Abstract
I show how Sir William Rowan Hamilton’s philosophical commitments led him to a causal interpretation of classical mechanics. I argue that Hamilton’s metaphysics of causation was injected into his dynamics by way of a causal interpretation of force. I then detail how forces are indispensable to both Hamilton’s formulation of classical mechanics and what we now call Hamiltonian mechanics (i.e., the modern formulation). On this point, my efforts primarily consist of showing that the contemporary orthodox interpretation of potential energy is the interpretation found in Hamilton’s work. Hamilton called the potential energy function the “force-function” because he believed that it represents forces at work in the world. Various non-historical arguments for this orthodox interpretation of potential energy are provided, and matters are concluded by showing that in classical Hamiltonian mechanics, facts about the potential energies of systems are grounded in facts about forces. Thus, if one can tolerate the view that forces are causes of motion, then Hamilton provides one with a road map for transporting causation into one of the most mathematically sophisticated formulations of classical mechanics, viz., Hamiltonian mechanics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing is not applicable to this article because I have not evaluated nor used any datasets.
Notes
(Kant 1998, 314–315 B253-254).
(Whewell 1858, 182–183).
From W.R. Hamilton to Aubrey De Vere, Observatory, May 7th, 1832. (Graves 1882, 554).
(Papineau 2013, 127). See (Earman 1976, 6); (Earman 2011, 494); (Field 2003, 435); (Ismael 2016, 134; cf. 113, 117, and 136); (Kutach 2013, 266, 272-273, 282 for “culpable causation”, the causation of interest to metaphysicians); (Loewer 2007); (Russell 1912-1913); (Schaffer 2008, 92); (Sider 2011, 15-17); (Spohn 2006, for whom all causal laws ultimately reduce to mere regularities ibid., 116); (van Fraassen 1989, 282). See also (Norton 2007a; 2007b, 2021). Farr and Reutlinger (2013, 216) state that “[m]any philosophers of physics today support…[the] claim that causal relations do not belong to the ontology suggested by fundamental physics.”.
Some resist the majority opinion. See the powers, capacities, and dispositions group: (Bird 2007, 168; Ellis 2002, 23-24, 159; Mumford 2004, 150, 188). Cf. (Ney 2009, 759-761); (Kistler 2013); and (Bartels 1996) inter alios.
Some members of the group that promotes the role of mechanism in natural philosophy hesitate to allow causation into fundamental physics: (Glennan 1996, 61, 64; 2010, 367); (Glennan 2017). On the other hand, some seem to encourage its insertion into physics: (Machamer et. al. 2000); (Andersen 2011).
Frisch (2005, 2007, 2009, 2014) gives one some reason to put causation in fundamental physics, but he admits his position is consistent with an instrumentalist attitude about causation there. Frisch’s position also seems compatible with a strictly functionalist theory of causation as in (Woodward 2014).
I believe Hamilton provides us with key insights that recommend how to get causation into the ontology of modern Hamiltonian mechanics in a manner that extends beyond mere instrumentalist and/or functional viewpoints about microphysical causation, if (as Hamilton believed) forces can be properly understood as causes of motions. While I do not spend time arguing for the emphasized antecedent, it is a significant enough task to show how forces are essential to the ontology of Hamiltonian mechanics, for Hamiltonian mechanics is usually regarded as an energy-based theory in which forces play no essential role (q.v., the quotation of North at note 21).
(Redhead 1990, 146).
(Ismael 2016, 114).
(Ismael 2016, 134–135).
(Norton 2007a, 15, 19–20).
Stan remarked:
“Thus an inflection point occurred circa 1750 as the science of motion exited its adolescence. Specifically, we see a deep shift in the form and status of the laws of motion. The shift is where early modern mechanics turns into classical mechanics as we know it.” (2022, 405).
(Stan 2022, 388).
(Stan 2022, 402–403).
See (Stan 2022, 403) for both quotations.
(Stan 2022, 403). Stan is best interpreted as discussing how the laws were regarded during the late Enlightenment era and onward. How else could the past (“counted”) epistemic opacity in question be caused by the failure of the laws to “mention bodies or their causal actions”? Stan is clearly committed to the hypothesis that the dynamical laws were no longer causally interpreted under the reconceptualization of them during the late-Enlightenment era.
(Stan 2022, 402).
One can show why some prominent eighteenth century thinkers were not at all “oblivious”, but were instead rationally justified in their continued promulgation of causal interpretations of physical laws even if those laws are restricted to Stan’s [1] (Euler’s statement of the second law of motion) and [3] (Lagrange’s version of the principle of least action) in (Stan 2022). Indeed, in this essay, I provide reasons for causally interpreting the principle of least action (q.v., n. 122).
See, e.g., (Weaver 2019) for the positive case. I’m not alone. James Woodward said,
“…to guard against misunderstanding, let me say unequivocally that it is not part of my argument that causal notions play no role in or are entirely absent from fundamental physics. I see no reason to deny, for example, that forces cause accelerations.” (Woodward 2007, 68 emphasis in the original).
See also (Loew 2017) who responds to the objections from locality and time-reversal.
As North wrote,
“Newtonian mechanics ‘describes the world in terms of forces and accelerations (as related by the second law)’ (Taylor 2005, 521), where ‘force is something primitive and irreducible’ (Lanczos, 1970, 27). Lagrangian and Hamiltonian mechanics describe systems in terms of energy, with force being ‘a secondary quantity’ derivable from the energy (Lanczos, 1970, 27). According to Newtonian mechanics, the world is fundamentally made up of particles that move around in response to various forces between them. According to Lagrangian and Hamiltonian mechanics, particles move around and interact as a result of their energies. Although energy and force functions are inter-derivable in ways that physics books will show (albeit under certain contestable assumptions…), these are nonetheless prima facie different pictures of the world, built up out of different fundamental quantities, with correspondingly different explanations of the phenomena.” (North 2022, 29).
Some Kant scholars may find some of my claims about Kant contentious. I have therefore devoted more time and space to Sect. 2.1.1 than any other historical section.
As Hamilton’s biographer and contemporary, Robert Perceval Graves’s (1810–1893) summarized:
Hankins (1980, 460) agrees: Hamilton
“built his metaphysics of mathematics on a direct reading of Kant’s works. An appreciation of Hamilton’s arguments about the foundations of algebra therefore requires a plunge into Kant’s critical philosophy.”
When Hamilton spoke of “Algebra” (he liked to capitalize the term) he was referring to both algebra (as we know it) and calculus or analysis (Halberstam and Ingram 1967, xv). For more on Hamilton’s algebra including work on quaternions, see (Crowe 1985, 17–46, 117–124); (Fisch 1999); (Hamilton 1837; 1853 or MPH3, 117–155); (Hankins 1980, 268–275); (Mathews 1978); (Merzbach and Boyer 2011, 510–512); (Øhstrøm 1985); and (Winterbourne 1982).
Here’s some evidence for the claim in the main text. First, Hamilton quotes Kant, sometimes in the original German (Hamilton MPH3, 118). Hamilton appeals to Kant again at (Hamilton MPH3, 125, the lengthy footnote that extends to p. 126). Second, we learn from Hamilton’s personal correspondence and memoranda that Hamilton both read and studied Kant and secondary literature about Kant (see Hankins 1980, 268–275). Third, Hamilton explicitly acknowledged his indebtedness to Kant in the development of the algebra discussed at (Hamilton 1837) in the 1853 preface to the Lectures on Quaternions (1853), a work that is one of the most important contributions to the history of mathematical physics being in part a seed for the creation of modern vector calculus (Hendry 1984, 63). Fourth, Hamilton’s contemporaries understood him to be developing Kantian motifs in some of his scholarly work (Graves 1885, 141–143). Fifth, ideological dependence can be demonstrated in other ways besides pointing to direct quotations. If we group together reasons (1)-(4) and then see relevant ideological similarities and motifs throughout portions of the work of two individuals, the best explanation is that the temporally later thinker drew upon the former (especially if the temporally later thinker admits to doing so).
