Structure of the analytical physical theories

de Oliveira, Mário J

doi:10.1590/1806-9126-RBEF-2021-0160

Mário J de Oliveira ^** Correspondence email address: oliveira@if.usp.br

Universidade de São Paulo, Instituto de Física, Rua do Matão, 1371, 05508-090 São Paulo, SP, Brasil.

https://orcid.org/0000-0002-4139-1445

* Correspondence email address: oliveira@if.usp.br

Thumbnail

Figure 1
QLN is the trajectory of a body acted by a central force towards the center C. The point K is close to the point L. The distance CR and CK are equal and RP is perpendicular to LK.

Thumbnail

Figure 2
An illustration of Hydrodynamics [¹⁷17. D. Bernoulli, Hydrodynamica (Dulseckeri, Argentorati, 1738).] used in the demonstration of the Bernoulli equation. The vessel ABGC is kept full of water, and o is the hole through which the water escapes at constant velocity.

Thumbnail

Figure 3
A particle following the trajectory GAH is acted at A by a force in the direction AE. The line segment BA is tangent to the trajectory at A, and AO is perpendicular to BA. The point O is the center of the osculating circle.

Thumbnail

Figure 4
A particle following the trajectory FAG is acted at A by a force in the direction AC. The line segment DAB is tangent to the trajectory at A and AO is perpendicular to DAB. The point O is the center of the osculating circle.

Thumbnail

Figure 5
A particle following the trajectory BDAM is acted at D by a force in the vertical direction DF. The line segment DG is tangent to the trajectory at D and OD is perpendicular to DG. The point O is the center of the osculating circle.

Thumbnail

Figure 6
The particles oscillate along a straight line with the same period and the same amplitude. The arrangements of particles are shown at intervals of time equal to one fourth of the period of oscillations, from top to bottom. The vertical dotted lines indicate the central point of oscillation of each particle and they are equally spaced. The pulses are indicated by ellipses and correspond to regions where the particles are closer to each other.

Thumbnail

Figure 7
The wavy line represents the Fourier expansion of the square wave given by equation () truncated at the tenth term.

Thumbnail

Figure 8
The wires 𝒜 and ℬ carry electrical currents I and I′, respectively. The length of AC is ds, of BD is ds′, the angle CAB is θ′, the angle ABD is θ, and the angle CAE or FBD is γ.

Thumbnail

Figure 9
The line ECD is a bended wire carrying an electric current. The distance from its vertex C to the ring at A is ℓ. The distances BG, AG and AB are x, y and r, respectively. If the ring is replaced by a magnet, then the figure also illustrates the arrangement of the Biot experiment.

Thumbnail

Figure 10
The point C is a source of light waves that is partially intercepted by the opaque body GA. Two trajectories from C to a point P of the screen. Fringes are produced at the screen and are explained by the difference in the lengths of the two paths. The geometric shadow occurs at the left of point B.

Thumbnail

Figure 11
Intensity I of diffraction fringes produced by a straight edge as a function of the distance x to the limit of the geometric edge.

Thumbnail

Figure 12
The incident light, polarized along the plane PP′, traverses a crystalline plate with principal section represented by OO′ and a rhomb of calcite with principal section SS′. The incident beam is divided into two beams polarized in the CO and CE directions.

Thumbnail

Table 1
The author of a scientific theory is listed together with the abbreviated name of the main work where it is found, the year of its public presentation or publication, the subject of the theory, the main concepts including the primary concepts, and references.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

(59)

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68)

(69)

(70)

(71)

(72)

(73)

(74)

(75)

(76)

(77)

(78)

(79)

(80)

(81)

(82)

(83)

(84)

(85)

(86)

(87)

(88)

(89)

(90)

(91)

(92)

(93)

(94)

(95)

(96)

(97)

(98)

(99)

(100)

(101)

(102)

(103)

(104)

(105)

(106)

(107)

(108)

(109)

(110)

(111)

(112)

(113)

(114)

(115)

(116)

(117)

(118)

(119)

(120)

(121)

(122)

(123)

(124)

(125)

(126)

(127)

(128)

(129)

(130)

(131)

(132)

(133)

(134)

(135)

(136)

(137)

(138)

(139)

(140)

(141)

(142)

(143)

(144)

(145)

(146)

(147)

(148)

(149)

(150)

(151)

(152)

(153)

(154)

(155)

(156)

