## Abstract

The notions of conservation and relativity lie at the heart of classical mechanics, and were critical to its early development. However, in Newton’s theory of mechanics, these symmetry principles were eclipsed by domain-specific laws. In view of the importance of symmetry principles in elucidating the structure of physical theories, it is natural to ask to what extent conservation and relativity determine the structure of mechanics. In this paper, we address this question by deriving classical mechanics—both nonrelativistic and relativistic—using relativity and conservation as the primary guiding principles. The derivation proceeds in three distinct steps. First, conservation and relativity are used to derive the asymptotically conserved quantities of motion. Second, in order that energy and momentum be continuously conserved, the mechanical system is embedded in a larger *energetic framework* containing a massless component that is capable of bearing energy (as well as momentum in the relativistic case). Imposition of conservation and relativity then results, in the nonrelativistic case, in the conservation of mass and in the frame-invariance of massless energy; and, in the relativistic case, in the rules for transforming massless energy and momentum between frames. Third, a force framework for handling continuously interacting particles is established, wherein Newton’s second law is derived on the basis of relativity and a staccato model of motion-change. Finally, in light of the derivation, we elucidate the structure of mechanics by classifying the principles and assumptions that have been employed according to their explanatory role, distinguishing between symmetry principles and other types of principles (such as compositional principles) that are needed to build up the theoretical edifice.

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## Notes

See Sect. 5.3.1 for further details.

See Sect. 5.3.1 for background on this shift.

The

*mass*of a body is here taken as a measure of substance from which a body is composed. In particular, no connection between mass and inertia (degree of resistance to force) is implied. The mass is assumed to be independent of the body’s state of motion, and hence frame-independent.For simplicity, this energy-additivity is taken as given here, but is in fact a manifestation of the compositional symmetry of associativity (see Sect. 5.2).

We assume here that isolated bodies move at constant velocity.

Although the indicated collision is a premise of the following argument, it can be traced to more primitive symmetry requirements: (a) The possibility of post-collisional motion of at least one body along the

*y*-axis (vertical) can be traced to the requirement of continuity together with the fact that both grazing and head-on collisions are possible. (b) The fact that the post-collisional velocities must then be equal and opposite can be traced to the requirement that relatively-rotated reference frames are physically equivalent and the requirement that the same initial conditions lead to the same final conditions. Specifically, suppose that, as viewed in frame*S*, one particle post-collisionally travels along the positive*y*-axis, but the other*not*along the negative*y*-axis. Then an observer in a reference frame (\(\widetilde{S}\)) rotated about the*z*-axis (perpendicular to plane of collision, and passing through the midpoint of the particles’ initial positions) of*S*by \(\pi\) would see a collision with the same initial positions and velocities, but with one of the post-collisional velocities along the*negative**y*-axis and the other*not*along the positive*y*-axis. That is, the*same*initial positions and initial velocities would lead to*different*post-collisional velocities for observers in*S*and \(\widetilde{S}\). This inconsistency can be avoided if the post-collisional velocities are along the positive and negative*y*-axes. Moreover, the same requirements imply that the post-collisional*speeds*of the two particles are equal. (c) The fact that, additionally, the post-collisional speeds coincide with the pre-collisional speeds then follows from the assumptions that (i) relatively-rotated reference frames are physically equivalent, and (ii) the time-reversed version of an elastic collision (*viz.*a collision that asymptotically conserves the total scalar quantity of motion) is also possible.That the mass of the composite is equal to the sum \(m_1 + m_2\) can either be assumed, or derived from the compositional symmetry of associativity (see Sect. 5.2).

By definition, massless energy refers to any form of energy other than the two forms—namely, rest energy and kinetic energy—that are explicitly associated with a massive body.

In Sect. 4.2, we discuss a derivation of nonrelativistic kinetic energy due to Maimon in which the frame-invariance of massless energy is implicitly assumed.

