In science, Hamiltonian (TR:10) is []

Lagrange
In 1788, French mathematician Joseph Lagrange, in his Analytical Mechanics, introduced his ‘central function’, aka Lagrangian.

Hamilton
In Mar 1834, William Hamilton, in his “Introductory Remarks”, to his two-part On a General Method in Dynamics, opened to the following: [1]

“The theoretical development of the laws of motion of bodies is a problem of such interest and importance, that it has engaged the attention of all the most eminent mathematicians, since the invention of dynamics as a mathematical science by Galileo, and especially since the wonderful extension which was given to that science by Newton. Among the successors of those illustrious men, Lagrange has perhaps done more than any other analyst, to give extent and harmony to such deductive researches, by showing that the most varied consequences respecting the motions of systems of bodies may be derived from one radical formula; the beauty of the method so suiting the dignity of the results, as to make of his great work a kind of scientific poem. But the science of force, or of power acting by law in space and time, has undergone already another revolution, and has become already more dynamic, by having almost dismissed the conceptions of solidity and cohesion, and those other material ties, or geometrically imaginably conditions, which Lagrange so happily reasoned on, and by tending more and more to resolve all connections and actions of bodies into attractions and repulsions of points: and while the science is advancing thus in one direction by the improvement of physical views, it may advance in another direction also by the invention of mathematical methods. And the method proposed in the present essay, for the deductive study of the motions of attracting or repelling systems, will perhaps be received with indulgence, as an attempt to assist in carrying forward so high an inquiry.

In the methods commonly employed, the determination of the motion of a free point in space, under the influence of accelerating forces, depends on the integration of three equations in ordinary differentials of the second order; and the determination of the motions of a system of free points, attracting or repelling one another, depends on the integration of a system of such equations, in number threefold the number of the attracting or repelling points, unless we previously diminish by unity this latter number, by considering only relative motions. Thus, in the solar system, when we consider only the mutual attractions of the sun and the ten known planets, the determination of the motions of the latter about the former is reduced, by the usual methods, to the integration of a system of thirty ordinary differential equations of the second order, between the coordinates and the time; or, by a transformation of Lagrange, to the integration of a system of sixty ordinary differential equations of the first order, between the time and the elliptic elements: by which integrations, the thirty varying coordinates, or the sixty varying elements, are to be found as functions of the time. In the method of the present essay, this problem is reduced to the search and differentiation of a single function, which satisfies two partial differential equations of the first order and of the second degree: and every other dynamical problem, respecting the motions of any system, however numerous, of attracting or repelling points, (even if we suppose those points restricted by any conditions of connection consistent with the law of living force,) is reduced, in like manner, to the study of one central function, of which the form marks out and characterizes the properties of the moving system, and is to be determined by a pair of partial differential equations of the first order, combined with some simple considerations. The difficulty is therefore at least transferred from the integration of many equations of one class to the integration of two of another: and even if it should be thought that no practical facility is gained, yet an intellectual pleasure may result from the reduction of the most complex and, probably, of all researches respecting the forces and motions of body, to the study of one characteristic function, [N1] the unfolding of one central relation.

N1. Lagrange and, after him, Laplace and others, have employed a single function to express the different forces of a system, and so to form in an elegant manner the differential equations of its motion. By this conception, great simplicity has been given to the statement of the problem of dynamics; but the solution of that problem, or the expression of the motions themselves, and of their integrals, depends on a very different and hitherto unimagined function, as it is the purpose of this essay to show.

The present essay does not pretend to treat fully of this extensive subject, task which may require the labors of many years and many minds; but only to suggest the thought and propose the path to others. Although, therefore, the method may be used in the most varied dynamical researches, it is at present only applied to the orbits and perturbations of a system with any laws of attraction or repulsion, and with one predominant mass or center of predominant energy; and only so far, even in this one research, as appears sufficient to make the principle itself understood. It may be mentioned here, that this dynamical principle is only another form of that idea which has already been applied to optics in the ‘Theory of Systems of Rays’ [1833], and that an intention of applying it to the motion of systems of bodies was announced at the publication of that theory. And besides the idea itself, the manner of calculation also, which has been thus exemplified in the sciences of optics and dynamics, seems not confined to those two sciences, but capable of other applications; and the peculiar combination which it involves, of the principles of variations with those of partial differentials, for the determination and use of an important class of integrals, may constitute, when it shall be matured by the future labors of mathematicians, a separate branch of analysis.”

Hamilton then enters into a derivation of a force function of a system; to quote the key statement:

“The function which has been here called U may be named the ‘force-function’ of a system: it is of great utility in theoretical mechanics, into which it was introduced by Lagrange.”

Clausius
In 1850 to 1865, Rudolf Clausius, while penning the science of thermodynamics, implicitly used the framework of Lagrangian-based Hamiltonian force function mathematics and dynamics in his formulation of the first and second law of thermodynamics; in 1875, in retrospect discussion of the symbol U, Clausius outlined how his internal energy formulation derives, thereabouts, from the function of Hamilton.

