i}+\pi_{4}\dot{p}-\frac{1}{2}\mathcal{R}_{i}\mathcal{R}_{i}-\frac{1}{2}\mathcal{R}_{4}\mathcal{R}_{4}.$
(44)

Again, this $H^{*}$ has nothing to do with the total mechanical energy of the
system, although it _is_ a conserved quantity, since $H^{*}=0$ for the actual
motion—just as in the example of Section 1.1. In order to write down
Hamilton’s equations, we must express $\mathcal{H}^{*}$ in terms of the fields
and the conjugate momenta, eliminating $\dot{u}_{i}$ and $\dot{p}$.

Observe that $\mathcal{R}_{i}=\pi_{i}/\hat{\rho}$, and ignoring for the moment
the incompressible limit, we may write
$\mathcal{R}_{4}=\pi_{4}/\hat{\rho}^{\prime}$. In this way, using the
functional expressions for the residuals given by (24) and (25), we find that

$\dot{u}_{i}=\frac{1}{(\hat{\rho})^{2}}{\pi_{i}}-\frac{1}{\hat{\rho}}\biggl{(}\hat{\rho}u_{i,j}u_{j}+{p_{,i}-\mu
u_{i,jj}-(\mu+\lambda)u_{j,ji}}-\hat{\rho}b_{i}\biggr{)},$ (45)
$\dot{p}=\frac{1}{(\hat{\rho}^{\prime})^{2}}{\pi_{4}}-\frac{1}{\hat{\rho}^{\prime}}\biggl{(}{\hat{\rho}^{\prime}p_{,i}u_{i}+\hat{\rho}u_{i,i}}\biggr{)},$
(46)

and

$\displaystyle\mathcal{H}^{*}[t,x_{i},u_{i},p,\pi_{i},\pi_{4}]=$
$\displaystyle\frac{1}{2}\frac{1}{(\hat{\rho})^{2}}{\pi_{i}\pi_{i}}-\frac{1}{\hat{\rho}}\biggl{(}\hat{\rho}u_{i,j}u_{j}+{p_{,i}-\mu
u_{i,jj}-(\mu+\lambda)u_{j,ji}}-\hat{\rho}b_{i}\biggr{)}\pi_{i}$
$\displaystyle+\frac{1}{2}\frac{1}{(\hat{\rho}^{\prime})^{2}}{\pi_{4}\pi_{4}}-\frac{1}{\hat{\rho}^{\prime}}\biggl{(}{\hat{\rho}^{\prime}p_{,i}u_{i}+\hat{\rho}u_{i,i}}\biggr{)}\pi_{4}.$
(47)

Hamilton’s equations [2, 3] are as follows:

$\displaystyle\dot{u}_{i}$ $\displaystyle=\frac{\delta H^{*}}{\delta\pi_{i}},$
$\displaystyle\dot{p}$ $\displaystyle=\frac{\delta H^{*}}{\delta\pi_{4}},$
(48) $\displaystyle\dot{\pi}_{i}$ $\displaystyle=-\frac{\delta H^{*}}{\delta
u_{i}},$ $\displaystyle\dot{\pi}_{4}$ $\displaystyle=-\frac{\delta
H^{*}}{\delta p},$ (49)

where $\delta H^{*}/\delta u_{i}$, $\delta H^{*}/\delta p$, $\delta
H^{*}/\delta\pi_{i}$, and $\delta H^{*}/\delta\pi_{4}$ are the functional
derivatives of $H^{*}$ with respect to the field quantities and the conjugate
momenta. The latter equations, (49), reproduce the Euler-Lagrange equations
(35), (36) of the second-order formulation.