Considering the motion of a body under the influence of a sequence of discrete impulses, between which the particle moves inertially, as a way of deducing results concerning the motion of the body when under the influence of a corresponding continuous force is a tactic employed extensively by Huygens, Newton, and others (see, for example, Ref. [2], where it is noted that “From Newton to Laplace, impulses were usually regarded as more fundamental, and continuous forces were assumed to be equivalent, in their observable effects, to a very rapid succession of impulses”). The specific argument given here is inspired by the derivation given in §2.1 of Ref. [2]

In Ref. [2], this is referred to as the

*secular principle*.Einstein alludes to the arbitrariness that is involved in extending ‘force = mass \(\times\) acceleration’ to the relativistic setting in [17, §10].

Specifically, \(\mathbf {F} = \gamma ^3 m \mathbf {a}_\parallel + \gamma m \mathbf {a}_\perp\), while \(\mathbf {f} = \mathbf {a}_\parallel + \mathbf {a}_\perp\). Thus, for a particle of given mass, the velocity \(\mathbf {u}\) must be given in order to convert \(\mathbf {F}\) to \(\mathbf {f}\) or \(\mathbf {f}\) to \(\mathbf {F}\).

Buridan speculates that, if such resistance were entirely absent, a body would continue its motion indefinitely, and that such a condition might obtain with heavenly bodies.

For details of the broader context of Bernoulli’s submission, see [7], Chapter 7. Bernoulli’s alternative derivations are discussed in Chapter 8.

See, for example, Ref. [26], Chapter 9

The classification given here is extracted from [29]. The full classification described therein contains additional types of principle which are not required in the present discussion.

For example, using Galilean relativity, one can explain what happens in an elastic head-on collision of equal bodies moving at unequal speeds \(u_1, u_2\) in terms of what happens when those some bodies collide at equal speeds \((u_1 + u_2)/2\). See also footnote 31.

The first principle acts as a constraint on which start- and end-states can be dynamically connected; the second as a constraint on allowable paths connecting given initial and final configurations; and the third as a constraint on allowable quantum numbers (‘old’ quantum theory) or on possible multiparticle states (‘new’ quantum theory).

For example, in the case of conservation of energy, the transformation under consideration is time evolution of the system; the ‘object’ transformed is the physical state of the system; and the equivalence relation between states is that they ‘possess’ the same total energy. The conservation principle thus posits that time evolution is a symmetry transformation of physical states with respect to this equivalence relation.

More precisely, on the assumption that

*h*is differentiable at a point, one can show [32] that \(h(a, b) = f^{-1}(f(a) + f(b))\), where*f*is a continuous, monotonic function. Hence, if one*regraduates*the masses \(m_i\) via*f*, so that \(\mu _i \equiv f(m_i)\), then \(\mu = \mu _1 + \mu _2\) is the total*regraduated*mass of the system of two bodies. However, since*f*is monotonic, one can just as well quantify the ‘amount of matter’ via the \(\mu _i\) rather than the \(m_i\). Hence, without loss of generality, one can say that mass is additive. The same line of argument applies to any scalar quantity, such as kinetic energy, associated with the bodies, provided that one has clear physical ground for believing that the total quantity for a system of bodies is a function of the quantities associated with each of the bodies.See, for instance, Ref. [33], Chapter 1. The core assumptions here are: (i) the resultant of two parallel forces has magnitude equal to the sum of the magnitudes of these forces, and points in the same direction; (ii) the resultant of a number of forces is commutative and associative; (iii) the resultant of two forces is rotationally covariant; (iv) the resultant of two equal forces varies continuously with the angle between these forces.

In Principles II 36 [34], Descartes asserts: “there is a fixed and determined quantity of [motion] ...always the same in the universe as a whole even though there may at times be more or less motion in certain of its individual parts”, and that “when one part of matter moves twice as fast as another twice as large, there is as much motion in the smaller as in the larger”, roughly interpreted as the assertion that \(\sum _i m_i u_i\) is the conserved quantity, where

*m*is a measure of the ‘size’ of a body.Descartes’ conservation principle was insufficient to account for collisional behaviour. Lacking another principle of similar scope capable of rectifying this insufficiency, Descartes introduced other considerations in a rather

*ad hoc*manner. The defects of the resultant laws of collision were readily apparent. For example, Leibniz showed these laws to be inconsistent with the requirement of continuity [35, pp. 290–291]. Nevertheless, these laws were a spur to development of the correct laws.Huygens’ laws of collisions can be be divided into two cases: (i) For equal bodies in head-on collision (whether elastic or not), all collisions involving bodies with unequal incident speeds follows via relativity from the case of equal incident speeds, the behaviour in this latter case being taken as axiomatic. (ii) For unequal bodies in head-on elastic collision, the additional assumption of the asymptotic conservation of total scalar quantity of motion, where the quantity of motion is a function of speed, and an auxiliary assumption (if one mass undergoes a change, so must the other) implies that the relative speed of the two masses is the same before and after the collision.

