Extension of Hamiltonian Mechanics to Non-Conservative Systems Via Higher-Order Dynamics

Sanders, John W.; Becker, Eric T.; DeVoria, Adam C.

doi:10.1115/1.4067376

Abstract

This paper presents a detailed review of the emerging topic of higher-order dynamics and their intrinsic variational structure, which has enabled—for the very first time in history—the general application of Hamiltonian formalism to non-conservative systems. Here the general theory is presented alongside several interesting applications that have been discovered to date. These include the direct modal analysis of non-proportionally damped dynamical systems, a new and more efficient algorithm for computing the resonant frequencies of damped systems with many degrees-of-freedom, and a canonical Hamiltonian formulation of the Navier–Stokes problem. A significant merit of the Hamiltonian formalism is that it leads to the transformation theory of Hamilton and Jacobi, and specifically the Hamilton–Jacobi equation, which reduces even the most complicated of problems to the search for a single scalar function (or functional, for problems in continuum mechanics). With the extension of the Hamiltonian framework to non-conservative systems, now every problem in classical mechanics can be reduced to the search for a single scalar. This discovery provides abundant opportunities for further research, and here we list just a few potential ideas. Indeed, the present authors believe there may be many more applications of higher-order dynamics waiting to be discovered.

1 Introduction

The techniques of Hamiltonian mechanics—Hamilton’s principle of stationary action [1–4], Hamilton’s canonical equations [2–4], and the transformation theory of Hamilton and Jacobi [5,6]—are known to apply only to “Hamiltonian” systems (systems that are both conservative and holonomic) [7,8]. And yet, for almost as long as the principle of stationary action has been known, researchers have been attempting to extend it to non-Hamiltonian systems [9]. This paper reviews an emerging development on that subject: the recent discovery that higher-order dynamics are intrinsically variational, in the sense that higher-derivative versions of the classical equations of motion can always be derived from a stationary action principle, even when the physical system is non-conservative [10–14]. This discovery has already led to three interesting applications: the direct modal analysis of non-proportionally damped systems [10–12], a new and more efficient algorithm for computing a damped system’s resonant frequencies [15,16], and a novel canonical Hamiltonian formulation of the Navier–Stokes problem [17,18].

The fundamental obstacle to applying Hamiltonian formalism to non-conservative systems is the fact that the virtual work done by dissipative forces cannot be expressed as the exact variation of a work function [8]. For simplicity of notation, consider a point mass

m

with a single-degree-of-freedom

x

⁠. The force of inertia

m \ddot{x}

can always be obtained from the variation of the kinetic energy

T

δ T = δ \frac{1}{2} m {\dot{x}}^{2} = m \dot{x} δ \dot{x} ≅ - m \ddot{x} δ x

(1)

(Note: Here and in what follows, we will use the congruence operator

≅

as a shorthand to denote the results of integration by parts underneath a time integral.) Likewise, conservative forces

F (x)

can always be obtained from the variation of the potential energy

V (x)

δ V = V^{'} (x) δ x = - F δ x

(2)

hence the conventional Hamilton action

S = \int (T - V) d t

[1–4]. However, it happens that no force depending on

\dot{x}

can be obtained by variation of a work function. Indeed, the most general first-order Lagrangian

L (t, x, \dot{x})

gives upon variation

δ L ≅ (\frac{\partial L}{\partial x} - \frac{\partial}{\partial x} \frac{\partial L}{\partial \dot{x}} \dot{x} - \frac{\partial}{\partial t} \frac{\partial L}{\partial \dot{x}} - \frac{\partial^{2} L}{\partial {\dot{x}}^{2}} \ddot{x}) δ x

(3)

In order for this to yield the proper inertial term,

- m \ddot{x} δ x

⁠, and no other terms involving

\ddot{x}

⁠, it must be the case that

\partial^{2} L / \partial {\dot{x}}^{2} = m

⁠, implying that

L (t, x, \dot{x}) = \frac{1}{2} m {\dot{x}}^{2} + f (x, t) \dot{x} + g (x, t)

(4)

But in that case

\frac{\partial}{\partial t} \frac{\partial L}{\partial \dot{x}} = \frac{\partial f (x, t)}{\partial t}

(5)

does not depend on

\dot{x}

⁠, and neither does

\frac{\partial L}{\partial x} - \frac{\partial}{\partial x} \frac{\partial L}{\partial \dot{x}} \dot{x} = \frac{\partial f}{\partial x} \dot{x} + \frac{\partial g}{\partial x} - \frac{\partial f}{\partial x} \dot{x} = \frac{\partial g (x, t)}{\partial x}

(6)

Hence, it is impossible to derive both the inertial force $m \ddot{x}$ and a force depending on $\dot{x}$ from a first-order Lagrangian without also obtaining non-physical terms involving $\ddot{x}$ ⁠. This mathematical incongruity of odd-order derivatives—which lies at the very core of this paper and will be a recurring theme throughout—coincides with the fundamental physical distinction between conservative and non-conservative forces, as it is well known that dissipative forces only arise from a failure to account for the motion of individual atoms and molecules.

Despite this fundamental challenge, over the last two centuries many researchers have attempted to extend the stationary action principle to dissipative systems [9,19–24]. It is not within the scope of this paper to give a detailed account of every such attempt (see, for example, de Leon et al. [25], Limebeer et al. [26], Sanders et al. [17,18], and the references cited therein). In what follows, we focus on bona fide variational principles, and we exclude, for example, the d’Alembert-Lagrange principle [27,28] and non-canonical dissipative brackets [25], among others.

One of the earliest attempts at a variational theory of dissipation was made by Gauss [9]. Consider a system of

N

particles (masses

m_{j}

and positions

r_{j}

⁠, where

j = 1, 2, \dots, N

⁠), and define Gauss’s constraint function as

G = \sum_{j = 1}^{N} \frac{1}{2 m_{j}} {(m_{j} {\ddot{r}}_{j} - N_{j})}^{2}

(7)

where

N_{j}

is the net force on particle

j

arising from all non-constraint forces [8,9]. The quantity

m_{j} {\ddot{r}}_{j} - N_{j}

is simply the net constraint force on particle

j

⁠. Thus,

G

is a weighted average of the squares of the constraint force magnitudes. Upon varying the accelerations

{\ddot{r}}_{j}

(not the positions

r_{j}

⁠), Gauss [9] observed that

δ_{{\ddot{r}}_{j}} G = \sum_{j = 1}^{N} (m_{j} {\ddot{r}}_{j} - N_{j}) \cdot δ {\ddot{r}}_{j} = 0

(8)

In other words, of all conceivable accelerations the particles might take, the actual accelerations are those for which $G$ achieves a local minimum. This is referred to as Gauss’s principle of least constraint [8,9]. One may interpret Gauss’s principle as stating that the actual accelerations of a constrained system are as close as possible to the unconstrained accelerations, and in the absence of constraints, $G$ is free to assume its absolute minimum value of zero [8].