This idea appears in Hamilton’s 1853 exposition of his earlier views (Hamilton MPH3, 117).
(Kant 1998, 304; B232). I have removed the emphasis of the first quotation in this sentence.
(Kant 1992, 37). For Kant, this is a principle of “metaphysical cognition” (ibid.). Its flavor reminds one of Isaac Newton’s (1643–1727) third law of motion (i.e., the action-reaction principle). In the Prolegomena to Any Future Metaphysics, Kant presented three a priori principles about the cognition of appearances in experience. The last of these invokes an action-reaction principle (reminiscent of Newton’s third law of motion and the reciprocity mentioned in the quotation in the main text) understood as a type of epistemic guide or tool (Kant 1933, 66). Of course, all of this resembles ideas found within Kant’s third analogy of experience in the Critique of Pure Reason.
I quote here the translation of Kant’s Prolegomena to Any Future Metaphysics (A91-92/B123-124) provided by Michael Friedman in (Friedman, Causal Laws 1992, 161–162) emphasis in the original. For standard English translations of Kant’s Prolegomena, see (Kant 1933) and (Kant 2004).
Compare: “the concept of cause implies a rule, according to which one state follows another necessarily” (Kant 1933, 76).
There is a question about whether Kant’s laws of mechanics are identical to Newton’s laws of motion. For the view that they are identical, see (Friedman 2013). For the view that they are distinct, but perhaps similar, see (Stan 2009, 43–44); (Watkins 2005). I should add that Watkins’s views have undergone an evolution. Compare (ibid.) to (Watkins 2019).
My reading is within the confines of well-regarded Kant scholarship. See (Guyer 1987); (Friedman Causal Laws 1992); (Friedman 2013, 118, 265); (Watkins 2019, 90). According to Friedman (2014), Kant also maintained that Newton’s universal law of gravitation was a kind of causal law endowed with a kind of necessity.
(Hamilton, 1837, 7).
(Melnick 2006, 227).
In one place, Hamilton refers to (Lagrange 1788) as “a kind of scientific poem” (as quoted by Truesdell 1968, 86).
(Lagrange 1997, 11) emphasis in the original.
Save some philosophers, virtually everyone believes that for Newton forces are causes of motion (q.v., n. 46 if you require source citations). Both (Ismael 2016) and (Norton 2007a) were wrong to see in Newton an abandonment of causation-laden laws of physics. Ronald S. Calinger, a foremost Euler scholar, has recently said “[t]he concept of force was crucial to Euler’s mechanics, and he treated it as an external entity to a body causing change in motion.” (Calinger 2016, 126).
(Archibald 2003, 198).
(Truesdell 1968, 133). Truesdell also listed other problems with Lagrange’s equations of motion, problems that result from “obscuring the forces” (ibid.).
(Panza 2003, 151–152). For a sample of Lagrange’s high view of Newtonian forces within the Analytical Mechanics, q.v., n. 37 and n. 40.
(Lagrange 1997, 169) emphasis mine.
Q.v., n. 37.
(Whewell 1858, 205) emphasis in the original.
The paper was received by Philosophical Transactions of the Royal Society on April 1st, 1834 and read April 10th, 1834. However, at the end of Hamilton’s introductory remarks, Hamilton includes a date of March, 1834, and because his letter to Whewell references (Hamilton 1834 although it was not yet published) we can infer that Hamilton’s letter (dated March 31st, 1834) was authored after Hamilton had completed but not yet published (Hamilton 1834).
From W.R. Hamilton to Dr. Whewell, Observatory, Dublin, March 31, 1834 in (Graves 1885, 82).
As quoted in (Graves 1885, 83).
(Whewell 1967, 573–594).
(Whewell 1967, 574) emphasis in the original.
(Whewell 1967, 574) emphasis in the original.
(Whewell 1967, 575) emphasis in the original.
The idea is all over the essay, but see (ibid., 581) for just one (more) example among many.
Cf. the conclusions in (Hankins 1980, 178).
See (Leibniz 1989, 223).
Hamilton stated, “[i]n seeking for absolute objective reality I can find no rest but in God…” From Hamilton to H.F.C. Logan, June 27, 1834 in (Graves 1885, 87).
(Hankins 1980, 179), although Hankins does not relate Hamilton’s work to that of Leibniz.
Leibniz’s Monadology maintained that corporeal substances are phenomenal depending for their existence upon quasi-mental simple entities called monads. Leibniz said, “simple things alone are true things, the rest are only beings through aggregation, and therefore phenomena, and, as Democritus used to say, exist by convention not in reality.” (Leibniz’s letter to Burchard de Volder (1643–1709) (June 20th of 1703, sent a second time). As quoted by (Garber 2009, 368)). And elsewhere Leibniz wrote,
“…if there are only monads with their perceptions, primary matter will be nothing other than the passive power of the monads, and an entelechy will be their active power…” (Leibniz and Des Bosses 2007, 274–275 emphasis mine).
The equivalent of monads in Hamilton’s system are fundamental simple things called energies or powers (see the main text). Like Leibniz’s monads, Hamilton’s simple powers/energies give rise to the external world. Hankins calls Hamilton’s view an “idealized version of atomism” (Hankins 1977, 182).
From Hamilton to H.F.C. Logan, June 27, 1834 in (Graves 1885, 87).
All discussion of modern physics will use the SI unit system.
Generalized momenta can also be stated in terms of the Lagrangian \(\mathscr{L}\) (the difference between kinetic and potential energy) and generalized velocity (Penrose 2005, 476, I’m citing in this case because some suggest otherwise):
$$\mathbf{p}=\frac{\partial \mathscr{L}}{\partial \dot{q}}$$In technical discussions of Hamiltonian mechanics in the work of physicists, mathematicians, and some philosophers, one will see: (a) higher-dimensional phase spaces the points of which represent possible states of the system modeled (because they encode information about the positions and momenta of constituents of the system), (b) phase space orbits or flows tracing out (c) curves in phase space understood as representations of possible evolutions of the modeled system given by solutions to Hamilton’s equations, (d) Liouville’s theorem, (e) measures, (f) Poisson brackets, etc. For all of that, see (Dürr and Teufel 2009, 12–26); (Mann 2018, 167–201); (Torres del Castillo 2018, 103–228) and pair it with (Healey 2007, 248–251). I skip that stuff here in the interest of brevity. My central argument will remain unaffected by details about cotangent bundles or phase spaces that are symplectic manifolds, measure preserving flows, and symplectic geometry. What gives you the curves that represent evolutions in the phase space are solutions to Hamilton’s equations. In addition, the “dynamical evolution of a system can…be geometrically encapsulated in a single scalar function (namely the Hamiltonian)” (Penrose 2005, 484). So, the important questions are: How should one interpret Hamilton’s equations and their solutions? How should one interpret the Hamiltonian?
Later, I will make much of Galilean invariance in classical mechanics. Some might therefore object to precluding a discussion of the geometry of Hamiltonian mechanics because both canonical transformations and canonical invariants (or canonical form-invariants) are important to Hamiltonian and Hamilton–Jacobi mechanics. In order to appreciate canonical invariants and transformations (especially those that have to do with time), one must study symplectomorphisms and that study will require that one give attention to cotangent bundles and symplectic geometry. Canonical invariants and transformations have to do with tracking systems in a higher-dimensional phase space. But we need not worry about any of that. Hamilton’s equations of motion are canonical form-invariant, and (again) their solutions provide one with the motions of systems modeled by the Hamiltonian apparatus (points orbiting in the higher-dimensional phase space). Once again, the question is, how should we interpret Hamilton’s equations and their solutions?
According to (Lützen 1995, 15–16), higher dimensional spaces were referenced by Carl Friedrich Gauss (1777–1855) in 1816, by his student August Ritter (1826–1908) in 1853 (with some reservation), and by Jean-Gaston Darboux (1842–1917) in 1869. However, systematic treatments of mechanics with the equipment of higher dimensional geometry cannot be found until after 1870 (Lützen 1995, 18).