*Proof sketch*: in any such a collision, there is a Galilean frame of reference in which the speed of one body does not change (comparing the initial and final states), only its direction of motion. Hence, its quantity of motion does not change. But, asymptotic conservation of total quantity of motion then implies that the speed of the*other*body also does not change. But if the direction of one mass changes, so must the other (by the auxiliary assumption). Hence, relative speed in this chosen frame is same before and after. But relative speed is frame-independent. Therefore, irrespective of the (inertial) frame in which the collision is viewed, the relative speed is unchanged. For details, see [23, pp. 313–317] and also [36, §9.4].Huygens’ law of conservation of relative speed of two bodies in head-on elastic collision (see footnote 31) implies that the conserved quantity of motion cannot be

*mv*(as can be seen by considering a body of mass \(m < M\) striking a body of mass*M*initially at rest). Furthermore, appealing to Galileo’s law of free fall and Torricelli’s principle (that the centre of gravity of a system of interacting bodies cannot rise), Huygens showed that the conserved quantity of motion is, in fact, \(mv^2\).Newton (amongst others) asserted that atoms were hard bodies that collide completely

*inelastically*[38, pp. 4–5]. Hence the fundamental importance of formulating laws applicable to inelastic collisions.Although Leibniz championed the conservation of

*vis viva*, a compelling account of the ‘missing’ quantity of motion at the stillpoint of an elastic collision, or at the end-point of an inelastic collision, was lacking. As a consequence, scalar conservation principles were marginalized. For example, in textbooks through to the end of the eighteenth century, elastic collisions were handled using a situation-specific law (Huygens’ conservation of the masses’ relative speed—see Footnote 31), rather than the asymptotic conservation of*vis viva*—see [7] (Appendix) and [8].See Ref. [2] for a detailed historical investigation into these derivations.

We note that this implies that the (nonrelativistic) kinetic theory of gases is inconsistent—insofar as ‘heat’ is regarded as a form of massless energy, it is frame-invariant, and so cannot be represented by the kinetic energy of a set of particles, which is not frame-invariant.

We give here the some of the relevant quotes from [44]: “The conserved quantities of classical mechanics are Noether charges only because the classical equations of motion are what they are. But whether or not the classical equations of motion hold is something that needs to be established...”. And: “Given what the equations of motion are, and that they hold where they do, it is indeed necessary that the conservation laws hold, but that’s just a conditional necessity. The connection between the symmetries of the equations of motion and conservation laws is shown by Noether’s theorem. That these are the correct equations of motion, however, is a completely different matter.”

The assumptions underlying the least-action approach to nonrelativistic particle mechanics can be broken down as follows: (i) the (configuration-space) trajectory,

*x*(*t*), of a particle system between times \(t_1, t_2\) has an associated action*S*[*x*(*t*)]; (ii) the actual trajectory between given configurations at times \(t_1, t_2\) is one that extremizes*S*[*x*(*t*)]; (iii) the action is given by the time integral of a function,*L*, of*x*(*t*) and a finite number of temporal derivatives thereof; (iv) the function*L*has the form \(L = T - V\), where*T*,*V*are the kinetic and potential energies of the system. Of these assumptions, the first*three*can be posited independently from Newton’s equations of motion. However, the common view is that the*fourth*—\(L = T - V\)—arises through a transformation of Newton’s equations of motion via d’Alembert’s principle (a more direct approach is given in [45]). Although it is possible to use fundamental symmetries (homogeneity of space and time, isotropy of space, and Galilean invariance) to show that*L*is proportional to*T*for a single isolated particle [46, §4]; and, further, to use compositional symmetries to show that \(L = \sum _i T_i\) for a set of noninteracting particles, we are not aware of a derivation of \(L = T - V\) that avoids presuming Newton’s equations of motion.