It should be noted that, because Gauss’s principle [9] employs variations in the accelerations—not the coordinates—it does not lead to a set of canonical equations [2–4] nor to an associated theory of canonical transformations [5,6]. Even so, we will have reason to return to Gauss’s principle in Sec. 5.

The twentieth century saw a renewed interest in extending Lagrangian and Hamiltonian formalism to non-conservative systems. Bateman [20] observed that any equation of motion can be trivially obtained from a stationary action principle if one introduces dual degrees-of-freedom and, consequently, additional equations. For instance, to obtain the damped harmonic oscillator, Bateman [20] proposed a Lagrangian of the form

L (x, \dot{x}, \ddot{x}, y) = y (\ddot{x} + 2 ω ζ \dot{x} + ω^{2} x)

(9)

where

x

and

y

are to be varied independently, and

ω

and

ζ

are the usual undamped natural frequency and damping ratio, respectively. This observation developed into the dual system method of Bateman, Morse, Feshbach, and Tikochinsky [20,23,24]. In this case, the Euler–Lagrange equations yield two oscillators: the damped harmonic oscillator, and a dual anti-damped oscillator

\ddot{y} - 2 ω ζ \dot{y} + ω^{2} y = 0

(10)

A different approach is the Kanai–Caldirola oscillator [21,22], which employs the time-dependent Lagrangian

L (t, x, \dot{x}) = e^{2 ω ζ t} (\frac{1}{2} m {\dot{x}}^{2} - \frac{1}{2} m ω^{2} x^{2})

(11)

with the resulting Euler–Lagrange equation

e^{2 ω ζ t} m (\ddot{x} + 2 ω ζ \dot{x} + ω^{2} x) = 0

(12)

The Kanai–Caldirola approach [21,22] suffers from a non-physical inertial term $e^{2 ω ζ t} m \ddot{x}$ ⁠, consistent with our earlier observation regarding the incompatibility of the inertial force $m \ddot{x}$ with forces depending on $\dot{x}$ ⁠.

The remainder of this paper presents a chronological review of the emerging topic of higher-order dynamics and their intrinsic variational structure [13,14]. The general framework is presented alongside various applications that have been discovered to date, including the direct modal analysis of damped dynamical systems [10–12], a new and more efficient algorithm for computing damped resonant frequencies [15,16], and a novel canonical Hamiltonian formulation of the Navier–Stokes problem [17,18]. The present authors believe there may be more such applications waiting to be discovered, and we will conclude with some potential ideas in Sec. 7.

2 Fourth-Order Dynamics

We may take from the discussion at the beginning of Sec. 1 that even-order derivatives (⁠ $x$ ⁠, $\ddot{x}$ ⁠) are easily derivable from an action integral, while odd-order derivatives (⁠ $\dot{x}$ ⁠) are impossible to derive without incurring non-physical terms. Although we cannot discard the odd-order terms from an equation indiscriminately, it is possible to “hide” them in even-order terms by doubling the order of the equation [10,11].

Following Sanders [10,11], consider the damped harmonic oscillator

\ddot{x} + 2 ω ζ \dot{x} + ω^{2} x = 0

(13)

and differentiate twice with respect to time

\overset{⃛}{x} + 2 ω ζ \ddot{x} + ω^{2} \dot{x} = 0

(14)

\overset{⃜}{x} + 2 ω ζ \overset{⃛}{x} + ω^{2} \ddot{x} = 0

(15)

From the fourth-order equation , eliminate

\overset{⃛}{x}

in favor of

\dot{x}

using the third-order equation . Then, eliminate

\dot{x}

in favor of

\ddot{x}

and

x

using the original second-order equation . This yields

\overset{⃜}{x} + 4 ω^{2} (\frac{1}{2} - ζ^{2}) \ddot{x} + ω^{4} x = 0

(16)

which is a Pais–Uhlenbeck equation [29] (well known in quantum gravity physics as a toy model for higher-derivative theories of quantum gravity [30–34]). As noted by Sanders [10,11], the

\ddot{x}

-term vanishes for

ζ = 1 / \sqrt{2}

⁠, which is the optimal damping ratio in terms of the tradeoff between rise time and percent overshoot in response to a step input [35,36]. It is a curious feature of the fourth-order dynamics that they can sometimes single out particularly useful values of the model parameters [10,11].

The action corresponding to Eq. is

S^{*} = \int_{t_{1}}^{t_{2}} \frac{1}{2} [{\ddot{x}}^{2} - 4 ω^{2} (\frac{1}{2} - ζ^{2}) {\dot{x}}^{2} + ω^{4} x^{2}] d t

(17)

where the asterisk on

S^{*}

serves to differentiate it from the classical Hamilton action

S

[10,11]. Sanders [11] observed that the “Lagrangian” (i.e., the action integrand) can be obtained by squaring the residual

R = \ddot{x} + 2 ω ζ \dot{x} + ω^{2} x

of the second-order equation and discarding terms with odd numbers of derivatives (e.g.,

x \dot{x}

and

\dot{x} \ddot{x}

⁠), since these are exact differentials and can therefore be integrated out of the action [8]. Interestingly, even if we do not manually eliminate those terms, we will still obtain the same Euler–Lagrange equation, with only even-order terms [13,14]. Here again we see the fundamental incompatibility of odd-order derivatives with stationary action principles. Indeed, the higher-order action integral acts as a filter for terms with odd numbers of derivatives [11,13,14,17,18].

We note in passing that, with

ζ = 1 / \sqrt{2}

⁠, Eq. reduces to

S^{*} = \int_{t_{1}}^{t_{2}} \frac{1}{2} ({\ddot{x}}^{2} + ω^{4} x^{2}) d t, ζ = \frac{1}{\sqrt{2}}

(18)

This is consistent with the optimal control problem of Newton et al. [35,36].