Hence, the logical ordering of the equations in the main text. We can also affirm:
$$-\frac{\partial \mathscr{L}}{\partial t}=\frac{\partial H}{\partial t}$$My gloss on the formalism here follows (Hamilton 1834; 1835) only in part. It is more in line with the notational style of contemporary historians of mathematics, Hamilton scholars, and historians of physics (e.g., Goldstine 1980, 176–189; Lützen 1995; Nakane and Fraser 2002 etc. on which my exposition leans). My discussion will rely upon my own reading of Hamilton, but it owes much to the sources just cited along with (Fraser 2003); (Graves 1842); (Cayley 1890); and (Hankins 1980, 181–198).
Hamilton also calls this the “equation of the characteristic function” (Hamilton 1834, 252).
(Hankins 1980, 186).
(Hankins 1980, 183). This was a common assumption at the time.
(Dürr and Teufel 2009, 16).
(Hamilton 1834, 250).
(Hamilton 1834, 251–252). Hamilton had already written about the characteristic function in (Hamilton 1828) a work on optics published when Hamilton was only 21 years of age. He said there, “[i]n every optical system, the action may be considered as a characteristic function, from the form of which function may be deduced all the other properties of the system” (Hamilton 1828, 79 emphasis in the original). According to Sir Edmund Whittaker (1873–1956), Hamilton discovered the function at the age of 16 (Whittaker 1954, 82).
(Hamilton 1834, 250).
(Hamilton 1834, 249) emphasis in the original. It was very common during Hamilton’s time to call the potential function U, the “force-function” (Nakane and Fraser 2002, 163). One primary example cited by Nakane and Fraser is Jacobi who would in some ways improve Hamilton’s work. Interestingly, Euler identified what Lagrange thought of as the potential with force effort (Euler 1753, 173–175); (Boissonnade and Vagliente 1997, xxxvii).
(Hankins 1980, 184).
Following (Nakane and Fraser 2002, 163). Hamilton’s statement is restricted at (Hamilton 1834, 273) to a conservative system of but two point-masses. At (Hamilton 1835a), the formula for the variation of the force-function is suitably generalized. Moreover, at (ibid.) it is stated (as a well-known fact) that the equations of motion follow from considering the variation of the force-function.
Just to be clear, Hamilton did not use the Lagrangian. He used its mathematical equivalent.
See (Feynman, Leighton and Sands 2010, 19–8); (Taylor 2005, 239). There are many titles and names of principles thrown about in the literature. For example, Richard Feynman (1918–1988) called the specification of the action integral \(S=\underset{0}{\overset{t}{\int }}\mathscr{L} dt\) “the principle of least action” at (Feynman, Leighton and Sands 2010, 19–8). There Feynman is concerned with the relativistic limit, but \(\mathscr{L}\) becomes the difference between kinetic and potential energy in the classical and non-relativistic limit (compare Feynman, Leighton and Sands 2010, 19–3).
(Nakane and Fraser 2002, 163).
The Oxford Dictionary of Physics defines energy as “[a] measure of a system’s ability to do work” (Rennie 2015, 180).
See, e.g., (Taylor 2005, 111) who says that “[w]e define \(U\left(\mathbf{r}\right)\)” in terms of, \(-W({\mathbf{r}}_{0}\to \mathbf{r})\). But,
$$-W({\mathbf{r}}_{0}\to \mathbf{r})\equiv -\underset{{\mathbf{r}}_{0}}{\overset{\mathbf{r}}{\int }}\mathbf{F}({\mathbf{r}}^{\mathbf{^{\prime}}})\cdot d\mathbf{r}\mathbf{^{\prime}}$$where \(W\) is the work function. That we can define \(U\left(\mathbf{r}\right)\) in terms of work is enough for my purposes because, again, work has to do with force. Hecht remarked,
“…only changes in \(PE\) are defined, and these are defined as the work done on a system by conservative forces. Such work is measurable as it is being done. However, once work is done it no longer exists and is no longer measurable; if \(\Delta PE\) is only defined by work done (e.g., mgh, or \(1/2k{x}^{2}\), or \(1/2C{V}^{2}\)) it cannot be measured in stasis while it supposedly exists.” (Hecht 2019a, 500 emphasis in the original).
Some maintain that potential energy and work are identical. Coopersmith stated, “[p]otential energy and work were eventually seen to be one and the same (an integration of force over distance)” (Coopersmith 2015, 115).
It is common to understand the work quantity in such a way that it is deemed more fundamental than potential energy. In his classic graduate level text on classical mechanics, Cornelius Lanczos (1893–1974) stated,
“[t]he really fundamental quantity of analytical mechanics is not the potential energy but the work function…In all cases where we mention the potential energy, it is tacitly assumed that the work function has the special form \(W=W({q}_{1},{q}_{2},\dots ,{q}_{n})\), together with the connection \(U=-W\)” (Lanczos 1970, 34)
I have rephrased Lanczos’s equations by using my own notation. The first equation in the quotation is inserted here in place of Lanczos’s reference to Eq. (17.6). I argue that forces are more fundamental than potential energy in Sect. 4.3 below. Lanczos argues that work is more fundamental than force (Lanczos 1970, 27). I disagree. Work is technically defined in terms of force.
See (Archibald 2003, 198) and (Dunnington 1955, 160).
(Archibald 2003, 198).
(Green 1871, p. 9) emphasis mine.
(Gray 2008, 71).
If the reader is tempted to understand potential energy in such a way that it is basic, I ask that that reader digest the arguments of Sect. 4.3 as well as the analogical consideration in Sect. 5. I am here only trying to show that the potential function represents forces. This is a claim about the mathematical object and not a claim about potential energies in the world.
(Coopersmith 2015, 337).
(Taylor 2005, 531–532).
(ibid., 532).
(North 2022, 29).
I will assume that generalized force \({Q}_{i}\) has the dimension of a force and not that of the moment of a force. On this distinction, see (Langhaar 1962, 17). For more on generalized forces in general, see (Fitzpatrick 2011); (Lanczos 1970, 27-31); (Langhaar 1962, 14-23); (Peacock and Hadjiconstantinou 2007); (Sommerfeld vol. 1 1964, 187-189); (Stewart 2016, 16–21). In places, my discussion follows these sources.
Following (Lanczos 1970, 28).
(Lanczos 1970, 28).
We could use \({\delta }_{n}\) instead of \({d}_{n}\) to highlight the fact that the involved force impression can be virtual (as in Langhaar 1962, 16).
(Marion and Thornton 1988, 72).
What is said in the main text holds true for both internal and external potential energy. None “of these potential energies is independently measurable” (Hecht 2016, 10); (Ryder 2007, 58). What I say here also holds true for static electric field energy (Hecht 2019b, 3) (q.v., Sect. 5).
Albert Einstein (1879–1955) said,
“[b]ut if every gram of material contains this tremendous energy, why did it go so long unnoticed? The answer is simple enough: so long as none of the energy is given off externally, it cannot be observed. It is as though a man who is fabulously rich should never spend or give away a cent; no one could tell how rich he was.” (Einstein 1954, 340).
Sometimes it is interpreted as the gravitational force field.
Absolute potential energy does not fail to make sense because it is not measurable. There is no verificationism afoot here.
It is therefore no surprise that the SI unit of measurement for both weight and force is one and the same, viz., the newton.
All of this remains as I am presenting it in both Newtonian and Hamiltonian mechanics. See (Meyer and Offin 2017, 61–62) for a discussion of gravitation and classical non-relativistic Hamiltonian mechanics.
I should add that I believe that the arbitrariness is to blame for the negativity. The arbitrariness is also the reason why negative gravitational potential energies should not confound the metaphysician of physics (Mann 2018, 17). If energies are but measures or convenient ways of representing what’s really happening with forces, then the signs of the convenient devices need not bother one. It’s what’s fundamental (or, in this case at least, what’s more fundamental) that matters.
The reference to Gaussian surfaces wasn’t just for the purposes of being humorous. Gaussian surfaces are arbitrarily specified closed surfaces introduced by the physicist to help with calculations in electrodynamics.