We return now to our previous observation concerning the vanishing of the
residuals. While $H^{*}$ vanishes for the _particular_ fields $(u_{i},p)$ that
satisfy the governing field equations (24), (25) of the first-order
formulation, it does not vanish for _every conceivable_ $(u_{i},p)$. The
latter would imply, according to Equations (48), that $\dot{u}_{i}\equiv 0$
and $\dot{p}\equiv 0$, which is not generally the case. This observation, and
the fact that Equations (49) faithfully reproduce the Euler-Lagrange equations
(35), (36) of the second-order formulation, confirm that the Hamiltonian
formulation described above is, in fact, a legitimate reformulation of the
problem. [In the following section, we develop the Hamilton-Jacobi theory as
it relates to the present formulation, the goal being to find a canonical
transformation to a new set of fields ($\phi_{i}$, $\phi_{4}$) and conjugate
momenta for which the Hamiltonian _does_ vanish identically.]

To obtain the Hamiltonian for incompressible flow, we set
$\hat{\rho}^{\prime}\equiv 0$ from the beginning, in which case
$\mathcal{R}_{4}$ reduces to $\hat{\rho}u_{i,i}$ and $\pi_{4}$ vanishes
identically, consistent with the fact that $\mathcal{R}_{4}$ becomes
independent of $\dot{p}$. The Hamiltonian density in turn reduces to

$\mathcal{H}^{*}=\pi_{i}\dot{u}_{i}-\frac{1}{2}\mathcal{R}_{i}\mathcal{R}_{i},$
(50)

or, in terms of the conjugate momenta,

$\mathcal{H}^{*}[t,x_{i},u_{i},p,\pi_{i}]=\frac{1}{2}\frac{1}{{\rho}^{2}}{\pi_{i}\pi_{i}}-\frac{1}{{\rho}}\biggl{(}{\rho}u_{i,j}u_{j}+p_{,i}-\mu
u_{i,jj}-{\rho}b_{i}\biggr{)}\pi_{i},$ (51)

where the density $\hat{\rho}=\rho$ is a constant and we have used the fact
that $u_{i,i}=0$. Hamilton’s equations $\dot{u}_{i}=\delta
H^{*}/\delta\pi_{i}$, $\dot{\pi}_{i}=-\delta H^{*}/\delta u_{i}$, and
$0\equiv\dot{\pi}_{4}=-\delta H^{*}/\delta p$ still apply, but $\dot{p}=\delta
H^{*}/\delta\pi_{4}$ must be replaced by the constraint that $u_{i,i}=0$,
since for incompressible flow $\dot{p}$ is not determined by the governing
equations. That the incompressibility condition should take the place of the
pressure equation $\dot{p}=\delta H^{*}/\delta\pi_{4}$ is consistent with the
well known result that the pressure usually serves as the Lagrange multiplier
for the incompressibility constraint [10].

### 3.4 Hamilton-Jacobi equation

One of the most significant aspects of the Hamiltonian formalism is that it
leads to the transformation theory of Hamilton and Jacobi [1, 2, 3, 11, 12,
13, 10], celebrated both for unifying particle mechanics with wave optics [1]
and for its relationship to the Schrödinger equation of quantum mechanics [72,
73]. Here we will obtain a Hamilton-Jacobi equation representing the Navier-
Stokes problem.

In the context of discrete mechanics, Hamilton’s principal function is
obtained as the solution to the Hamilton-Jacobi equation, which is in turn
defined by the functional form of the Hamiltonian. Hamilton’s principal
function provides the generating function for a canonical transformation to a
new set of generalized coordinates and conjugate momenta for which the
Hamiltonian vanishes identically, in which case Hamilton’s equations do, in
fact, become trivial. The new coordinates and their conjugate momenta are
simply equal to their initial values.