3 Applications to Modal Analysis and the Computation of Resonant Frequencies

An interesting feature of the fourth-order equation is that it can be written as a system of two coupled second-order equations [10,11,32]

\frac{d^{2}}{d t^{2}} [\begin{matrix} x \\ y \end{matrix}] + [\begin{matrix} + μ_{1} & - ρ_{1} \\ - ρ_{2} & + μ_{2} \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}]

(19)

where

μ_{1}

⁠,

μ_{2}

⁠,

ρ_{1}

⁠, and

ρ_{2}

are generally complex-valued constants satisfying

μ_{1} + μ_{2} = 4 ω^{2} (\frac{1}{2} - ζ^{2})

(20)

μ_{1} μ_{2} - ρ_{1} ρ_{2} = ω^{4}

(21)

and

y (t)

is a dual-oscillator coordinate. As noted by Sanders [10,11], this shows that linear, viscous damping is mathematically equivalent to a spring-like coupling between the original oscillator and a fictitious particle, where the stiffnesses of the springs may be complex-valued or negative [10,11]. For instance, with the optimal damping ratio

ζ = 1 / \sqrt{2}

⁠, these fictitious coupling springs would have purely imaginary stiffnesses

\pm i k

⁠, where

k

is the actual stiffness of the physical system [10,11].

Applying similar methods, Sanders [11] showed that any forced linear system with

N

degrees-of-freedom

\ddot{x} + H \dot{x} + L x = f (t)

(22)

where

H

is the

N \times N

mass-normalized damping matrix,

L

is the

N \times N

mass-normalized stiffness matrix,

x

is an

N \times 1

column matrix containing all of the oscillator coordinates, and

f (t)

is the

N \times 1

mass-normalized force matrix, has an equivalent fourth-order formulation

\begin{aligned} \overset{⃜}{x} + (H L H^{- 1} - H^{2} + L) \ddot{x} + H L H^{- 1} L x \\ = \ddot{f} (t) - H \dot{f} (t) + H L H^{- 1} f (t) \end{aligned}

(23)

This fourth-order system may in turn be written as two coupled second-order systems

\frac{d^{2}}{d t^{2}} [\begin{matrix} x \\ y \end{matrix}] + [\begin{matrix} + U & - P \\ - R & + V \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} I & 0 \\ A & B \end{matrix}] [\begin{matrix} f (t) \\ \dot{f} (t) \end{matrix}]

(24)

where

y

is an

N \times 1

column matrix of dual-oscillator coordinates;

U

⁠,

P

⁠,

R

⁠,

V

⁠,

A

⁠, and

B

are complex-valued constant

N \times N

matrices related to

H

and

L

⁠; and

I

is the

N \times N

identity (for the details, see Sanders [11] and Sanders and Inman [15]). With

A = 0

and

B = I

⁠, the square stiffness matrix in Eq. takes the form

\tilde{Ω} \equiv [\begin{matrix} + U & - P \\ - R & + V \end{matrix}] = [\begin{matrix} L - H^{2} & H \\ - L H & L \end{matrix}]

(25)

The absence of odd-order derivatives in Eq. enables direct modal analysis and decoupling of the augmented system, even when the original second-order system is non-proportionally damped and cannot be decoupled via simultaneous diagonalization of

H

and

L

[11,15]. Indeed, the system’s resonant frequencies

ω_{j}^{*}

(for underdamped modes with

ζ_{j} < 1 / \sqrt{2}

⁠) are obtained from the complex conjugate pair eigenvalues of

\tilde{Ω}

ω_{j}^{*} = \sqrt{Re [Λ_{\tilde{Ω}, j}]}, j = 1, 2, \dots, P

(26)

where the

Λ_{\tilde{Ω}, j}

are the eigenvalues of

\tilde{Ω}

and

P

is the number of resonant modes, which may be less than or equal to the total degrees-of-freedom

N

[11,15].

The modal analysis technique described above is referred to as the dual-oscillator approach to complex-stiffness damping [11]. Sanders and Inman [15] have demonstrated empirically that the dual-oscillator approach is significantly faster at computing a system’s resonant frequencies than the traditional approach (pioneered by Foss [37] and later published by Traill-Nash [38]), specifically for systems with large degrees-of-freedom. For example, with $145$ degrees-of-freedom, Sanders and Inman [15] found that the dual oscillator approach was 25% faster, with attained significance levels below machine precision.

Precisely why the dual-oscillator approach is more efficient is still unclear. In a separate paper, Becker et al. [16] showed that the efficiency is due almost entirely to a decrease in the amount of time required to compute the eigenvalues of the complex-stiffness matrix $\tilde{Ω}$ ⁠. A rigorous mathematical explanation remains elusive, but the present authors believe it must have to do with the Hamiltonian structure of the dual-oscillator system (24), which it inherits from the fourth-order dynamics (23).

It should be mentioned here that the dual-oscillator approach is not restricted to linear systems. Indeed, Sanders [11] shown that the nonlinear power-law hardening oscillator

\ddot{x} + α \dot{x} + β x + γ x^{η} = f (t), η \neq 1

(27)

where

α

⁠,

β

⁠,

γ

⁠, and

η \neq 1

are constants, is mathematically equivalent to the nonlinear dual oscillator system [11]

\ddot{x} + (β + \frac{α^{2}}{η - 1}) x - ρ y + γ x^{η} = f (t)

(28)

\begin{aligned} \ddot{y} - ρ x + (β - \frac{α^{2} η}{η - 1}) y + η γ x^{η - 1} y \\ = [\frac{α^{2} η}{(η - 1) ρ}] f (t) - (\frac{α}{ρ}) \dot{f} (t) \end{aligned}

(29)

with

ρ

given by [11]

ρ = i α \sqrt{β + \frac{η α^{2}}{(η - 1)^{2}}}

(30)

The undamped (but complex-valued) nonlinear normal modes of Eqs. (28) and (29) should be mathematically equivalent to the damped nonlinear normal modes of the original oscillator (27) [11]. It may also be possible to extend the approach to different kinds of nonlinearities (other than power-law hardening) and nonlinear systems with multi-degrees-of-freedom.

4 General Hamiltonian Framework

Sanders [11] recognized the intrinsic variational structure of the fourth-order formulation and subsequently established the general form of the associated action [12]

S^{*} [q_{i} (t)] = \int_{t_{1}}^{t_{2}} L^{*} d t, L^{*} = \frac{1}{2} R_{i} R_{i}

(31)

where the

q_{i} = q_{i} (t)

are suitable generalized coordinates (note that, in order to maintain dimensional consistency, the generalized coordinates may need to be nondimensionalized),

L^{*}

is the Lagrangian,

R_{i} \equiv {\ddot{q}}_{i} - f_{i} (q_{j}, {\dot{q}}_{j}, t), i, j = 1, 2, \dots, n

(32)

are the residuals of the second-order formulation,

f_{i} = f_{i} (q_{j}, {\dot{q}}_{j}, t)

are generalized forces,

n

is the system’s degrees-of-freedom, and we employ the Einstein summation convention on repeated subscript indices. As noted by Sanders [12], the minimization of

L^{*}

is the strong form of the well known and widely applied principle of least squares [39]. The minimization of

S^{*}

is thus a time-averaged version of the principle of least squares [12].