(Sullivan 1934, 247–248).
As Robert Mills (of Yang-Mills theory) said, “the idea of potential energy is not truly fundamental and that it breaks down in the relativistic world…” (Mills 1994, 152). Cf. (Lanczos 1970, 34) already quoted; and (Thomson 1888, 15). Coopersmith (2015, 339) argues that because kinetic energy has the same form for every system and potential energy does not, the former is more fundamental than the latter (citing (Maxwell 1871) in support).
As Jennifer Coopersmith remarked in correspondence:
“Even outside of Einstein's Relativity theories, it is not always true that the potential energy is invariant. A requirement for Galilean invariance is that the potential energy depends only on 'relative' coordinates (e.g. the difference between two positions) and not on 'absolute' coordinates (the position of the old oak tree at the corner of the street).” (11/08/2020).
See (Diaz et. al. 2009, 271–272) who remarked,
“…we illustrated that after a change of reference frame, the work done by each force also changes (even if the transformation is Galilean). Consequently, the corresponding potential energies change when they exist.” (ibid., 272).
These authors give an argument for their conclusions at (ibid., 271–272).
Horzela et. al. (1991) stated that,
“The explicit expressions of the potential energy as functions of the position \(\overrightarrow{x}(t)\) all have noncovariant meaning and therefore may be valid only in one inertial reference frame.” (ibid., 12, their argument for this starts on page 11).
With respect to force and acceleration, Tefft and Tefft (2007) state,
“…quantities, such as acceleration and force, are invariant or, to use Newton’s term, ‘absolute’ between inertial reference frames. Such quantities have the same values in any inertial frame.” (ibid., 220).
Cf. the discussion in (Earman 2004, 1230).
(Nozick 2001, 81). Earman (2004) likes the type of inference invited by Nozick in the context of classical mechanics, but believes it suffers important setbacks when dealing with general relativity.
“The implementation of part of Nozick’s formula objectivity = invariance by means of the constrained Hamiltonian formalism goes swimmingly: in case after case it yields intuitively satisfying results. But the application to Einstein’s GTR yields some surprising and seemingly unpalatable consequences.” (ibid., 1234).
As Coopersmith stated, “energy is not an invariant quantity” (2015, 342). The Hamiltonian is not invariant under boost operations (Butterfield 2007, 6). Butterfield said, “the Hamiltonian of a free particle is just its kinetic energy, which can be made zero by transforming to the particle’s rest frame; i.e. it is not invariant under boosts.” (ibid.) This is true even in quantum mechanics (Lombardi et. al. 2010, 99). In classical mechanics, the Lagrangian is invariant under rotations and translations, but not under boosts (Finkelstein 1973, 106–107). Be careful. Landau and Lifshitz’s famous text on mechanics says that “the Lagrangian is [Galilean] invariant”, but their argument only demonstrates that Lagrange’s equations of motion are covariant under Galilean transformations (Landau and Lifshitz 1976, 7).
Interestingly, Coopersmith (2015, 239) says that the “potential energy function” just “is the Hamiltonian…and it is the function that determines the entire dynamics of the system.” If Coopersmith is right, it would not be surprising that potential energy fails to be a Galilean invariant quantity (cf., ibid., 313).
I must add here that Green’s function is not guaranteed to be invariant either (Appel 2007, 165).
(Schroeren 2020, 52) emphasis in the original. Schroeren goes on to correctly note how “every property linked to a symmetry in the relevant sense is invariant under that symmetry” (ibid.). So, some type of thesis regarding the invariance of energy may be saved. However, that type of invariance cannot be indicative of that which is one of “the most objective thing[s] there actually is” for, again, total mechanical energy is conserved but not objective at least because of the arbitrariness that sneaks into potential energy (q.v., Sect. 4.2). I do not know if Schroeren would agree with my conclusions.
(Nozick 2001, 76).
Someone may ask: But what about the action that is minimized (or taken to equal an extremum) in Lagrangian and Hamiltonian mechanics? That quantity is invariant under Galilean transformations, and it is typically understood in terms of the Lagrangian multiplied by a small change in time (usually flanked by the integration symbol (the “action integral”)). It is difficult to discern the metaphysical nature of that quantity, but it is far from potential energy alone. My view of the relationship between force and action belonged to both Euler and Lagrange. The action and action integral track the evolution of the system by indirectly representing its dynamical force interactions. That is why when you shift from one system with an operating force Fn to a system with different operating forces, the form of the action changes. Euler recognized that if the system involves accelerations, the action is a minimum (or extremum), given that the system being modeled is acted upon by forces (Euler 1744, 311–312). The correct act of integration yields the system’s trajectory only under the assumption that a force has acted and that the form of the integral is appropriately specified in light of that force-action. This same point was made by Lagrange. See (Lagrange 1867, 365–468) and the translated quotation at (Boissonnade and Vagliente 1997, xxxiii). So, the explanation of the invariance of the action integral arises from the invariance of the acting forces, those same forces that the form of the action integral is sensitive to.
The incorporation of work into the list should not be surprising. As I’ve already argued, energy should be technically defined in terms of work. Energy is not Galilean invariant, and so neither is work.
(Maudlin 2011, 174).
Please note that I am not here discussing the Lorentz force of classical relativistic electromagnetism, but the simple electric force of classical electrostatics (see SEF in the main text).
(Shankar 2016, 97).
I assume that SYSE is one for which it is true that the curl of the electric field equals zero.
(Jackson 1999, 24–25, 29).
(Jackson 1999, 40).
(Griffiths 2017, 81).
(Feynman, Leighton and Sands 2010, 4–4). Q.v., Appendix 1.
(Feynman, Leighton and Sands 2010, 4–7). The fact that the curl of the electric field equals zero, or:
$$\nabla \times \mathbf{E}=0$$can be derived from a generalized version of Coulomb’s law:
$$\mathbf{E}(\mathbf{r})=\frac{1}{4\pi {\epsilon }_{0}}\int \rho ({\mathbf{r}}^{\boldsymbol{^{\prime}}})\frac{\mathbf{r}-{\mathbf{r}}^{\mathbf{^{\prime}}}}{{\left|\mathbf{r}-{\mathbf{r}}^{\mathbf{^{\prime}}}\right|}^{3}}{d}^{3}r{\prime}$$where \(d^{3} r^{\prime}\) gives the 3D volume element at \({\mathbf{r^{\prime}}}\), and where \(\rho \left( {{\mathbf{r^{\prime}}}} \right)\) gives the volume charge density at \({\mathbf{r}}^{\mathbf{^{\prime}}}\) (a well-known fact mentioned in numerous places, but see (ibid, 4–3, 4–7) and (Jackson 1999, 24–25, 29)). This derivation-fact supports Feynman’s claim.
Charges near conductors likewise create electrostatic potentials (Appel 2007, 165).
(Shankar 2016, 82).
(Rankine 1853, 106) emphasis in the original.
(Thomson 1860, 334) emphasis in the original.
(Klein and Nellis 2012, 237).
(Klein and Nellis 2012, 350).
References
Andersen, H. K. (2011). Mechanisms, laws, and regularities. Philosophy of Science., 78, 325–331.
Appel, W. (2007). Mathematics for physics and physicists. Princeton University Press.
Atkins, P., de Paula, J., & Keeler, J. (2019). Atkins’ physical chemistry: Molecular thermodynamics and kinetics. Oxford University Press.
Bartels, A. (1996). Modern essentialism and the problem of individuation of spacetime points. Erkenntnis, 45(1), 25–43.
Bird, A. (2007). Nature’s Metaphysics: Laws and Properties. Oxford: Oxford University Press.
Boissonnade, A. and Victor N. Vagliente. (1997) “Translator’s introduction”. In J.L. Lagrange. Analytical Mechanics: Translated from the Mécanique analytique, novelle édition of 1811, Auguste Boissonnade, Victor N. Vagliente (Eds.). Translated by Auguste Boissonnade and Victor N. Vagliente. xi-xli. Dordrecht: Kluwer Academic Publishers.