In the present context, the role of Hamilton’s principal function is played by
a characteristic functional $\text{S}^{*}=\text{S}^{*}[t,u_{i},p]$ (not to be
confused with the action $\mathcal{S}^{*}$, although they are related; see
Appendix A), which is the solution to the following Hamilton-Jacobi equation:

${H}^{*}\left[t,u_{i},p,\frac{\delta\text{S}^{*}}{\delta
u_{i}},\frac{\delta\text{S}^{*}}{\delta
p}\right]+\frac{\partial\text{S}^{*}}{\partial t}=0,$ (52)

where $\delta\text{S}^{*}/\delta u_{i}$ and $\delta\text{S}^{*}/\delta p$ are
the functional derivatives of $\text{S}^{*}$ with respect to the field
quantities. Interested readers will find the derivation of (52) in Appendix A.
Henceforth we will refer to $\text{S}^{*}$ as “Hamilton’s functional.”
Substituting for the conjugate momenta in (47), we obtain for the Hamilton-
Jacobi equation

$\displaystyle\int\text{d}^{3}x\left[\frac{1}{2}\frac{1}{(\hat{\rho})^{2}}\frac{\delta\text{S}^{*}}{\delta
u_{i}}\frac{\delta\text{S}^{*}}{\delta
u_{i}}-\frac{1}{\hat{\rho}}\biggl{(}\hat{\rho}u_{i,j}u_{j}+{p_{,i}-\mu
u_{i,jj}-(\mu+\lambda)u_{j,ji}}-\hat{\rho}b_{i}\biggr{)}\frac{\delta\text{S}^{*}}{\delta
u_{i}}\right.$
$\displaystyle\left.+\frac{1}{2}\frac{1}{(\hat{\rho}^{\prime})^{2}}\frac{\delta\text{S}^{*}}{\delta
p}\frac{\delta\text{S}^{*}}{\delta
p}-\frac{1}{\hat{\rho}^{\prime}}\biggl{(}{\hat{\rho}^{\prime}p_{,i}u_{i}+\hat{\rho}u_{i,i}}\biggr{)}\frac{\delta\text{S}^{*}}{\delta
p}\right]+\frac{\partial\text{S}^{*}}{\partial t}=0.$ (53)

In contrast to the four original field equations—(21) and (22)—the Hamilton-
Jacobi equation (53) is a _single_ equation in Hamilton’s functional
$\text{S}^{*}$. This constitutes an equivalent formulation of the problem, as
a complete and nontrivial solution to (53) is tantamount to an integration of
Hamilton’s equations (48) and (49) (note that it is not appropriate to take
$\text{S}^{*}\equiv 0$ for the same reason that it is not appropriate to take
$H^{*}\equiv 0$). In this way, we have reduced the problem of finding four
separate field quantities to that of finding a single functional in those
field quantities. One need only deduce (or even guess) the general form of
$\text{S}^{*}$ in order to solve the problem. If an analytical expression for
$\text{S}^{*}$ can be obtained, it will lead via canonical transformation to a
new set of fields ($\phi_{i}$, $\phi_{4}$) and conjugate momenta which are
simply equal to their initial values, giving analytical expressions for the
four original fields $(u_{i},p)$.

The case of incompressible flow requires care, as $\hat{\rho}^{\prime}\equiv
0$ and $\hat{\rho}^{\prime}$ appears in the denominators of terms in (53).
Even so, the Hamiltonian formulation remains well posed in the incompressible
limit. Recall that, with $\hat{\rho}^{\prime}\equiv 0$, $\mathcal{R}_{4}$
reduces to $\hat{\rho}u_{i,i}$, $\pi_{4}$ vanishes identically, and the
Hamiltonian density reduces to (51). Substituting for the conjugate momenta
$\pi_{i}$ in (51), the corresponding Hamilton-Jacobi equation is

$\int\text{d}^{3}x\left[\frac{1}{2}\frac{1}{{\rho}^{2}}\frac{\delta\text{S}^{*}}{\delta
u_{i}}\frac{\delta\text{S}^{*}}{\delta
u_{i}}-\frac{1}{{\rho}}\biggl{(}{\rho}u_{i,j}u_{j}+p_{,i}-\mu
u_{i,jj}-{\rho}b_{i}\biggr{)}\frac{\delta\text{S}^{*}}{\delta
u_{i}}\right]+\frac{\partial\text{S}^{*}}{\partial t}=0,$ (54)

with $\delta\text{S}^{*}/\delta p=0$, since again $\pi_{4}$ vanishes
identically for incompressible flow. This is the form of the Hamilton-Jacobi
equation as it relates to the traditional Navier-Stokes problem. In this case,
the pressure is determined last of all, and is whatever it needs to be to
enforce the incompressibility constraint $u_{i,i}=0$ (again consistent with
the role of pressure as Lagrange multiplier [10]).