With this action integral, it is finally possible to apply the full theoretical framework of Hamiltonian mechanics—including Hamilton’s canonical equations [2–4] and the transformation theory of Hamilton and Jacobi [5,6]—to non-Hamiltonian systems [12–14].

4.1 Euler–Lagrange Equations.

Varying the coordinates

q_{i} (t)

⁠, Sanders [13,14] obtained

δ S^{*} = \int_{t_{1}}^{t_{2}} (R_{i} δ {\ddot{q}}_{i} - R_{j} \frac{\partial f_{j}}{\partial {\dot{q}}_{i}} δ {\dot{q}}_{i} - R_{j} \frac{\partial f_{j}}{\partial q_{i}} δ q_{i}) d t

(33)

and after integration by parts

\begin{aligned} δ S^{*} ≅ \int_{t_{1}}^{t_{2}} & [{\ddot{R}}_{i} + \frac{d}{d t} (R_{j} \frac{\partial f_{j}}{\partial {\dot{q}}_{i}}) - R_{j} \frac{\partial f_{j}}{\partial q_{i}}] δ q_{i} d t \\ + {[- ({\dot{R}}_{i} + R_{j} \frac{\partial f_{j}}{\partial {\dot{q}}_{i}}) δ q_{i} + R_{i} δ {\dot{q}}_{i}]}_{t_{1}}^{t_{2}} \end{aligned}

(34)

The Euler–Lagrange equations are [13,14]

{\ddot{R}}_{i} + \frac{d}{d t} (R_{j} \frac{\partial f_{j}}{\partial {\dot{q}}_{i}}) - R_{j} \frac{\partial f_{j}}{\partial q_{i}} = 0

(35)

These equations are fourth-order in time, and are mathematically equivalent to the original second-order equations

R_{i} = 0

with identical initial conditions on

q_{i}

⁠,

{\dot{q}}_{i}

⁠,

{\ddot{q}}_{i}

⁠, and

{\overset{⃛}{q}}_{i}

⁠. Specifically, if

q_{i} (0) = q_{i}^{0}

and

{\dot{q}}_{i} (0) = v_{i}^{0}

⁠, then we require

{\ddot{q}}_{i} (0) = f_{i} (q_{j}^{0}, v_{j}^{0}, 0)

(36)

and

{\overset{⃛}{q}}_{i} (0) = {\dot{f}}_{i} (q_{j}^{0}, v_{j}^{0}, 0)

(37)

to recover the solution to the original second-order problem [13,14]. Indeed,

R_{i} = 0

constitutes an equilibrium solution of the fourth-order formulation, so that, provided

R_{i}

and

{\dot{R}}_{i}

vanish at time

t = 0

⁠, they will remain zero for all future times, thus recovering the actual motion [10–14,17,18].

4.2 Hamilton’s Equations.

Corresponding to each of the coordinates

q_{i}

and the associated velocities

v_{i} = {\dot{q}}_{i}

are canonically conjugate momenta, which can be identified directly from the boundary term in Eq. . Specifically, the momenta conjugate the coordinates

q_{i}

are [13,14]

p_{i} = - ({\dot{R}}_{i} + R_{j} \frac{\partial f_{j}}{\partial {\dot{q}}_{i}})

(38)

and the momenta conjugate to the velocities

v_{i}

are [13,14]

P_{i} = R_{i}

(39)

The associated Hamiltonian is obtained via the Legendre transform

H^{*} = p_{i} v_{i} + P_{i} {\dot{v}}_{i} - L^{*} = \frac{1}{2} P_{i} P_{i} + P_{i} f_{i} (q_{j}, v_{j}, t) + p_{i} v_{i}

(40)

where

L^{*} = \frac{1}{2} R_{i} R_{i} = \frac{1}{2} P_{i} P_{i}

and

{\dot{v}}_{i} = {\ddot{q}}_{i} = R_{i} + f_{i} (q_{j}, v_{j}, t) = P_{i} + f_{i} (q_{j}, v_{j}, t)

(41)

have been expressed in terms of the coordinates, velocities, and conjugate momenta [13,14]. Unlike the classical Hamiltonian, this

H^{*}

has nothing to do with the total mechanical energy of the system. Nevertheless, it is still a conserved quantity, since it vanishes for the actual motion (⁠

R_{i} = 0

⁠).

Hamilton’s canonical equations are given by [13,14]

{\dot{q}}_{i} = \frac{\partial H^{*}}{\partial p_{i}}, {\dot{v}}_{i} = \frac{\partial H^{*}}{\partial P_{i}}

(42)

{\dot{p}}_{i} = - \frac{\partial H^{*}}{\partial q_{i}}, {\dot{P}}_{i} = - \frac{\partial H^{*}}{\partial v_{i}}

(43)

The first set of equations, ${\dot{q}}_{i} = \partial H^{*} / \partial p_{i}$ ⁠, recovers ${\dot{q}}_{i} = v_{i}$ ⁠. The second, ${\dot{v}}_{i} = \partial H^{*} / \partial P_{i}$ ⁠, recovers the right-hand side of Eq. (41), which is mathematically equivalent to Eq. (39). The third, ${\dot{p}}_{i} = - \partial H^{*} / \partial q_{i}$ ⁠, recovers the Euler–Lagrange equations (35). The fourth and last set of equations, ${\dot{P}}_{i} = - \partial H^{*} / \partial v_{i}$ ⁠, recovers Eq. (38). These canonical equations are therefore mathematically equivalent to the fourth-order problem, which is in turn mathematically equivalent to the original second-order problem [13,14].