Butterfield, J. (2007). On symplectic reduction in classical mechanics. In J. Butterfield & J. Earman (Eds.), Philosophy of Physics: Handbook of the Philosophy of Science (pp. 1–132). North-Holland an imprint of Elsevier.
Butts, R. E. (1968). Introduction. In R. E. Butts (Ed.), William Whewell’s Theory of Scientific Method (pp. 3–29). University of Pittsburgh Press.
Calinger, R. S. (2016). Leonhard Euler: Mathematical genius in the enlightenment. Princeton University Press.
Caparrini, S., & Fraser, C. (2013). Mechanics in the Eighteenth Century. In J. Z. Buchwald & R. Fox (Eds.), The Oxford Handbook of the History of Physics (pp. 358–405). Oxford University Press.
del Castillo, T., & Gerardo, F. (2018). An introduction to hamiltonian mechanics. Birkhäuser.
Cayley, A. (1890) “Report on the recent progress of theoretical dynamics”, Report of the British Association for the Advancement of Science, 1857, pp. 1–42 in Arthur Cayley, in Mathematical Papers: The Collected Mathematical Papers of Arthur Cayley. Vol. 3. Cambridge: Cambridge University Press., pp. 156–204.
Chatti, S. (2011). Extensionalism and scientific theory in quine’s philosophy. International Studies in the Philosophy of Science., 25(1), 1–21.
Conway, A. W., & Synge, J. L. (1931). Introduction. In A. W. Conway & J. L. Synge (Eds.), William Rowan Hamilton The Mathematical Papers of Sir William Rowan Hamilton Geometrical Optics. Cambridge University Press.
Coopersmith, J. (2015). Energy, the subtle concept: the discovery of Feynman’s Blocks from Leibniz to Einstein (Revised). Oxford University Press.
Crowe, M. J. (1985). A history of vector analysis: The evolution of the idea of a vectorial system. Dover Publications Inc.
Darrigol, O. (2000). Electrodynamics from Ampère to Einstein. Oxford University Press.
Diaz, R., Herrar, W. J., & Manjarrés, D. A. (2009). Work and energy in inertial and noninertial reference frames. American Journal of Physics., 77, 270–273.
Dirac, PA.M. (1995). In R. H. Dalitz (Eds.), The Collected Works of P.A.M Dirac: 1924–1948. Cambridge: Cambridge University Press.
DobbsTeeter, B. J. (1992). The janus faces of genius: the role of alchemy in newton’s thought. Cambridge University Press.
Dunnington, G. W. (1955). Carl Friedrich Gauss: Titan of science: A study of his life and work. Exposition Press.
Dürr, D., & Teufel, S. (2009). Bohmian mechanics: The physics and mathematics of quantum theory. Springer.
Earman, J. (1976). Causation: A matter of life and death. The Journal of Philosophy, 73(1), 5–25.
Earman, J. (2004). Laws, symmetry, and symmetry breaking: Invariance, conservation principles, and objectivity. Philosophy of Science., 71, 1227–1241.
Earman, J. (2011). Sharpening the Electromagnetic Arrow(s) of Time. In C. Callender (Ed.), The Oxford Handbook of Philosophy of Time (pp. 485–527). Oxford University Press.
Einstein, A. (1954). Ideas and opinions. In C. Seelig (Ed.), New Translations and Revisions by Sonja Bargmann. Crown Publishers.
Ellis, B. D. (2002). The Philosophy of Nature: A Guide to the New Essentialism. Montreal: McGill-Queen’s University Press.
Euler, L. (1744) Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti”. Euler Archive—All Works. 65. https://scholarlycommons.pacific.edu/euler-works/65
Euler, L. (1753) Harmonie entre les principes généraux de repos et de mouvement de M. de Maupertuis. Euler Archive—All Works. 197. https://scholarlycommons.pacific.edu/euler-works/197
Farr, M., & Reutlinger, A. (2013). A relic of a bygone age? causation, time symmetry and the directionality argument. Erkenntnis, 78, 215–235.
Fecko, M. (2006). Differential geometry and lie groups for physicists. Cambridge University Press.
Feynman, R., Leighton, R. B., & Sands, M. (2010). Feynman: The Feynman lectures on physics (Vol. 2). Basic Books.
Field, H. (2003). Causation in a physical world. In M. J. Loux & D. W. Zimmerman (Eds.), The Oxford Handbook of Metaphysics (pp. 435–460). Oxford University Press.
Finkelstein, R. J. (1973). Nonrelativistic mechanics. W.A. Benjamin Inc.
Fisch, M. (1999). The Making of Peacock’s Treatise on Algebra: A case of creative indecision. Archive for History of Exact Sciences., 54, 137–179.
Fisch, M., & Schaffer, S. (Eds.). (1991). William Whewell: A composite portrait. Clarendon Press.
Fitzpatrick, R. (2011) Generalized Forces. https://farside.ph.utexas.edu/teaching/336k/lectures/node76.html University of Texas at Austin. Downloaded: 11/10/2020.
Fraser, C. (2003) The calculus of variations: A historical survey. In Hans Niels Jahnke (Eds.) A History of Analysis. Providence, Rhode Island: American Mathematical Society.
Fraser, C. (1985). J.L. Langrange’s Changing Approach to the Foundations of the Calculus of Variations. Archive for History of Exact Sciences, 32, 151–191.
French, A. P. (1971). Newtonian mechanics. W.W. Norton & Company.
Friedman, M. (1992). Causal laws and the foundations of natural science. In P. Guyer (Ed.), The Cambridge Companion to Kant (pp. 161–199). Cambridge University Press.
Friedman, M. (2013). Kant’s construction of nature: A reading of the metaphysical foundations of natural science. Cambridge University Press.
Friedman, M. (2014). Laws of nature and causal necessity. Kant Studien., 105(4), 531–553.
Frisch, M. (2005). Inconsistency, asymmetry, and non-locality: A philosophical Investigation of classical electrodynamics. Oxford University Press.
Frisch, M. (2007). Causation, Counterfactuals, and Entropy. In H. Price & R. Corry (Eds.), Causation, Physics, and the Constitution of Reality (pp. 351–395). Oxford University Press.
Frisch, M. (2009). The most sacred tenet? Causal reasoning in physics. The British Journal for the Philosophy of Science., 60(3), 459–474.
Frisch, M. (2014). Causal reasoning in physics. Cambridge University Press.
Garber, D. (2009). Leibniz: Body, substance, Monad. Oxford University Press.
Glennan, S. (1996). Mechanisms and the Nature of Causation. Erkenntnis, 44(1), 49–71.
Glennan, S. (2010). Mechanisms, causes, and the layered model of the world. Philosophy and Phenomenological Research., 81(2), 362–381.
Glennan, S. (2017). The new mechanical philosophy. Oxford University Press.
Goldstine, H. H. (1980). A History of the Calculus of Variations from the 17th through the 19th Century. Springer-Verlag.
Goldstine, H., Poole, C. P., Jr., & Safko, J. (2002). Classical mechanics. Addison-Wesley.
Graves, R. P. (1842) Our Portrait Gallery—NO. XXVI. Sir William R. Hamilton. The Dublin University Magazine 19 (January to June): 94–110.
Graves, R. P. (1882). Life of Sir William Rowan Hamilton (Vol. 1). Hodges, Figgis, & CO., Grafton-Street.
Graves, R. P. (1885). Life of Sir William Rowan Hamilton (Vol. 2). Hodges, Figgis, & CO., Grafton-Street.
Graves, R. P. (1889). Life of Sir William Rowan Hamilton (Vol. 3). Hodges, Figgis, & CO., Grafton-Street.
Gray, J. (2008). Plato’s ghost: The modernist transformation of mathematics. Princeton University Press.
Gray, J. (2015). The real and the complex: A history of analysis in the 19th Century. Springer.
Green, G. (1871). Mathematical papers of the late George Green, N.M. Ferrers. London: MacMillan and Co.
Greiner, W. (1998). Classical electrodynamics. Springer.