## 4 Discussion

In this section we provide some qualitative interpretations of the
developments of Section 3. More specifically, we investigate the
incompressible form (via constant, uniform density) of the Euler-Lagrange
equations (35) and (36) when the residuals $\mathcal{R}_{i}$ and
$\mathcal{R}_{4}$ are substituted.

Our motivation is again the simple example of Section 1 for which the first-
order non-Hamiltonian system $\dot{v}=-v$ was converted to the second-order
Hamiltonian system $\ddot{v}=v$ by (manual) _elimination_ of the non-
conservative ‘damping’ term $\dot{v}$ (see [6] for a similar result for the
damped harmonic oscillator converting from second- to fourth-order dynamics).
Sanders [9] showed that the elimination process is ‘automated’ by the
definition of the action in the first integral of (3), which is generalized to
the action in (23) for our current continuum dynamics problem containing
fields.

First consider the pressure equation (36) and corresponding natural boundary
condition (41), which take the following incompressible forms:

$-\mathcal{R}_{i,i}=0\quad\forall x_{j}\in\mathcal{V},\quad\text{subject to
}\quad\mathcal{R}_{i}n_{i}=0\quad\forall x_{j}\in\partial\mathcal{V}$ (55)

This higher-order field equation is simply the divergence of the residual
$\mathcal{R}_{i}$. Upon substituting for $\mathcal{R}_{i}$ from (21) and
subsequently imposing the incompressible continuity condition
$u_{i,i}=\mathcal{R}_{4}/\rho=0$ from (22), we obtain:

$p_{,ii}=-[\rho u_{j}u_{i,j}]_{,i}+\rho b_{i,i},$ (56)

which is a Poisson equation for the pressure. The boundary condition is a
Neumann type requiring the specification of the normal pressure gradient,
$n_{i}p_{,i}=p_{,n}\equiv f(x_{j},t)$, on the boundary:

$f(x_{j},t)=-n_{i}\big{[}\rho\dot{u}_{i}+\rho u_{j}u_{i,j}-\mu u_{i,jj}-\rho
b_{i}\big{]}$ (57)

Equation (56) and boundary condition (57) evolve the pressure in a manner that
ensures the velocity field is solenoidal. This is a well known pressure-
velocity based formulation commonly used in the numerical solution of
incompressible flows (e.g. [74, 33]).

Next, we consider the velocity equations (35) which, at present, have a more
elusive physical interpretation. Here, we instead begin with the natural
boundary conditions (39) and (40), which are due to the $\delta u_{i}$ and
$\delta u_{i,j}$ variations. The incompressible versions of these equations
are:

$\displaystyle\rho\mathcal{R}_{i}u_{j}n_{j}+\mu\mathcal{R}_{i,j}n_{j}=0\quad\text{and}\quad-\mu\mathcal{R}_{i}n_{j}=0\quad\forall
x_{j}\in\partial\mathcal{V}$ (58)

The boundary conditions involving the residual $\mathcal{R}_{i}$ are those
compatible with the first-order Navier-Stokes equations, such as the no-slip
and no-penetration conditions. Indeed, if we specify the velocity vector of
the _actual motion_ on the boundary, then $\mathcal{R}_{i}\equiv 0$ there.
Note that the pressure of the actual motion on the boundary will be known from
the simultaneous solution of (56).