4.3 Hamilton–Jacobi Theory.

Associated with the Hamiltonian is a corresponding Hamilton–Jacobi equation [13,14]

\frac{1}{2} \frac{\partial S^{*}}{\partial v_{i}} \frac{\partial S^{*}}{\partial v_{i}} + \frac{\partial S^{*}}{\partial v_{i}} f_{i} (q_{j}, v_{j}, t) + \frac{\partial S^{*}}{\partial q_{i}} v_{i} + \frac{\partial S^{*}}{\partial t} = 0

(44)

where Hamilton’s principal function

S^{*} = S^{*} (q_{i}, v_{j}, t)

serves as the generating function for a canonical transformation to a new set of coordinates, velocities, and conjugate momenta for which the Hamiltonian

H^{*}

vanishes identically (i.e., not just for the actual motion, but for every conceivable motion), in which case Hamilton’s equations are trivial: the new coordinates, velocities, and their conjugate momenta are simply equal to their initial values. The merit of the Hamilton–Jacobi equation is that reduces the problem of finding the (generally multitudinous) generalized coordinates

q_{i} (t)

and velocities

v_{i} (t)

to that of finding a single scalar function. Indeed,

S^{*}

contains within it the entire solution to the initial value problem, and to have found the former is to have solved the latter.

Unlike the classical Hamilton–Jacobi equation [2–8], Eq. (44) is valid for all mechanical systems, both Hamiltonian and non-Hamiltonian [13,14]. It reduces every single problem in classical mechanics to the search for a single scalar function, $S^{*} (q_{i}, v_{j}, t)$ [13,14].

5 Comparison to Gauss’s Principle

The Lagrangian $L^{*} = \frac{1}{2} R_{i} R_{i}$ of the fourth-order formulation bears a superficial resemblance to Gauss’s constraint function $G$ [8,9] in that both are sums of squares of residuals. In fact, the two are so similar for unconstrained systems that Sanders [11] originally mistook them for the same quantity, identifying $S^{*}$ with the time average of $G$ and calling $S^{*}$ the “Gauss action” [11]. However, a simple counterexample demonstrates that $G$ is generally not a valid Lagrangian for the fourth-order dynamics.

Following a suggestion in Ref. [12], consider a nondimensionalized simple pendulum with angle

ϕ (t)

⁠. On the one hand, we find that

L^{*} = \frac{1}{2} (\ddot{ϕ} + \sin ϕ)^{2} = \frac{1}{2} ({\ddot{ϕ}}^{2} + 2 \ddot{ϕ} \sin ϕ + \sin^{2} ϕ)

(45)

and upon varying

ϕ

δ L^{*} ≅ (\overset{⃜}{ϕ} + 2 \ddot{ϕ} \cos ϕ - {\dot{ϕ}}^{2} \sin ϕ + \sin ϕ \cos ϕ) δ ϕ

(46)

gives a mathematically valid fourth-order formulation of the simple pendulum, since

\overset{⃜}{ϕ} + 2 \ddot{ϕ} \cos ϕ - {\dot{ϕ}}^{2} \sin ϕ + \sin ϕ \cos ϕ = 0

is equivalent to

\frac{d^{2}}{d t^{2}} (\ddot{ϕ} + \sin ϕ) = \overset{⃜}{ϕ} + \ddot{ϕ} \cos ϕ - {\dot{ϕ}}^{2} \sin ϕ = 0

with

\ddot{ϕ} + \sin ϕ = 0

⁠.

On the other hand, we find that

G = \frac{1}{2} ({\ddot{ϕ}}^{2} + {\dot{ϕ}}^{4} + 2 \ddot{ϕ} \sin ϕ + 2 {\dot{ϕ}}^{2} \cos ϕ + 1)

(47)

In order for

G

to yield a valid equation of motion, we must vary the acceleration

\ddot{ϕ}

⁠, in which case we find that

δ_{\ddot{ϕ}} G = (\ddot{ϕ} + \sin ϕ) δ \ddot{ϕ}

(48)

and setting this to zero gives the correct second-order equation

\ddot{ϕ} + \sin ϕ = 0

⁠. However, we do not obtain a valid fourth-order equation from

G

by varying

ϕ

⁠. This is easiest to see by considering the difference

G - L^{*} = \frac{1}{2} {\dot{ϕ}}^{4} + {\dot{ϕ}}^{2} \cos ϕ + \frac{1}{2} \cos^{2} ϕ

(49)

Upon varying

ϕ

⁠, we have that

δ (G - L^{*}) ≅ (- 6 {\dot{ϕ}}^{2} \ddot{ϕ} - 2 \ddot{ϕ} \cos ϕ + {\dot{ϕ}}^{2} \sin ϕ - \sin ϕ \cos ϕ) δ ϕ

(50)

which does not vanish for the actual motion satisfying

\ddot{ϕ} + \sin ϕ = 0

⁠. Indeed, setting

\sin ϕ = - \ddot{ϕ}

⁠, may be simplified to

δ (G - L^{*}) ≅ (- 7 {\dot{ϕ}}^{2} - \cos ϕ) \ddot{ϕ} δ ϕ

(51)

and it is not the case that

- 7 {\dot{ϕ}}^{2} - \cos ϕ = 0

⁠. We conclude that

G

is generally not a valid Lagrangian for the fourth-order dynamics.

6 Hamiltonian Formulation of the Navier–Stokes Problem

The general framework outlined in Sec. 4 has given rise to the very first canonical Hamiltonian formulation of the Navier–Stokes problem [17,18].

Starting from the action

S^{*} = \int d^{4} x (\frac{1}{2} R_{i} R_{i} + \frac{1}{2} R_{4} R_{4})

(52)

where

R_{i} \equiv ρ {\dot{u}}_{i} + ρ u_{i, j} u_{j} + p_{, i} - μ u_{i, j j} - (μ + λ) u_{j, j i} - ρ b_{i} = 0

(53)

R_{4} \equiv \dot{ρ} + ρ_{, i} u_{i} + ρ u_{i, i} = 0

(54)

are the residuals of the compressible Navier–Stokes equations, all quantities are as defined in Refs. [17,18], and the integral is carried out over the spacetime occupied by the fluid, Sanders et al. [17,18] were able to construct a conserved Hamiltonian functional

H^{*}

for which Hamilton’s equations constitute a mathematically equivalent second-order formulation of the problem [17,18].