Griffiths, D. J. (2017). Introduction to electrodynamics (4th ed.). Oxford University Press.
Guyer, P. and Allen W. Wood. (1998). Critique of Pure Reason: The Cambridge Edition of the Works of Immanuel Kant. In: Paul Guyer and Allen W. Wood. Translated by Paul Guyer and Allen W. Wood. New York: Cambridge University Press.
Guyer, P. (1987). Kant and the claims of knowledge. Cambridge University Press.
Haar Ter, Dirk. (1961). Elements of Hamiltonian Mechanics. North-Holland Publishing Company.
Hamilton, W. R. (1828). Theory of systems of rays. Transactions of the Royal Irish Academy, 15, 69–174.
Hamilton, W. R. (1931). The mathematical papers of Sir William Rowan Hamilton. In A. W. Conway & J. L. Synge (Eds.), Geometrical Optics. Cambridge University Press. Abbreviated: “MPH1”.
Hamilton, W. R. (1834). On a general method in dynamics; by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function. Philosophical Transactions of the Royal Society., 124, 247–308.
Hamilton, W. R. (1835). Second essay on a general method in dynamics. Philosophical Transactions of the Royal Society, 125, 95–144.
Hamilton, W. R. (1837). Theory of conjugate functions, or algebraic couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time. The Transactions of the Royal Irish Academy, 17, 293–423.
Hamilton, W. R. (1940). The Mathematical Papers of Sir William Rowan Hamilton. In A. W. Conway & A. J. McConnell (Eds.), Dynamics. Cambridge University Press. Abbreviated: “MPH2”.
Hamilton, W. R. (1853) “Preface to ‘Lectures on Quaternions. In MPH3, pp. 117–155.
Hamilton, W. R. (1967). The Mathematical Papers of Sir William Rowan Hamilton. In A. H. Halberstam & R. E. Ingram (Eds.), Algebra. Cambridge University Press. Abbreviated: “MPH3”.
Hankins, T. L. (1977). Triplets and Triads: Sir William Rowan Hamilton on the Metaphysics of Mathematics. Isis, 68(2), 174–193.
Hankins, T. L. (1980). Sir William Rowan Hamilton. The Johns Hopkins University Press.
Healey, R. (2007). Gauging What’s Real: The Conceptual Foundations of Contemporary Gauge Theories. Oxford University Press.
Hecht, E. (2003). An historico-critical account of potential energy: Is PE really real? The Physics Teacher., 41, 486–493.
Hecht, E. (2007). Energy and change. The Physics Teacher., 45, 88–92.
Hecht, E. (2016). Relativity, potential energy, and mass. European Journal of Physics., 37, 065804.
Hecht, E. (2019a). Understanding energy as a subtle concept: a model for teaching and learning energy. American Journal of Physics., 87, 495.
Hecht, E. (2019b). Reply to comment on ‘Relativity, potential energy, and mass. European Journal of Physics, 40, 028002.
Hendry, J. (1984). The evolution of William Rowan Hamilton’s view of algebra as the science of pure time. Studies in the History and Philosophy of Science, 15(1), 63–81.
Hitchcock, C. (2007). What russell got right. In H. Price & R. Corry (Eds.), Causation, Physics, and the Constitution of Reality: Russell’s Republic Revisited (pp. 66–105). Oxford University Press.
Horzela, A., Edward K. and Jaroslaw K. (1991) “On the Galilean covariance of classical mechanics”. Report INP No 1556/PH. Downloaded from the INIS Repository: https://inis.iaea.org/collection/NCLCollectionStore/_Public/23/088/23088077.pdf 10/09/2020.
Ismael, J. T. (2016). How physics makes us free. Oxford University Press.
Jackson, J. D. (1999). Classical electrodynamics (3rd ed.). John Wiley & Sons Inc.
Jammer, M. (1957). Concepts of force: A study in the foundations of dynamics. Harvard University Press.
Kant, I. (1903). Kant’s Prolegomena and Metaphysical Foundations of Natural Science. Translated from the Original with a Biography and Introduction by Ernest Belfort Bax. London: George Bell and Sons.
Kant, I. (1933). Kant’s Prolegomena to Any Future Metaphysics. Edited in English by Dr. Paul Carus. Chicago: Open Court Publishing Company.
Kant, I. (1992). Theoretical Philosophy: 1755–1770. Translated and edited by David Walford in collaboration with Ralf Meerbote. Cambridge: Cambridge University Press.
Kant, I. (1998). Critique of pure reason: The cambridge edition of the works of Immanuel Kant. In P. Guyer & A. W. Wood (Eds.), Translated by Paul Guyer and Allen W Wood. Cambridge University Press.
Kant, I. (2004). Prolegomena to any future metaphysics that will be able to come forward as science: with Selections from the Critique of Pure Reason. Cambridge University Press.
Kargon, R. (1965). William Rowan Hamilton and Boscovichian Atomism. Journal of the History of Ideas., 26(1), 137–140.
Kistler, M. (2013). The interventionist account of causation and non-causal association laws. Erkenntnis, 78, 65–84.
Klein, S., & Nellis, G. (2012). Thermodynamics. Cambridge University Press.
Kutach, D. (2013). Causation and its Basis in Fundamental Physics. Oxford University Press.
Lagrange, J-L. (1997). Analytical mechanics: Translated from the Mécanique analytique, novelle édition of 1811, edited by Auguste Boissonnade and Victor N. Vagliente. Translated by Auguste Boissonnade and Victor N. Vagliente. Dordrecht: Kluwer Academic Publishers.
Lagrange, J.-L. (1867). Œuvres De Lagrange (Vol. 1). Gauthier-Villars, Imprimeur-Libraire.
Lanczos, C. (1970). The variational principles of mechanics (4th ed.). University of Toronto Press.
Landau, Lev D. and Evgeny M. Lifshitz. (1976). Mechanics: Volume 1. Translated by J.B. Sykes and J.S. Bell. New York: Pergamon Press Inc.
Lange, M. (2017). Because without Cause: Non-Causal Explanations in Science and Mathematics. Oxford University Press.
Langhaar, H. L. (1962). Energy methods in applied mechanics. John Wiley and Sons Inc.
Leibniz, G. W. (1898) The monadology and other philosophical writings. Translated with Introduction and Notes by Robert Latta. Oxford: Clarendon Press.
Leibniz, G. W. (1989). Philosophical Essays. In R. Ariew & D. Garber (Eds.), Translated by Roger Ariew and Daniel Garber. Hackett Publishers.
Leibniz, G. W., & Bartholomeus, D. B. (2007). The Leibniz-Des Bosses Correspondence. In B. Look & D. Rutherford (Eds.), Translated by Brandon Look and Donald Rutherford. Yale University Press.
Loew, C. (2017). Causation, physics, and Fit. Synthese, 194(6), 1945–1965.
Loewer, B. (2007). Counterfactuals and the second law. In H. Price & R. Corry (Eds.), Causation, Physics, and the Constitution of Reality: Russell’s Republic Revisited (pp. 293–326). Oxford University Press.
Lombardi, O., Castagnino, M., & Ardenghi, J. S. (2010). The modal-Hamiltonian interpretation and the galilean covariance of quantum mechanics. Studies in History and Philosophy of Modern Physics., 41, 93–103.
Lützen, J. (1995). Interactions between Mechanics and Differential Geometry in the 19th Century. Archive for History of Exact Sciences, 49(1), 1–72.
Machamer, P., Darden, L., & Craver, C. F. (2000). Thinking about mechanisms. Philosophy of Science., 67(1), 1–25.
Mann, P. (2018). Lagrangian and hamiltonian dynamics. Oxford University Press.
Marion, J. B., & Thornton, S. T. (1988). Classical dynamics of particles and systems (3rd ed.). Harcourt Brace Javanovich Inc.
Mathews, J. (1978). William Rowan Hamilton’s Paper of 1837 on the Arithmetization of analysis. Archive for History of Exact Sciences., 19, 177–200.
