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].
Hence, it is impossible to derive both the inertial force and a force depending on from a first-order Lagrangian without also obtaining non-physical terms involving . 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.
In other words, of all conceivable accelerations the particles might take, the actual accelerations are those for which 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, 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 Kanai–Caldirola approach [21,22] suffers from a non-physical inertial term , consistent with our earlier observation regarding the incompatibility of the inertial force with forces depending on .
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 (, ) are easily derivable from an action integral, while odd-order derivatives () 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].
3 Applications to Modal Analysis and the Computation of Resonant Frequencies
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 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 . 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).
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
4.1 Euler–Lagrange Equations.
4.2 Hamilton’s Equations.
The first set of equations, , recovers . The second, , recovers the right-hand side of Eq. (41), which is mathematically equivalent to Eq. (39). The third, , recovers the Euler–Lagrange equations (35). The fourth and last set of equations, , 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.
5 Comparison to Gauss’s Principle
The Lagrangian of the fourth-order formulation bears a superficial resemblance to Gauss’s constraint function [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 with the time average of and calling the “Gauss action” [11]. However, a simple counterexample demonstrates that 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].
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 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 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.