From that

H^{*}

⁠, Sanders et al. [17,18] formulated the associated Hamilton–Jacobi equation. In the incompressible limit, that Hamilton–Jacobi equation takes the form [17,18]

\begin{aligned} \int d^{3} x [\frac{1}{2} \frac{1}{ρ^{2}} & \frac{δ S^{*}}{δ u_{i}} \frac{δ S^{*}}{δ u_{i}} - \frac{1}{ρ} (ρ u_{i, j} u_{j} + \dots \\ \dots + p_{, i} - μ u_{i, j j} - ρ b_{i}) \frac{δ S^{*}}{δ u_{i}}] + \frac{\partial S^{*}}{\partial t} = 0 \end{aligned}

(55)

where

S^{*} = S^{*} [u_{i}, p, t]

is Hamilton’s principal functional, and

δ S^{*} / δ u_{i}

are the Volterra [40] functional derivatives of

S^{*}

with respect to the velocities (⁠

δ S^{*} / δ p = 0

for incompressible flow) [17,18]. This Hamilton–Jacobi equation reduces the problem of finding four separate field quantities

(u_{i}, p)

to that of finding a single scalar functional

S^{*} [u_{i}, p, t]

⁠. The solution of such equations has received little attention since the first half of the twentieth century [41–47], posing both a challenge and, at the same time, a significant opportunity for the greater community. If a complete analytical solution

S^{*}

to Eq. can be found, it will provide the generating function for a canonical transformation to a new set of fields which are equal to their initial values, thereby providing analytical expressions for the original velocity and pressure fields [17,18]. Alternatively, if an analytical solution for

S^{*}

cannot be found, but one can nevertheless establish that a complete solution does or does not exist under the usual assumptions, that will also resolve the question of existence of solutions [17,18]. Thus it is possible that this new formulation may help solve the Navier–Stokes problem once and for all.

7 Conclusion and Ideas for Future Work

The intrinsic variational structure of higher-order dynamics has enabled, for the first time in history, the general application of Hamiltonian formalism to non-Hamiltonian systems [10–18]. The present paper has given a chronological review of this relatively recent development, with attention to both the general mathematical framework and various applications that have been identified to date. This discovery provides ample opportunities for future work, and below we sketch a few such ideas:

Rekindle the analytical study of equations containing Volterra [40] functional derivatives, with the goal of either finding a complete solution $S^{*} [u_{i}, p, t]$ to Eq. (55), or otherwise establishing that a complete solution to Eq. (55) does or does not always exist under the usual assumptions. Either of these would provide a resolution to the Navier–Stokes existence problem.
Develop a method to quantify the number of critical points an action integral possesses based on the functional form of the Lagrangian. If, for example, one can establish that the action integral $S^{*}$ as given by Eq. (52) always has exactly one local minimum, or that there exist conditions under which it fails to achieve a local minimum, that can also resolve questions of existence and uniqueness of solutions to the Navier–Stokes problem.
Use a higher-order formulation as the basis for a symplectic integration scheme [26,48–60]. As noted by Becker et al. [16], it seems plausible that the intrinsic Hamiltonian structure of higher-order dynamics [13,14] would lend itself to a symplectic integrator for dissipative systems. If so, such schemes may find applications in computational fluid dynamics via the recently discovered canonical Hamiltonian formulation of the Navier–Stokes problem [17,18].
Provide a rigorous mathematical explanation for the improved efficiency of the dual-oscillator approach to computing damped resonant frequencies [15,16] over the traditional approach of Foss [37] and Traill-Nash [38]. An answer to that question may point to additional computational applications for higher-order dynamics.
Use the dual-oscillator approach to perform direct modal analysis of damped nonlinear systems [11].

Acknowledgment

This paper is based on work supported by the Lt. Col. James B. Near, Jr., USAF, ’77 Center for Climate Studies at The Citadel under the Climatological Research Studies Grant (CRSG) entitled “Toward more accurate and reliable weather predictions via progress on the Navier–Stokes problem”.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

No data, models, or code were generated or used for this paper.

References

1.

Lagrange

,

J. L.

,

1811

,

Mécanique Analytique (Analytical Mechanics)

,

Gauthier-Villars

,

Paris, France

.

2.

Hamilton

,

W. R.

,

1833

, “

On a General Method of Expressing the Paths of Light, and of the Planets, by the Coefficients of a Characteristic Function

,”

Dublin Univ. Rev. Q. Mag.

,

1

, pp.

795

–

826

.

3.

Hamilton

,

W. R.

,

1834

, “

On a General Method in Dynamics

,”

Philos. Trans. R. Soc.

,

124

, pp.

247

–

308

.

4.

Hamilton

,

W. R.

,

1835

, “

Second Essay on a General Method in Dynamics

,”

Philos. Trans. R. Soc.

,

125

, pp.

95

–

144

.

5.

Jacobi

,

C. G. J.

,

1837

, “

Zür Theorie der Variations-Rechnung und der Differential-Gleichungen (On the Theory of the Calculus of Variations and the Differential Equations)

,”

J. die reine und angew. Math. (J. Pure Appl. Math.)

,

1837

(

17

), pp.

68

–

82

.

6.

Jacobi

,

C. G. J.

,

1842–1843

, Vorlesungen über Dynamik (Lectures on Dynamics). Königsberg lectures of 1842–1843.

7.

Whittaker

,

E. T.

,

1904

,

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies

,

Cambridge University Press

,

Cambridge

.

8.

Lanczos

,

C.

,

1970

,

The Variational Principles of Mechanics

, 4th ed.,

University of Toronto Press

,

New York

.

9.

Gauss

,

C. F.

,

1829

, “

Über ein neues allgemeines grundgesetz der mechanik

,”

J. die reine Angew. Mathe.

,

4

, pp.

232

–

235

.

10.

Sanders

,

J. W.

,

2021

, “Fourth-Order Dynamics of the Damped Harmonic Oscillator,” Preprint https://arxiv.org/abs/2109.06034.

11.

Sanders

,

J. W.

,

2022

, “

A Dual-Oscillator Approach to Complex-Stiffness Damping Based on Fourth-Order Dynamics

,”

Nonlinear Dyn.

,

109

(

2

), pp.

285

–

301

.

Google Scholar

Crossref

12.

Sanders

,

J. W.

,

2023

, “

Correction to: A Dual-Oscillator Approach to Complex-Stiffness Damping Based on Fourth-Order Dynamics

,”

Nonlinear Dyn.

,

111

, pp.

2959

–

2960

.

Google Scholar

Crossref

13.

Sanders

,

J. W.

,

2023

, “Intrinsic Variational Structure of Higher-Derivative Formulations of Classical Mechanics,” Preprint. https://arxiv.org/abs/2301.10175.

14.

Sanders

,

J. W.

,

2024

, “A Canonical Theory of Non-Conservative Dynamics”. In Preparation.

15.

Sanders

,

J. W.

, and

Inman

,

D. J.

,

2023

, “

Rapid Computation of Resonant Frequencies for Non-Proportionally Damped Systems Using Dual Oscillators

,”

ASME J. Vib. Acoust.

,

145

(

3

), p.

031008

.

Google Scholar

Crossref

16.

Becker

,

E. T.

,

Inman

,

D. J.

, and

Sanders

,

J. W.

,

2024

, “

On the Improved Efficiency of Higher-Order Dynamics for Computing Resonant Frequencies

,”

Proceedings of the ASME 2024 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC-CIE)

,

Washington, DC

,

Aug. 25–28

.