Maudlin, T. (2011). Quantum Non-Locality and Relativity (3rd ed.). Wiley-Blackwell.
Maxwell, J. C. (1888). An elementary treatise on Electricity. Edited by William Garnett. Clarendon Press.
Maxwell, J. C. (1871). Theory of heat. Longmans.
McGuire, J. E. (1977). Neoplatonism and Active Principles: Newton and the Corpus Hermeticum. In: Robert S. Westman and J.E. McGuire (Eds.) Hermeticism and the Scientific Revolution: Papers Read at the Clark Library Seminar, March 9, 1974. Los Angeles: Clark Memorial Library.
McGuire, J. E. (1968). Force, active principles, and Newton’s invisible realm. Ambix, 15, 154–208.
Melnick, A. (2004). Themes in Kant’s Metaphysics and Ethics. The Catholic University of America Press.
Melnick, A. (2006). Kant’s proofs of substance and causation. In P. Guyer (Ed.), The Cambridge Companion to Kant and Modern Philosophy (pp. 203–237). Cambridge University Press.
Merzbach, U. C., & Boyer, C. B. (2011). A History of mathematics (3rd ed.). John Wiley & Sons Inc.
Meyer, K. R., & Offin, D. C. (2017). Introduction to Hamiltonian dynamical systems and the n-body problem (3rd ed.). Springer.
Mills, R. (1994). Space, time and quanta. W.H. Freeman.
Mumford, S. (2004). Laws in Nature. New York: Routledge Publishers.
Nakane, M., & Fraser, C. G. (2002). The early history of Hamilton-Jacobi dynamics 1834–1837. Centaurus, 44, 161–227.
Newton, I. (1999). The Principia: Mathematical Principles of Natural Philosophy. Translated by I. Bernard Cohen, Anne Whitman and Julia Budenz. Berkeley and Los Angeles, CA: University of California Press.
Ney, A. (2009). Physical causation and difference-making. The British Journal for the Philosophy of Science., 60(4), 737–764.
North, J. (2022). Formulations of classical mechanics. In E. Knox & A. Wilson (Eds.), A Companion to the Philosophy of Physics (pp. 21–32). Routledge Publishers.
Norton, JD. (2021) The metaphysics of causation: An Empiricist Critique”, https://sites.pitt.edu/~jdnorton/papers/Causation_empiricist.pdf downloaded 12/02/2021
Norton, J. D. (2007a). Causation as folk science. In H. Price & R. Corry (Eds.), Causation, Physics, and the Constitutionof Reality: Russell’s Republic Revisited (pp. 11–44). Oxford University Press.
Norton, J. D. (2007b). Do the causal principles of modern physics contradict causal anti-fundamentalism? In P. Machamer & G. Wolters (Eds.), Thinking about Cause: From Greek Philosophy to Modern Physics (pp. 222–234). University of Pittsburgh Press.
Nozick, R. (2001). Invariance: The structure of the objective World. Harvard University Press.
Øhstrøm, P. (1985). W.R. Hamilton’s View of Algebra as the Science of Pure Time and His Revision of This View. Historia Mathematica., 12, 45–55.
Olness, F., & Scalise, R. (2011). Regularization, renormalization, and dimensional analysis: Dimensional regularization meets freshman E&M. American Journal of Physics., 79, 306–312.
Panza, M. (2003) The Origins of Analytic Mechanics in the 18th Century. In A History of Analysis, 137–153. American Mathematical Society.
Papineau, D. (2013). Causation is macroscopic but not irreducible. In G. Sophie, E. J. Lowe, & R. D. Ingthorsson (Eds.), Mental Causation and Ontology. Oxford University Press.
Peacock, T. and Nicolas H. (2007) “Course Materials for 2.003J/1.053J Dynamics and Control I, Spring 2007”. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded 11/10/2020.
Penrose, R. (2005). The road to reality: A complete guide to the laws of the universe. Vintage Books.
Pletser, V. (2018). Lagrangian and Hamiltonian analytical mechanics: Forty Exercises Resolved and Explained. Springer.
Pourciau, B. (2006). Newton’s Interpretation of Newton’s Second Law. Archive for History of Exact Sciences, 60(2), 157–207.
Pourciau, B. (2020). The principia’s second Law (as Newton Understood It) from Galileo to Laplace. Archive for History of Exact Sciences., 74, 183–242.
Quine. W. V. (2004). Quintessence: Basic Readings from the Philosophy of W.V. Quine, (eds.) Roger F. Gibson Jr. Cambridge, MA: Harvard University Press.
Quine, W. V. (1976). The ways of Paradox and other essays. Harvard University Press.
Raju, C. K. (1994). Time: Towards a consistent theory. Springer.
Rankine, W. (1853). On the general law of the transformation of energy. Philosophical Magazine., 5(30), 106–117.
Redhead, M. (1990). “Explanation.” Edited by Dudley Knowles. Royal Institute of Philosophy Supplement, 27, 135–154.
Rennie, R. (Ed.). (2015). Oxford dictionary of physics (7th ed.). Oxford University Press.
Russell, B. 1912–1913. “On the Notion of Cause.” Proceedings of the Aristotelian Society 13: 1–26.
Ryder, P. (2007). Classical mechanics. Shaker Verlag.
Saunders, S. (2003a). “Indiscernibles, General Covariance, and Other Symmetries”. In Revisiting the Foundations of Relativistic Physics: Festschrift in Honour of John Stachel. (pp. 151–173). Dordrecht: Springer.
Saunders, S. (2003b). “Physics and Leibniz’s Principles”. In Symmetries in Physics, edited by Katherine Brading and Ellena Castellani. 289–308. Cambridge: Cambridge University Press.
Saunders, S. (2013a). “Rethinking Newton’s Principia”. Philosophy of Science, 80(1), 22–48.
Saunders, S. (2013b). “Indistinguishability”. In R. Batterman (ed.) The Oxford Handbook of Philosophy of Physics (pp. 340–380). New York: Oxford University Press.
Schaffer, J. (2008). Causation and laws of nature: Reductionism. In T. Sider, J. Hawthorne, & D. W. Zimmerman (Eds.), Contemporary Debates in Metaphysics (pp. 82–107). Blackwell Publishers.
Schaffer, J. (2009). On what grounds what. In D. J. Chalmers, D. Manley, & R. Wasserman (Eds.), Metametaphysics: New Essays on the Foundations of Ontology (pp. 347–383). Oxford University Press.
Schroeren, D. (2020). The metaphysics of invariance. Studies in History and Philosophy of Modern Physics., 70, 51–64.
Shankar, R. (2016). Fundamentals of physics II: Electromagnetism, optics, and quantum mechanics. Yale University Press.
Sider, T. (2011). Writing the book of the world. Oxford University Press.
Smith, C. (1998). The science of energy: A cultural history of energy physics in Victorian Britain. University of Chicago Press.
Smith, N. K. (1918). A Commentary to Kant’s ‘Critique of Pure Reason.’ MacMillan and Co., Limited.
Smith, C., & Wise, N. M. (1989). Energy & empire: A biography study of lord kelvin: Part 1. Cambridge University Press.
Sommerfeld, A. (1964). Lectures on Theoretical Physics: Volume 1 Mechanics., translated by Martin O. Stern. New York: Academic Press.
Spohn, W. (2006). Causation: An alternative. The British Journal for the Philosophy of Science, 57, 93–119.
Stan, M. (2022). From Metaphysical Principles to Dynamical Laws. In M. M. David & J. Dana (Eds.), The Cambridge History of Philosophy of the Scientific Revolution. Cambridge University Press.
Stan, M. (2009). Kant’s early theory of motion: Metaphysical dynamics and relativity. The Leibniz Review., 19, 29–61.
Steane, A. M. (2012). Relativity made relatively easy. Oxford University Press.
Stewart, IW. 2016. “Advanced Classical Mechanics”. Edited by Francisco Machado, Thanawuth Thanathibodee, and Prashanth S. Venkataram. https://ocw.mit.edu/courses/physics/8-09-classical-mechanics-iii-fall-2014/lecture-notes/MIT8_09F14_full.pdf Downloaded 11/10/2020.
