Abstract
A point mass {P; m} is in a uniform mechanical state if its velocity is constant. Departures from a uniform state occur only by variations of velocity caused by solicitations external to {P; m} and acting on it. Such external solicitations are called forces. The vector equation
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Notes
- 1.
Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum nisi quatenus illud a viribus impressis cogitur statum suum mutare, [123, §13, page 54]. The first law was perceived by Leonardo da Vincialthough in a nonmathematical formalism: “...ogni moto attende al suo mantenimento, ovvero ogni corpo mosso sempre si muove, in mentre che la potenzia del motore in lui si rinserra, …ogni corpo seguirà tanto la via del suo corso per linea retta quanto durerà in esso la natura della violenza fatta…” Codex Atlanticus (1478–1518). A physical notion of the first law appears in G. Galilei [61]... Una nave che vadìa movendosi per la bonaccia del mare …è disposta, quando le fusser rimossi tutti gli ostacoli accidentarii ed esterni, a muoversi, con l’inpulso concepito una volta, incessabilmente e uniformemente....
- 2.
Lex II: Mutationem motus proportionalem esse vi motrici impressae et fieri secundum lineam rectam qua vis illa imprimitur..., [123, §13, page 54].... Vis impressa est actio in corpus exercita, ad mutandum ejus statum vel quiescendi vel movendi uniformiter in directum..., [123, §2, page 40]; this is one of the Definitiones preceding the Axiomata, (Def. III). The vis impress is also called by Newton vis motrix in Def. VIII, book I of [123], page 44.
- 3.
Materiae vis insita et potentia resistendi qua corporis unumquodque, quantum in se est, perserverat in statu suo vel quiescendi vel movendi uniformiter in directum. Hanc autem quantitatem (materiae) sub nomine corporis vel massae in sequentibus passim intelligo... [123, §1, page 39].
- 4.
Newton set such an inertial system in the fixed stars, e.g., those stars whose relative position and configuration had not significantly changed up to the eighteenth century since the astronomical observations of Ptolemy, about 130 CE.
- 5.
Lex III: Actioni contrariam semper et aequalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales et in partes contrarias dirigi, [123, §14, page 55].
- 6.
Tempus absolutum, verum, et mathematicum, in se et natura sua, sine relatione ad esternum quodvis aequabiliter fluit, alioque nomine dicitur duratio…, [123, §11, page 52].
- 7.
The original terminology for momentum was quantitas motus, literally quantityof motion [123, Liber I, Def. II, page 40]. The kinetic energy was initially called vis viva, i.e., “living force,” by G. W. Leibniz, in his Theoria Motus Abstracti.Leibniz conceived an elemental motion as an instantaneous elemental insurgence of the vis mortua into vis viva.
- 8.
Observed first by Newton in [123, Corollarium III, § II, page 59].
- 9.
At sea level g = 9. 8066 m/s2. It is, however, a function of altitude [76, F–158].
- 10.
Here \(\gamma = 6.7 \cdot 1{0}^{-11}\)m3 Kg/s2 is the gravitational constant [76, F–87].
- 11.
A formal notion of integral of motion, is in §6.1 of Chapter 6.
- 12.
On a plane π fix a line ℓ and a point O not in ℓ. A conic is the geometric locus of all points in π such that the ratio of their distance to O and to ℓ is constant. The constant value of such ratios is denoted by e and is called the eccentricity of the conic. The point O is a focus and the line ℓ is the directrix. The elliptic, parabolic, or hyperbolic nature of these orbits was observed by Newton in [123, Liber I, De Motu Corporum §66, page 134].
- 13.
Theoretically they could be ellipses, parabolas, or hyperbolas. Astronomical observations confirm that they are ellipses. Parabolic and hyperbolic orbits are observed in the motion of comets [49, 28–104], [50, 105–251], [52].
- 14.