However, the _gradient_ terms $\mathcal{R}_{i,j}$ will introduce up to third-
order spatial derivatives that must be specified. These represent the
additional boundary conditions that must accompany the higher-order governing
equation, which will be seen shortly to be second-order in time and fourth-
order in space. Again, recall the example of Section 1, whereby the system (2)
must be appended with a second (initial) condition specifying the (time)
derivative of the coordinate $v(t)$. In the present context, these boundary
conditions are ostensibly tantamount to specification of the viscous stress on
the boundary by way of velocity gradients.

In general, the conditions at a boundary require two transition relations [30,
32] to ultimately describe the momentum transport. Mathematically speaking,
these conditions are the jump in velocity (momentum intensity) and the jump in
stress (momentum flux). Under ordinary physical circumstances the velocity and
stress are assumed to be continuous. However, this is one particular form of
the transition relations, and there are familiar examples to which they do not
apply. For example, at a liquid-gas interface the stress relation is modified
to account for a non-zero jump in the normal stress that is balanced by a
force due to surface tension (the tangential stress component usually still
taken to be continuous). Similarly, in the event that molecular slip occurs,
the typical transition relation gives an expression for the slip velocity
(e.g. [75, 76]). In the case of energy transport, analogous conditions are
needed regarding jumps in temperature (intensity) and heat flow (flux), which
are recognized as the concept of thermal contact resistance.

We now turn our attention to the Euler-Lagrange equations (35), which upon
imposing incompressibility and expanding derivatives of product terms yields:

$\displaystyle\rho\dot{\mathcal{R}}_{i}+\rho
u_{j}\mathcal{R}_{i,j}=\rho\mathcal{R}_{j}u_{j,i}-\mu\mathcal{R}_{i,jj}\quad\forall
x_{j}\in\mathcal{V}$ (59)

The left-hand side is the material derivative of the residual
$\mathcal{R}_{i}$. Our purpose here is to observe which terms from the first-
order Navier-Stokes equation are ‘eliminated’ in the higher-order formulation.
Specifically, we are interested in the non-conservative viscous terms; while
the body force $b_{i}$ could also be non-conservative, we will not concern
ourselves with this possibility. Direct substitution of $\mathcal{R}_{i}$ into
(59) generates many terms, but it is found that only one is canceled: the
viscous Laplacian of the (time derivative of the) velocity, namely
$\mu\dot{u}_{i,jj}$. This term mutually appears from the
$\rho\dot{\mathcal{R}}_{i}$ and $-\mu\mathcal{R}_{i,jj}$ terms in (59). To
maintain notional clarity, we write the residual as:

$\mathcal{R}_{i}=\rho\dot{u}_{i}-\mu u_{i,kk}+\tilde{\mathcal{R}}_{i}$ (60)

where index $k$ has been used to avoid confusion with gradient operators in
(59) having index $j$, and $\tilde{\mathcal{R}}_{i}=\rho u_{k}u_{i,k}-\rho
b_{i}$ are the remaining terms in the residual. Substituting the above into
the first and last terms of (59), canceling the aforementioned
$\mu\dot{u}_{i,jj}$ term, and then dividing out by the density gives:

$\rho\ddot{u}_{i}+\dot{\tilde{\mathcal{R}}}_{i}+u_{j}\mathcal{R}_{i,j}=\mathcal{R}_{j}u_{j,i}-\nu\big{[}-\mu
u_{i,kkjj}+\tilde{\mathcal{R}}_{i,jj}\big{]}$ (61)

where $\nu=\mu/\rho$ is the kinematic viscosity (recall that all variables are
non-dimensional). We see that this equation is second-order in time and
fourth-order in space. Viscous terms still appear in the equation including
second- and third-order spatial derivatives. Nevertheless, the technique
detailed by Sanders [9] and employed here evidently ensures that (61) has a
corresponding Hamiltonian structure.