Sullivan, J. W. N. (1934). The limitations of science. The Viking Press.
Taylor, J. R. (2005). Classical mechanics. University Science Books.
Tefft, B. J., & Tefft, J. A. (2007). Galilean relativity and the work-kinetic energy theorem. The Physics Teacher., 45, 218–220.
Thiele, R. (1997). On Some Contributions to Field Theory in the Calculus of Variations from Beltrami to Carathéodory. Historia Mathematica., 24, 281–300.
Thomson, J. J. (1888). Applications of dynamics to physics and chemistry. Macmillan Co.
Thomson, W. (1860). Measurement of the electromotive force required to produce a spark in air between parallel metal plates at different distances. Proceedings of the Royal Society of London., 10, 326–338.
Thorne, K. S., & Blandford, R. D. (2017). Modern classical physics: Optics, fluids, plasmas, elasticity, relativity, and statistical physics. Princeton University Press.
Todhunter, I. (1876a). William Whewell, D.D. Master of Trinity College, Cambridge. An Account of His Writings with Selections from His Literary and Scientific Correspondence. Volume 1. London: MacMillan and Co.
Todhunter, I. (1876b). William Whewell, D.D. Master of Trinity College, Cambridge. An Account of His Writings with Selections from His Literary and Scientific Correspondence. Volume 2. London: MacMillan and Co.
Trapp, R. (2019). Multivariable calculus. Oxford University Press.
Truesdell, C. (1968). Essays in the history of mechanics. Springer-Verlag.
Uffink, J. (2002). Essay review: Time and chance. Studies in History and Philosophy of Modern Physics., 33, 555–563.
van Fraassen, B. C. (1989). Laws and symmetry. Oxford University Press.
Watkins, E. (2005). Kant and the metaphysics of causality. Cambridge University Press.
Watkins, E. (2019). Kant on laws. Cambridge University Press.
Weaver, C. (2021). In Praise of Clausius entropy: Reassessing the foundations of Boltzmannian statistical mechanics. Foundations of Physics, 51(3), 1–64.
Weaver, C. G. (2019). Fundamental causation: Physics, metaphysics, and the deep structure of the world. Routledge Publishers.
Weinberg, S. (2015). To explain the world: The discovery of modern science. HarperCollins.
Wellek, R. (1931). Immanuel Kant in England 1793–1838. Princeton, NJ: Princeton University Press.
Westfall, R. S. (1971). Force in Newton’s Physics: The Science of Dynamics in the Seventeenth Century. American Elsevier.
Whewell, W. (1836). An elementary treatise on mechanics: Designed for the use of Students in the University (5th ed.). Cambridge: Printed at the Pitt Press by John Smith for J. & J.J. Deighton, Cambridge: and Whittaker & Arnot, London.
Whewell, W. (1858). History of Scientific Ideas: Being the First Part of the Philosophy of the Inductive Sciences (3rd ed.). John W. Parker and Son.
Whewell, W. (1967). The Philosophy of the inductive sciences: Part Two. Frank Cass & Co. LTD.
Whittaker, E. T. (1954). William Rowan Hamilton. Scientific American., 190, 82–87.
Winterbourne, A. T. (1982). Algebra and Pure Time: Hamilton’s Affinity with Kant. Historia Mathematica, 9, 195–200.
Woodward, J. (2003). Making things happen: A theory of causal explanation. Oxford University Press.
Woodward, J. (2007). Causation with a human face. In H. Price & R. Corry (Eds.), Causation, Physics, and the Constitution of Reality (pp. 66–105). Oxford University Press.
Woodward, J. (2014). A functional account of causation: Or, A Defense of the Legitimacy of Causal Thinking by Reference to the Only Standard that Matters—Usefulness (as Opposed to Metaphysics or Agreement with Intuitive Judgment). Philosophy of Science, 81(5), 691–713.
Acknowledgements
An earlier draft of this paper was presented at an online meeting of the California Institute of Technology’s philosophy of physics reading group on November 4th, 2020. I’d like to thank Joshua Eisenthal, Christopher Hitchcock, Mario Hubert, and Charles T. Sebens for their questions and comments on that earlier draft. The paper was presented at an online meeting of the Center for Philosophy of Science at the University of Pittsburgh on April 2nd, 2021. I’d like to thank Nick Huggett, Mike Schneider, Mark Wilson, and Siddharth Muthukrishnan for their comments and questions at that event. It was presented yet again for the Pacific APA on April 9th, 2021, where it benefited from the helpful commentary of Ned Hall. It was presented again for the Southern California Philosophy of Physics group on May 1st, 2021. I thank Jeffrey A. Barrett, Eddy Chen, Joshua Eisenthal, David Mwakima, and James O. Weatherall for their comments and questions. I thank Michael Townsen Hicks too for his comments on a very early draft. And finally, I’d like to give special thanks to Tom Banks, Jennifer Coopersmith, Nick Louzon, and Don Page for helpful correspondence and/or conversations.
Funding
None.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares no conflict of interest.
Appendix 1: The Potential Energy—Work—Forces—Causation Link: Nothing New
Appendix 1: The Potential Energy—Work—Forces—Causation Link: Nothing New
The orthodox interpretation of potential energy is provided by PEI, though it may need some tinkering in order to handle various other system-types. I maintain that because orthodoxy defines potential energy in terms of work, and work in terms of forces, causation enters mechanics through the potential energy function, if forces really are causes of motion. Because Hamilton believed forces are causes, he was able to causally interpret classical mechanics by looking to the potential energy function.
Interestingly, my (and Hamilton’s) analysis of matters can be found in some of the very earliest work in which potential energy was first used in physics. For example, William Rankine (1820–1872) who coined the term ‘potential energy’ in 1853, remarked,
In this investigation the term energy is used to comprehend every affection of substances which constitutes or is commensurable with a power of producing change in opposition to resistance, and includes ordinary motion and mechanical power, chemical action, heat, light, electricity, magnetism, and all other powers, known or unknown, which are convertible or commensurable with these. All conceivable forms of energy may be distinguished into two kinds; actual or sensible and potential or latent…Potential energy…is measured by the amount of a change in the condition of a substance, and that of the tendency or force whereby that change is produced (or, what is the same thing, of the resistance overcome in producing it), taken jointly.Footnote 139
Rankine is here associating energy (in general) with productive power (causation), but he is also associating potential energy with work, work with forces, and forces with productive power or causation. Seven years later, William Thomson’s (or Lord Kelvin’s) discussion of the electric potential function (which is used in classical electrostatics to state the potential energy function (q.v., Sect. 5)) strongly associated that potential with work. He wrote,
Electric potential. —The amount of work required to move a unit of electricity from any one position to any other position, is equal to the excess of the electric potential of the second position above the electric potential of the first position.Footnote 140
Thomson here says that differences are what matter, and that the electrostatic potential (and so the electrostatic potential energy) just is work (whether virtual or not) required to complete a task.
Both Rankine and Thomson’s views of energy are important because they influenced the work of Rudolf Clausius (1822–1888), James Clerk Maxwell, and Ludwig Boltzmann (1844–1906). These mechanicians thought of entropy as a quantity that tracks how the energy (as understood by Rankine and Thomson) transforms over time.Footnote 141 Modern thermodynamics and statistical mechanics has thereby inherited the work-laden and so also force-laden notion of energy. It is therefore unsurprising to see in the work of modern thermodynamicists, such as Klein and Nellis, the following: “…the property entropy is introduced in order to quantify the quality of energy”Footnote 142 and “[t]he Second Law states that the quality of energy, i.e., the capability to do work, is reduced in all real processes.”Footnote 143 All of this is as one would expect given the truth of my interpretation of potential energy.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Weaver, C.G. Hamilton, Hamiltonian Mechanics, and Causation. Found Sci (2023). https://doi.org/10.1007/s10699-023-09923-y
Accepted:
Published:
DOI: https://doi.org/10.1007/s10699-023-09923-y