Google Scholar

Crossref

17.

Sanders

,

J. W.

,

DeVoria

,

A. C.

,

Washuta

,

N. J.

,

Elamin

,

G. A.

,

Skenes

,

K. L.

, and

Berlinghieri

,

J. C.

,

2023

, “A Canonical Hamiltonian Formulation of the Navier-Stokes Problem,” Preprint. https://arxiv.org/abs/2310.07085.

18.

Sanders

,

J. W.

,

DeVoria

,

A. C.

,

Washuta

,

N. J.

,

Elamin

,

G. A.

,

Skenes

,

K. L.

, and

Berlinghieri

,

J. C.

,

2024

, “

A Canonical Hamiltonian Formulation of the Navier-Stokes Problem

,”

J. Fluid. Mech.

,

984

, p.

A27

.

Google Scholar

Crossref

19.

Rayleigh

,

J. W. S.

,

1877

,

The Theory of Sound

,

Macmillan and Co

,

London, UK

.

20.

Bateman

,

H.

,

1931

, “

On Dissipative Systems and Related Variational Principles

,”

Phys. Rev.

,

38

(

4

), pp.

815

–

819

.

Google Scholar

Crossref

21.

Caldirola

,

P.

,

1941

, “

Forze Non Conservative Nella Meccanica Quantistica

,”

Il Nuovo Cimento

,

18

(

9

), pp.

393

–

400

.

Google Scholar

Crossref

22.

Kanai

,

E.

,

1948

, “

On the Quantization of the Dissipative System

,”

Progr. Theor. Phys.

,

3

(

4

), pp.

440

–

442

.

Google Scholar

Crossref

23.

Morse

,

P. M.

, and

Feshbach

,

H.

,

1953

,

Methods of Theoretical Physics

,

McGraw-Hill

,

New York

.

24.

Feshbach

,

H.

, and

Tikochinsky

,

Y.

,

1977

, “

Quantization of the Damped Harmonic Oscillator

,”

Trans. New York Acad. Sci.

,

38

(

1

), pp.

44

–

53

.

Google Scholar

Crossref

25.

de Leon

,

M.

,

Lainz

,

M.

, and

Lopez-Gordon

,

A.

,

2022

, “

Geometric Hamilton-Jacobi Theory for Systems With External Forces

,”

J. Math. Phys.

,

63

(

2

), p.

022901

.

Google Scholar

Crossref

26.

Limebeer

,

D. J. N.

,

Ober-Blöbaum

,

S.

, and

Farshi

,

F. H.

,

2020

, “

Variational Integrators for Dissipative Systems

,”

IEEE Trans. Automat. Contr.

,

65

(

4

), pp.

1381

–

1396

.

Google Scholar

Crossref

27.

Baddour

,

N.

,

2007

,

Leading-Edge Applied Mathematical Modeling Research

,

Nova Science Publisher

,

Hauppauge, NY

, pp.

155

–

182

.

28.

Flannery

,

M. R.

,

2011

, “

The Elusive D’Alembert-Lagrange Dynamics of Nonholonomic Systems

,”

Am. J. Phys.

,

79

(

9

), pp.

932

–

944

.

Google Scholar

Crossref

29.

Pais

,

A.

, and

Uhlenbeck

,

G. E.

,

1950

, “

On Field Theories With Non-Localized Action

,”

Phys. Rev.

,

79

(

1

), pp.

145

–

165

.

Google Scholar

Crossref

30.

Bender

,

C.

, and

Mannheim

,

P. D.

,

2008

, “

No-Ghost Theorem for the Fourth-Order Derivative Pais-Uhlenbeck Oscillator Model

,”

Phys. Rev. Lett.

,

100

(

11

).

31.

Smilga

,

A. V.

,

2009

, “

Comments on the Dynamics of the Pais-Uhlenbeck Oscillator

,”

Symm. Integr. Geom. Methods Appl. (SIGMA)

,

5

(

17

), pp.

1

–

13

.

Google Scholar

Crossref

32.

Mostafazadeh

,

A.

,

2010

, “

A Hamilton Formulation of the Pais-Uhlenbeck Oscillator That Yields a Stable and Unitary Quantum System

,”

Phys. Lett. A

,

375

(

2

), pp.

93

–

98

.

Google Scholar

Crossref

33.

Baleanu

,

D.

,

Petras

,

I.

,

Asad

,

J. H.

, and

Velasco

,

M. P.

,

2012

, “

Fractional Pais-Uhlenbeck Oscillator

,”

Inter. J. Theor. Phys.

,

51

, pp.

1253

–

1258

.

Google Scholar

Crossref

34.

Chen

,

T.-J.

,

Fasiello

,

M.

,

Lim

,

E. A.

, and

Tolley

,

A. J.

,

2013

, “

Higher Derivative Theories With Constraints: Exorcising Ostrogradski’s Ghost

,”

J. Cosmology Astroparticle Phys.

,

2013

(

2

).

35.

Newton

,

G. C.

,

Gould

,

L. A.

, and

Kaiser

,

J. F.

,

1957

,

Analytic Design of Linear Feedback Controls

,

Wiley

,

New York

.

36.

Jacobs

,

O. L. R.

,

1965

, “

The Damping Ratio of an Optimal Control System

,”

IEEE Trans. Automat. Contr.

,

10

(

4

), pp.

473

–

476

.

Google Scholar

Crossref

37.

Foss

,

K. A.

,

1956

, “Coordinates Which Uncouple the Equations of Motion of Damped Linear Dynamic Systems,” Technical Report, Massachusetts Institute of Technology.

38.

Traill-Nash

,

R. W.

,

1983

, “

Modal Methods in the Dynamics of Systems With Non-Classical Damping

,”

Earth. Eng. Struct. Dyn.

,

9

(

2

), pp.

153

–

169

.

Google Scholar

Crossref

39.

Finlayson

,

B. A.

,

1972

,

The Method of Weighted Residuals and Variational Principles, With Application in Fluid Mechanics, Heat and Mass Transfer

,

Academic Press

,

Cambridge

.

40.

Volterra

,

V.

,

1930

,

Theory of Functionals and of Integral and Integro-Differential Equations

,

Blackie & Son Limited

,

Glasgow, UK

. (Republished by Dover in 1959 and Reissued in 2005).

41.

Michal

,

A. D.

,

1926

, “

Concerning Certain Solvable Equations With Functional Derivatives

,”

Proc. Natl. Acad. Sci. U. S. A.