## 5 Case Study

We can explore how this method can be applied by considering a simplified
example with a known field solution. In looking at the variety of cases in
which the Navier-Stokes equations have a known analytical solution, the
simplest are those involving steady flows. While the Euler-Lagrange equations
(35), (36) can be written for these cases, the corresponding Hamilton-Jacobi
equation is trivial because for steady flows the fields are already equal to
their initial values.

It is therefore worthwhile to examine the simplest unsteady flows, which
should result in a nontrivial Hamilton-Jacobi equation. Indeed, there exists a
class of flows for which the Navier-Stokes equations take the same simplified
form: those in which the flow is incompressible and unidirectional [30]. This
class of problems include both of Stokes’s flows [77], in which a semi-
infinite fluid is influenced by a boundary moving in its own plane. In the
first of these cases, the boundary is impulsively started and in the second,
the boundary oscillates. We can also include developing flow in a channel or
pipe. The only difference between these flows results from initial and
boundary conditions, but the Navier-Stokes equations and therefore the present
Hamilton-Jacobi equation take the same form.

Here we will examine the case in which there is motion only in the $x_{1}$
direction, and the velocities take the form
$\\{u_{i}\\}=\\{u_{1}(x_{2},t),0,0\\}$. In the absence of a body force, our
pressure gradient in the $x_{1}$ direction is solely a function of time and
the pressure gradients in the $x_{2}$ and $x_{3}$ directions are zero. There
are thus only two unknown field quantities: $u_{1}(x_{2},t)$ and $p(x_{1},t)$,
where $p$ is linear in $x_{1}$. The field equation of primary interest is

$\mathcal{R}_{1}\equiv\rho\dot{u}_{1}+p_{,1}-\mu u_{1,22}=0,$ (62)

and the remaining field equations are satisfied automatically by the assumed
form of the fields. Following the procedure described above, the momenta
conjugate to $u_{1}$ and $p$ are given by

$\pi_{1}\equiv\rho\mathcal{R}_{1},\quad\pi_{4}\equiv 0.$ (63)

This results in a Hamiltonian density given by:

$\mathcal{H}^{*}=\frac{1}{2}\frac{1}{{\rho}^{2}}\pi_{1}\pi_{1}-\frac{1}{{\rho}}\biggl{(}p_{,1}-\mu
u_{1,22}\biggr{)}\pi_{1}.$ (64)

Hamilton’s functional $\text{S}^{*}=\text{S}^{*}[t,u_{1},p]$ can be expressed
as an integral over $x_{2}$ only, since the other spatial coordinates do not
appear and may be integrated out. In this way, we may write the Hamilton-
Jacobi equation as follows:

$\int\text{d}x_{2}\left[\frac{1}{2}\frac{1}{{\rho}^{2}}\frac{\delta\text{S}^{*}}{\delta
u_{1}}\frac{\delta\text{S}^{*}}{\delta
u_{1}}-\frac{1}{{\rho}}\biggl{(}p_{,1}-\mu
u_{1,22}\biggr{)}\frac{\delta\text{S}^{*}}{\delta
u_{1}}\right]+\frac{\partial\text{S}^{*}}{\partial t}=0,$ (65)

with $\delta\text{S}^{*}/\delta p=0$. The solution to (65) would provide a
canonical transformation to a new set of coordinates, giving analytical
expressions for $(u_{1},p)$. Despite knowing the analytical solution for these
fields in this particular example, the present authors have not been able to
solve this Hamilton-Jacobi equation. Indeed, it would seem that solving a
differential equation containing functional derivatives represents an ongoing
field of study [78]. This example therefore appears to be a good place to
start for tackling the general problem.

## 6 Conclusion

This paper has presented a novel Hamiltonian formulation of the isotropic
Navier-Stokes problem for both compressible and incompressible fluids. This
canonical formulation opens several previously unexplored avenues toward a
final resolution of the problem, which we briefly describe below.