The famed "inequality" of the Clausius inequality, the core of the second law of thermodynamics, was introduced at first only verbally in 1856 by German physicist Rudolf Clausius who derived the following equality based (=) equation to hold good in every cyclical process: [2]

$\int \frac{dQ}{T} = - N \,$

“In non reversible processes, those in which uncompensated transformations necessarily arise, the magnitude of N has consequently a determinable and necessarily positive value.”

where N is the sum of the equivalence-values involved in the cyclical process. Thus, in mathematical notation, although Clausius, in this year, did not use the inequality (something he began doing in 1862), this would be:

$\left\vert N \right\vert > 0 \,$

and we would have the following version of the Clausius inequality (the version defined by Clausius in 1865):

$\int \frac{dQ}{T} \le 0 \,$

In any event, this seemingly innocuous mention that "the magnitude of N has a positive value" introduced the notion of irreversibility into Newtonian mechanics and the arrow of time into cosmology, thus bring about the thermodynamic revolution and thus the quantum revolution, the latter being a precipitate of the former, and likely several revolutions yet to come.

Gibbs
In 1873, American engineer Willard Gibbs, building on Clausius, began to apply Clausius' inequality methods to the graphical thermodynamic analysis of the behavior of fluids and began to formulate new thermodynamic "force functions" by taking the slopes of various thermodynamic curves, similar to how Isaac Newton calculated the "force" acting on a moving body as the product of the mass of the body by the time derivative (or slope) of velocity change:

$\mathbf{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \mathbf{v})$

In 1876, Gibbs then parlayed this graphical basis into what he called the "general theory of thermodynamic equilibrium" and in doing so formulated five new versions of partial derivative inequalities, in which certain variables are held constant and thus extended the Clausius inequality to more general varieties of systems, such as chemical systems, open systems, electrical systems, etc. The five central Gibbs inequalities are as follows:

 Inequality(1876 notation) Inequality (modern notation) Description $(\delta \eta)_{\epsilon} \leqq 0 \,$ $dS_U \le 0 \,$ For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations δ of the state of the system which do not alter its energy ε, the variation of its entropy η shall either vanish or be negative. $(\delta \epsilon)_{\eta} \geqq 0 \,$ $dU_S \ge 0 \,$ For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations δ of the state of the system which do not alter its entropy η, the variation of its energy ε shall either vanish or be positive. $(\delta \psi)_{t} \geqq 0 \,$ $dF_T \ge 0 \,$where($F = U - TS \,$) Where by letting psi ψ represent the available energy [free energy] of the system, expressed by the formula ψ = ε – tη, it is found that for the equilibrium of any isolated system it is necessary and sufficient that in all possible variations δ of the state of the system which do not alter its temperature t, the variation of its available energy ψ shall either vanish or be positive. The difference in the two values of available energy ψ for two different states of the system which have the same temperature represents the work which would have to be expended in bringing the system from one state to another by a reversible process and without change of temperature. $\delta \epsilon \geqq 0 \,$ $dU \ge 0 \,$ The reduced form of the second of equilibrium criterion for systems incapable of thermal changes, whereby we may regard the entropy as having the constant value of zero. $\delta \psi \geqq 0 \,$ $dF \ge 0 \,$ The reduced form of the third equilibrium criterion for systems incapable of thermal changes, whereby we may regard the entropy as having the constant value of zero.

Gibbs goes on to state that the "suffix letter" (subscript) in each case indicates that the quantity which it represents remains constant. Lastly, and most importantly, Gibbs outlines the basics of the so-called thermodynamic "driving forces" of chemical processes:

“The quantities ‘–ε’ (negative energy) and ‘–ψ’ (negative available energy), related to a system without sensible motion, may be regarded as a kind of force-function for the system.”

The first of these ‘–ε’, Gibbs says, is the force-function for constant entropy, i.e. when the states of the systems under consideration have constant entropy. The second of these ‘–ψ’, he says, is the force-function for constant temperature, i.e. when the states of the system under consideration have constant temperature.

Lewis
In 1923, American physical chemist Gilbert Lewis, starting with the foundation of Gibbs inequalities, which he says are applicable to "chemical processes which are in some way harnessed for the production of useful work", introduced a new inequality as follows:

 Inequality (1923 notation) Inequality (modern notation) Description where($F = E + PV - TS \,$) where($G = U + PV - TS \,$) Whereby "no actual isothermal process is possible unless it meets this condition", which applies to "the far more common case of a reaction which runs freely, like the combustion of a fuel, or the action of an acid upon a metal; or, in other words, a system which is subject to no external forces, except a constant pressure exerted by the environment."