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## Acknowledgements

I am most grateful to Jennifer Coopersmith, Olivier Darrigol, Yiton Fu, PierGianLuca Porta Mana, Carolyn Merchant, Jill North, Costas Papachristou, Sebastiano Sonego, and Marius Stan for discussion and/or supportive feedback. I would particularly like to thank Olivier Darrigol for reading several versions of the paper, for pointing out historical omissions, and making important suggestions. I would also like to thank the anonymous referee for several valuable suggestions.

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## Solution of Functional Equations

### Solution of Functional Equations

In this appendix, the functional equations needed in the derivation of the energy and momentum of bodies are solved. If possible, we transform the functional equation of interest into a standard functional equation. For interest, we sometimes provide more than one possible method of solution. In each case, certain mathematical conditions must be satisfied by the unknown function in order for a solution to be obtained.

### 1.1 Solution of \(f(u + v) + f(u - v) = 2f(\sqrt{u^2 + v^2})\)

We present two different solution methods for Eq. (1), one that transforms it into Jensen’s functional equation, the other a direct solution by removing one degree of freedom.

#### 1.1.1 Solution by Transformation into Jensen’s Functional Equation

Using the substitution \(k(w^2) = f(w)\), Eq. (1) becomes

Setting \(x = u^2 + v^2\), \(y = 2uv\), we obtain

which is Jensen’s equation, with *x*, *y* independently variable within \(x>0, y>0\). If *k* is continuous, this equation, under the stated conditions, has general solution \(k(z) = az + b\). As *k* is continuous whenever *f* is continuous,

is the general solution of Eq. (1) under the condition that *f* is continuous.

#### 1.1.2 Direct Solution by Removal of One Degree of Freedom

Alternatively, one can directly solve Eq. (1) by removing one degree of freedom, albeit at the cost of the stronger regularity condition that *f* is analytic. Setting \(u = v\) in Eq. (1) yields

If *f* is differentiable, then, for \(n \ge 1\),

This yields \(f^{(n)}(0) = 0\) whenever \(n \ne 2\). Hence, if *f* is analytic,

### 1.2 Solution of \(g(v + u) - g(v - u) = 2g( \sqrt{u^2 + v^2}) \cdot v / \sqrt{u^2 + v^2}\)

Solution of Eq. (6) is most readily obtained by removing one degree of freedom by setting \(v = u\). Thence,

Setting \(u = 0\) fixes \(g(0) = 0\). If *g* is differentiable, then, for \(n \ge 1\),

For \(n \ge 2\), this yields \(g^{(n)}(0) = 0\). Thus, if *g* is analytic,

### 1.3 Solution of \(\widetilde{F}(x) + \widetilde{F}(y) = 2\widetilde{F}((x+y)/2)\)

Equation (34), with \(x = \gamma (u \oplus -v)\) and \(y = \gamma (u \oplus v)\), has the form of Jensen’s equation, but it is not immediately apparent that *x*, *y* are independent in some region. To see that this is so, it is helpful to express *u*, *v* in terms of rapidities:

Then \(u \oplus v = c\tanh (\phi _1 + \phi _2)\), so that

where \(\tilde{\gamma }(\phi ) \equiv (1 - \tanh ^2\phi )^{-1/2}\).

Now, \(u > 0\) and \(|v| < c\), so that \(\phi _1 > 0\) and \(\phi _2\) is free. Consequently, \((\phi _1 + \phi _2)\) and \((\phi _1 - \phi _2)\) can be independently chosen. Further, since \(\tilde{\gamma }\) is monotonic, \(x = \tilde{\gamma }(\phi _1 + \phi _2)\) and \(y = \tilde{\gamma }(\phi _1 - \phi _2)\) are independent in some region. Therefore, Eq. (34) has the solution \(\widetilde{F}(x) = a + bx\).

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Goyal, P. Derivation of Classical Mechanics in an Energetic Framework via Conservation and Relativity.
*Found Phys* **50**, 1426–1479 (2020). https://doi.org/10.1007/s10701-020-00376-y

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DOI: https://doi.org/10.1007/s10701-020-00376-y