Let m E be the mass of Earth. The mass m J of Jupiter, the planet of largest mass in the solar system, is m J = 318 m E . The mass m M of Mercury, the planet of smallest mass in the solar system, is m M =. 05 m E . The mass of the Sun is m o = 331950 m E . For these values,
$$\frac{{m}_{J} - {m}_{M}} {{m}_{M} + {m}_{o}} \, \approx \, 0.958 \cdot 1{0}^{-3}.$$The numerical data are taken from [76, F–145 and F–165].
- 15.
See for example the formulation of the n-body problem in §4.5c of the Complements. Newton was well aware of such a mutual gravitationalinteraction …Coelos nostros infra coelos fixarum in orbem revolvi volunt, et planetas secum deferre; singuale coelorum partes et planetae qui relative quidem in coelis suis proximis quiescunt, moventur vere…; Newton [123], §11, page 52.
- 16.
From the previous numerical data, m S = m 0. 99999699.
- 17.
Newton elaborates on the approximate nature of such an inertial system and on the conceptual difficulty of identifying an inertial system other than as mathematical postulate: …Motus quidem veros corporum singulorum cognoscere et ab apparentibus actu discriminare difficillium est; propterea quod partes spatii illius immobilis in quo corpora vere moventur, non incurrunt in sensus, [123, §11, page 52]. Despite Newton’s attempt to ground mechanics on a purely rationalistic basis, mechanical phenomena are based on observations that are true only within some order of approximation. This interplay between rational mechanics and experimental mechanics was clear in Galileo’s thinking: …Prendiamo per ora questo come postulato, la verità assoluta del quale ci verrà poi stabilita dal vedere altre conclusioni, fabbricate sopra questa ipotesi, rispondere e puntualmente confrontarsi con l’esperienza… [61].
- 18.
The average speed of the center of Earth about the Sun is 29. 8 km/s, or 2. 98 ⋅106 cm/s [76, F–145]. Therefore \(\|{\mathbf{v}}_{\Sigma }(O)\| \approx 3 \cdot 1{0}^{6}\)cm/s. The direction of v Σ (O) is determined by the trajectory of O according to Kepler’s first law. Since Earth completes a self-revolution about u 3 in one sidereal day, e.g., 86, 164 s [76, F–103–105; F–146], one computes \(\omega = 2\pi /86,164\,\)s\({}^{-1} = 7.292 \cdot 1{0}^{-5}\,\)s− 1.
- 19.
This is the centripetal acceleration of O directed as Ω − O. Let R denote the average distance from Earth to the Sun. Its numerical value is R = 149, 5 ⋅106 km, or 1, 495 ⋅1013 cm [76, F–145]. Therefore, assuming that the trajectory is approximately circular, by (2.6) of Chapter 1, \(\|{\mathbf{a}}_{\Sigma }(O)\| =\|{ \mathbf{v}}_{\Sigma }{(O)\|}^{2}/R = 0.6\,\)cm/s2. This value is less than one-thousandth of the mean acceleration of gravity g = 9. 83225 m/s2 [76, F–147–148].
- 20.
For a range of values of g from the equator to the poles in terms of latitude, as well as in terms of altitude from sea level, see [76, F–133, F–151, F–158].
- 21.
Such a level might or might not be attainable. See (7.8) and the related remarks.
- 22.
The dynamic viscosity is a measure of a resistance offered by a fluid when forced to change its shape. It is a sort of internal friction measured as the resistance elicited by two ideal parallel planes immersed in the fluid when forced into a mutual sliding motion. The unit of measure is the poise, after J.L.M. Poiseuille. It is measured in dyne/s per cm2 and is the force distributed tangentially on a planar surface of 1 cm2, needed to cause a variation of velocity of 1 cm/s between two ideal parallel planes immersed in the fluid and separated by a distance of 1 cm. For water at 20∘C, the dynamic viscosity is 0.01002 poise. The kinematic viscosity is the ratio of the dynamic viscosity to the density of the fluid. The c.g.s. unit of kinematic viscosity is the stoke, after G.G. Stokes. Numerical values of dynamic and kinematic viscosity for several fluids are in [76, F–36–45.].
- 23.
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DiBenedetto, E. (2011). Dynamics of a Point Mass. In: Classical Mechanics. Cornerstones. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4648-6_3
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