In modern format, assuming that from hence forth we use the term "process" to mean all natural isothermal-isobaric processes that occur on the surface of the earth, such as human chemical reactions occurring between people, what Lewis states here is that:

“No actual natural process is possible unless it meets this inequality ΔG < 0 criterion.”

This is huge statement. Getting directly to the point: any hypothetical relationships will only be possible if the free energy change for the process is less than zero. Next, Lewis goes on to state, similar to Gibbs' description of the two force-functions, that:

“We may think of the quantity ‘– ΔF ’ as the driving force of a reaction; where, in a thermodynamic sense, a system is stable when no process can occur with a diminution in free energy.”

This, again, is a huge statement. Lewis introduce here not only the notion that free energy change is the driving force of earth-bound natural processes, as was described in 1718 by Isaac Newton in terms of respective values of the force of chemical affinity A as being the driving force behind reactions:

 Newton notation (1718) Lewis notation (1923) Partington notation (1924) Modern notation (1933) $A > 0 \,$ $A = - \Delta F \,$ $A = -\Delta Z \,$ $A = - \Delta G \,$
 The position of G1 is such that, in the words of Gilbert Lewis (1923), "no further process can occur with a diminution in free energy", and is thus representative of a state of maximal stability; whereas the position of G1 could decrease further in free energy, to the position of state one, and is thus not maximally stable.

but also introduces the notion of "stability", i.e. that when the variation of free energy for the process or reaction reaches its lowest value, or, as Lewis puts it, when "no further process can occur with a diminution in free energy", the system will be said to be stable, as would be indicated graphically by the lowest point of free energy on the reaction coordinate, as depicted adjacent.

Thims
In 2014, Libb Thims, in thread dialogue with Inderjit Singh, in respect to an assimilation of the above, stated the following:

“Re (#4): “where is the director?” [Kierkegaard], this brings to mind the so-called purpose question—which you commented somewhere, in some thread, was your point or apex of debate collision with me—and what Bruce Lindsay (1983) has to say about this in regards to physics, the teleology debate issue, and sense of purpose. The key statement being the following:

‘A rational individual is said to arrange his actions so as to be sure of achieving his fundamental desires, whether it be to accumulate wealth or gain power over his fellow men. In particular the aim here is almost always to try to attain the given desired end at minimum cost in human effort. This strongly suggests a heuristic connection with the minimum principles of physics.’

Prior to this, Lindsay cites the following so-called minimum principles: Newtonian (not exactly a minimum principle) → LagrangianHamiltonian, and how they have merged into each other, to increasingly quantify the dynamics of the motions of systems comprised of moving particles (e.g. social systems). What he doesn’t state, however, is that the Hamiltonian, via Clausius (1865) and Gibbs (1876), got transformed into the so-called ‘Gibbsian’ minimum principle, according to which Lindsay’s statement, in 2014 retrospect, is something along the lines of:

‘A rational individual is said to arrange his actions so as to be sure of achieving his fundamental desires, whether it be to accumulate wealth or gain power … the aim here is almost always to try to attain the given desired end at minimum cost in human effort. This strongly suggests a heuristic connection with the [Gibbsian] minimum principles of the [physicochemical sciences].’

And hence, if as a rational individual, as Lindsay says, you desire to understand your desires, you are led naturally enough into a study of the so-called ‘human free energy’ theorists, so as to better understand the ‘whys’ and ‘hows’ associate with ‘minimum principles’ as they related to the arrangements your actions. This is not to say, to note, that atoms, molecules, and concordantly humans have "purpose", so to say, in the sense of ‘the purpose of hydrogen is to bond with oxygen to form water’, etc., as this is a type of non-sensical statement, whereby so it must also be so with humans (or human molecules), or are governed by "teleological" principles, as the covert religion-siding scientists like to argue, as this is an model that only works in an Aristotelian universe, but rather that there seems to be some well-honed truth, in need of deeper clarification, in what Lindsay states above.”

Quotes
The following are related quotes:

“I have also a great respect for the elder of those celebrated acrobats, Virial and Ergal, the Bounding Brothers of Bonn …. But it is rare sport to see those learned Germans contending for the priority of the discovery that the 2nd law of θΔcs is the Hamiltonsche Princip, when all the time they assume that the temperature of a body is but another name for the vis viva of one of its molecules, a thing which was suggested by the labors of Gay-Lussac, Dulong, etc., but the first deduced from the dynamical statistical considerations by dp/dt [myself]. The Hamiltonsche Princip, the while, soars along in a region unvexed by statistical considerations, while the German Icari (Ѻ) flap their waxen wings in nephelococcygia [cloud-cuckoo-land] (Ѻ) amid those cloudy forms which the ignorance and finitude of human science have invested with incommunicable attributes of the invisible Queen of Heaven.”
James Maxwell (1873), “Letter to Peter Tait”, Dec (Number 483) [1]