Perhaps the most obvious route would be to solve the Hamilton-Jacobi
equation—either (53) for the compressible case or (54) for the incompressible
case—for Hamilton’s functional $\text{S}^{*}[t,u_{i},p]$ directly. If a
complete solution for $\text{S}^{*}$ can be found, it will lead via canonical
transformation to a new set of fields which are equal to their initial values,
thereby giving analytical expressions for the original velocity and pressure
fields. Or, failing that, if one can simply establish based on existing
analytical techniques that a complete solution to this Hamilton-Jacobi
equation does (or does not) always exist under the usual assumptions, that
will also settle the question of existence of solutions.

An alternative strategy might be to investigate the corresponding Lagrangian
formulation based on the action $\mathcal{S}^{*}$ as given by (23). Because
the first- and second-order formulations are mathematically equivalent (recall
the proof in Section 3.2), $\mathcal{S}^{*}$ must have as many local minima as
there are solutions to the traditional, first-order formulation. Intuitively,
it seems as though it ought to be possible to determine—or at least to
establish bounds on—the number of critical points an action has based on the
form of the Lagrangian [64, 65]. If one can establish that, under the usual
assumptions, $\mathcal{S}^{*}$ always has exactly one local minimum, or else
demonstrate that there are cases where it fails to achieve a local minimum,
that too will resolve the question of existence and uniqueness.

Finally, it is worth noting that the techniques employed here are by no means
specific to the Navier-Stokes problem, nor are they restricted to the field of
classical mechanics. The suitably-averaged principle of least squares [5, 6,
7, 8, 9] can be applied to any traditionally non-Hamiltonian dynamical system
in order to formulate a mathematically equivalent higher-order Hamiltonian
system. It is believed that this fundamental result will also find uses in
other branches of pure and applied mathematics.

## Acknowledgments

The authors wish to thank Maggie Sanders for posing thought-provoking
questions during the development of this paper.

## Statements and declarations

### Funding

The authors declare that no funds, grants, or other support were received
during the preparation of this manuscript.

### Declaration of interests

The authors report no conflict of interest.

### Data availability

Any data appearing in the present work are available from the corresponding
author upon reasonable request.

### Author ORCID

  * •

J. W. Sanders, https://orcid.org/0000-0003-3059-3815

  * •

A. C. DeVoria, https://orcid.org/0000-0001-5615-0807

  * •

N. J. Washuta, https://orcid.org/0000-0002-4575-0564

  * •

J. C. Berlinghieri, https://orcid.org/0000-0001-7921-1895

### Author contributions

Sanders conceived the original idea for the work, wrote the Abstract,
Introduction, Analysis, Conclusion, and Appendix, and contributed to the
Literature Review. DeVoria wrote the Discussion, contributed to the Analysis
and Literature Review, and provided expertise in the area of fluid mechanics.
Washuta wrote the Case Study, contributed to the Literature Review, and
provided expertise in the area of fluid mechanics. Elamin contributed to the
Literature Review and provided expertise in the area of fluid mechanics.
Skenes organized and wrote the Literature Review and provided expertise in the
area of fluid mechanics. Berlinghieri contributed to the Introduction and
Literature Review, and provided expertise in the areas of analytical mechanics
and variational calculus. All authors contributed equally in checking the
accuracy of the work, discussing the results together as a group on multiple
occasions, drawing conclusions, and proofreading the manuscript.

### Ethical guidelines

Not applicable.

##  Appendix A Derivation of the Hamilton-Jacobi equation

In what follows, it is important to distinguish between two sets of solutions:
solutions to the second-order Euler-Lagrange equations (35) and (36), and
solutions to the first-order field equations (24) and (25). The latter are a
subset of the former, but not vice versa.