,

12

(

2

), pp.

113

–

116

.

Google Scholar

Crossref

PubMed

42.

Jordan

,

P.

, and

Pauli

,

W.

,

1928

, “

Zur Quantenelektrodynamik ladungsfreier Felder (On the Quantum Electrodynamics of Charge-Free Fields)

,”

Z. Phys.

,

47

, pp.

151

–

173

.

Google Scholar

Crossref

43.

Levy

,

P.

,

1951

,

Problemes Concrets D’Analyse Fonctionnelle (Concrete Problems in Functional Analysis)

,

Gauthier-Villars

,

Paris, France

.

44.

Tatarskii

,

V. I.

,

1961

, “

The Primitive Functional and Its Application to the Integration of Some Equations With Functional Derivatives

,”

Russian Math. Surv.

,

16

(

4

), pp.

179

–

186

.

45.

Syavavko

,

M. S.

, and

Mel’nichak

,

P. P.

,

1974

, “

A Class of Equations With Functional Derivatives

,”

Ukrainian Math. J.

,

26

(

6

), pp.

690

–

694

.

Google Scholar

Crossref

46.

Dieudonne

,

J.

,

1981

,

History of Functional Analysis

,

Elsevier

,

North-Holland

.

47.

Koval’chik

,

I. M.

,

1993

, “

On the Method of Variation of Parameters for Equations With Functional Derivatives

,”

J. Sov. Math.

,

67

, pp.

3269

–

3273

.

Google Scholar

Crossref

48.

Kane

,

C.

,

Marsden

,

J. E.

,

Ortiz

,

M.

, and

West

,

M.

,

2000

, “

Variational Integrators and the Newmark Algorithm for Consevative and Dissipative Mechanical Systems

,”

Int. J. Num. Methods Eng.

,

49

(

10

), pp.

1295

–

1325

.

Google Scholar

Crossref

49.

Marsden

,

J.

, and

West

,

M.

,

2001

, “

Discrete Mechanics and Variational Integrators

,”

Acta Numerica

,

10

, pp.

357

–

514

.

Google Scholar

Crossref

50.

Lew

,

A.

,

Marsden

,

J. E.

,

Ortiz

,

M.

, and

West

,

M.

,

2003

, “

Asynchronous Variational Integrators

,”

Arch. Rational Mech. Anal.

,

167

, pp.

85

–

146

.

Google Scholar

Crossref

51.

Lew

,

A.

,

Marsden

,

J. E.

,

Ortiz

,

M.

, and

West

,

M.

,

2004

,

Finite Element Methods: 1970’s and Beyond

,

CIMNE

. An Overview of Variational Integrators, Barcelona, Spain, pp.

98

–

115

.

52.

Lew

,

A.

,

Marsden

,

J. E.

,

Ortiz

,

M.

, and

West

,

M.

,

2004

, “

Variational Time Integrators

,”

Int. J. Numer. Methods Eng.

,

60

(

1

), pp.

153

–

212

.

Google Scholar

Crossref

53.

Kale

,

K. G.

, and

Lew

,

A. J.

,

2007

, “

Parallel Asynchronous Variational Integrators

,”

Int. J. Numer. Methods Eng.

,

70

(

3

), pp.

291

–

321

.

Google Scholar

Crossref

54.

Ober-Blöbaum

,

S.

,

Junge

,

O.

, and

Marsden

,

J. E.

,

2011

, “

Discrete Mechanics and Optimal Control: An Analysis

,”

Control, Optimisation, Calculus Variat.

,

17

(

2

), pp.

322

–

352

.

Google Scholar

Crossref

55.

Kraus

,

M.

, and

Maj

,

O.

,

2015

, “

Variational Integrators for Nonvariational Partial Differential Equations

,”

Phys. D: Nonlinear Phenomena

,

310

, pp.

37

–

71

.

Google Scholar

Crossref

56.

Hall

,

J.

, and

Leok

,

M.

,

2015

, “

Spectral Variational Integrators

,”

Numer. Math.

,

30

(

4

), pp.

681

–

740

.

Google Scholar

Crossref

57.

Lew

,

A. J.

, and

Mata

,

P.

,

2016

, “

A Brief Introduction to Variational Integrators

.”

Nonlinear Structural Dynamics and Flexible Multibody Dynamics

, pp.

201

–

291

.

58.

Schmitt

,

J. M.

, and

Leok

,

M.

,

2018

, “

Properties of Hamiltonian Variational Integrators

,”

IMA J. Num. Anal.

,

38

, pp.

377

–

398

.

Google Scholar

Crossref

59.

Man

,

S.

,

Gao

,

Q.

, and

Zhong

,

W.

,

2020

, “

Variational Integrators in Holonomic Mechanics

,”

Mathematics

,

8

(

8

), p. 1358.

60.

Kraus

,

M.

, and

Tyranowski

,

T. M.

,

2021

, “

Variational Integrators for Stochastic Dissipative Hamiltonian Systems

,”

IMA J. Num. Anal.

,

41

(

2

), pp.

1318

–

1367

.

Google Scholar

Crossref

Extension of Hamiltonian Mechanics to Non-Conservative Systems Via Higher-Order Dynamics

Abstract

1 Introduction

2 Fourth-Order Dynamics

3 Applications to Modal Analysis and the Computation of Resonant Frequencies

4 General Hamiltonian Framework

4.1 Euler–Lagrange Equations.

4.2 Hamilton’s Equations.

4.3 Hamilton–Jacobi Theory.

5 Comparison to Gauss’s Principle

6 Hamiltonian Formulation of the Navier–Stokes Problem

7 Conclusion and Ideas for Future Work

Acknowledgment

Conflict of Interest

Data Availability Statement

References

Get Email Alerts

Cited By

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

Extension of Hamiltonian Mechanics to Non-Conservative Systems Via Higher-Order Dynamics

Abstract

1 Introduction

2 Fourth-Order Dynamics

3 Applications to Modal Analysis and the Computation of Resonant Frequencies

4 General Hamiltonian Framework

4.1 Euler–Lagrange Equations.

4.2 Hamilton’s Equations.

4.3 Hamilton–Jacobi Theory.

5 Comparison to Gauss’s Principle

6 Hamiltonian Formulation of the Navier–Stokes Problem

7 Conclusion and Ideas for Future Work

Acknowledgment

Conflict of Interest

Data Availability Statement

References

Get Email Alerts

Cited By

Related Articles

Related Proceedings Papers

Related Chapters

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

This Feature Is Available To Subscribers Only