We define _Hamilton’s functional_ $\text{S}^{*}[t,u_{i},p]$ as

$\text{S}^{*}[t,u_{i},p]\equiv\int_{t_{0}}^{t}\text{d}t\left(\tilde{L}^{*}\right),$
(A.1)

where $t$ is the current (variable) time, $t_{0}$ is an arbitrary initial
time, and $\tilde{L}^{*}$ denotes the Lagrangian (26) evaluated for fields
satisfying the _second-order_ Euler-Lagrange equations (35) and (36)—not
necessarily the first-order field equations (24) and (25). Crucially, because
the fields do not necessarily satisfy the first-order field equations, it is
not appropriate to take $\text{S}^{*}\equiv 0$. We imagine that the time
integral has already been carried out, so that the functional $\text{S}^{*}$
may be regarded as an integral over $\mathcal{V}$, that is,

$\text{S}^{*}[t,u_{i},p]=\int\text{d}^{3}x\left(s^{*}\right),$ (A.2)

where $s^{*}$ is the Lagrangian density (27) evaluated for fields satisfying
the second-order Euler-Lagrange equations (which will be denoted
$\tilde{\mathcal{L}}^{*}$) integrated over time from $t_{0}$ to $t$.

Starting from (A.1) and evaluating the first variation of $\text{S}^{*}$ as we
did the action $\mathcal{S}^{*}$ in Section 3, we find that

$\delta\text{S}^{*}=\int\text{d}^{3}x\biggl{[}\pi_{i}\delta{u}_{i}+\pi_{4}\delta{p}\biggr{]}_{t_{0}}^{t}+\int\text{d}^{2}x\left(\cdots\right),$
(A.3)

where we have used the fact that the second-order Euler-Lagrange equations
(35) and (36) are satisfied by definition. But, by the definition of
functional derivatives, we have that

$\delta\text{S}^{*}=\int\text{d}^{3}x\biggl{[}\frac{\delta\text{S}^{*}}{\delta
u_{i}}\delta{u}_{i}+\frac{\delta\text{S}^{*}}{\delta
p}\delta{p}\biggr{]}+\int\text{d}^{2}x\left(\cdots\right).$ (A.4)

In this way, we may identify the conjugate momenta with the functional
derivatives of $\text{S}^{*}$:

$\pi_{i}=\frac{\delta\text{S}^{*}}{\delta
u_{i}},\quad\pi_{4}=\frac{\delta\text{S}^{*}}{\delta p}.$ (A.5)

Now, starting from (A.1) and evaluating the time derivative, we find that

$\frac{\text{d}\text{S}^{*}}{\text{d}t}=\tilde{L}^{*}=\int\text{d}^{3}x\left(\tilde{\mathcal{L}}^{*}\right).$
(A.6)

But, by the chain rule,

$\frac{\text{d}\text{S}^{*}}{\text{d}t}=\frac{\partial\text{S}^{*}}{\partial
t}+\int\text{d}^{3}x\biggl{[}\frac{\delta\text{S}^{*}}{\delta
u_{i}}\dot{u}_{i}+\frac{\delta\text{S}^{*}}{\delta p}\dot{p}\biggr{]}.$ (A.7)

In this way, we find that

$\int\text{d}^{3}x\biggl{[}\frac{\delta\text{S}^{*}}{\delta
u_{i}}\dot{u}_{i}+\frac{\delta\text{S}^{*}}{\delta
p}\dot{p}-\tilde{\mathcal{L}}^{*}\biggr{]}+\frac{\partial\text{S}^{*}}{\partial
t}=0.$ (A.8)

Here the integral is simply the Hamiltonian $H^{*}$, with the conjugate
momenta replaced by the functional derivatives in accordance with (A.5). Hence
we arrive at the Hamilton-Jacobi equation

${H}^{*}\left[t,u_{i},p,\frac{\delta\text{S}^{*}}{\delta
u_{i}},\frac{\delta\text{S}^{*}}{\delta
p}\right]+\frac{\partial\text{S}^{*}}{\partial t}=0,$ (A.9)

as claimed in (52).

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