(157)

(158)

(159)

(160)

(161)

(162)

(163)

(164)

(165)

(166)

(167)

(168)

(169)

(170)

(171)

(172)

(173)

(174)

(175)

(176)

(177)

(178)

(179)

(180)

(181)

(182)

(183)

(184)

(185)

(186)

(187)

(188)

(189)

(190)

(191)

(192)

(193)

(194)

(195)

(196)

(197)

(198)

(199)

(200)

(201)

(202)

(203)

(204)

(205)

(206)

(207)

(208)

(209)

(210)

(211)

(212)

(213)

(214)

(215)

(216)

(217)

(218)

(219)

(220)

(221)

(222)

(223)

(224)

(225)

(226)

(227)

(228)

(229)

(230)

(231)

(232)

(233)

(234)

(235)

(236)

(237)

(238)

(239)

(240)

(241)

(242)

(243)

(244)

(245)

(246)

(247)

(248)

(249)

(250)

(251)

(252)

(253)

(254)

(255)

(256)

(257)

(258)

(259)

(260)

(261)

(262)

(263)

(264)

(265)

(266)

(267)

(268)

(269)

(270)

(271)

(272)

(273)

(274)

(275)

(276)

(277)

(278)

(279)

(280)

(281)

(282)

(283)

Author	Work	Year	Subject	Concepts	References
Varignon	Motion in general	1700	particle dynamics	space, time, force	[¹²12. P. Varignon, Mémoires de l’Académie Royale des Sciences, année 1700, p. 22 (1760)., ¹³13. P. Varignon, Mémoires de l’Académie Royale des Sciences, année 1700, p. 83 (1760)., ¹⁴14. P. Varignon, Mémoires de l’Académie Royale des Sciences, année 1700, p. 224 (1760).]
Euler	Mechanics	1736	particle dynamics	space, time, force	[²³23. L. Euler, Mechanica sive Motus Scientia Analytice Exposita (Academiae Scientiarum, Petropoli, 1736).]
D. Bernoulli	Hydrodynamics	1738	hydrodynamics	hydrodynamic pressure	[¹⁷17. D. Bernoulli, Hydrodynamica (Dulseckeri, Argentorati, 1738)., ¹⁸18. D. Bernoulli and J. Bernoulli, Hydrodynamics and Hydraulics (Dover, New York, 1968)., ¹⁹19. D. Bernoulli, Hydrodynamique (Blanchard, Paris, 2004).]
Euler	Principles of fluid motion	1757	fluid mechanics	pressure, density	[²⁸28. L. Euler, Mémoires de l’académie des sciences de Berlin 11, 217 (1757)., ²⁶26. L. Euler, Mémoires de l’académie des sciences de Berlin 11, 274 (1757)., ²⁷27. L. Euler, Mémoires de l’académie des sciences de Berlin 11, 316 (1757).]
Euler, Lagrange	various papers	1759	sound propagation	pressure, density	[^{36; 37; 38; 39; 40; 41}36. J.L. Lagrange, Miscellanea Taurinensia 1, 1 (1759).]
Lagrange	Analytical Mechanics	1788	mechanics	space, time, mass, force	[³¹31. J.L. Lagrange, Méchanique Analitique (Veuve Desaint, Paris, 1788).]
Laplace	Celestial Mechanics	1799	mechanics	space, time, mass, force	[⁵⁷57. P.S. Laplace, Traité de Mécanique Céleste (Duprat, Paris, 1799), tome premier.]
Laplace, Biot, Poisson	Theory of sound	1808	sound propagation	pressure, density	[^{64; 65; 66; 67}64. P.S. Laplace, Annales de Chimie et de Physique 3, 238 (1816).]
Poisson	Electricity distribution	1812	electricity	electric fluid	[⁷³73. S.D. Poisson, Mémoires de la Classe de Sciences Mathématiques et Physiques de l’Institut Impérial de France, première partie année 1811, premiére partie, p. 1 (1812)., ⁷⁴74. S.D. Poisson, Mémoires de la Classe de Sciences Mathématiques et Physiques de l’Institut Impérial de France, seconde partie année 1811, premiére partie, p. 183 (1812).]
Fresnel	Diffraction of light	1819	light diffraction	light wave	[⁸⁷87. A. Fresnel, Mémoires de l’Académie Royale des Sciences de l’Institut de France, années 1820 et 1821, 5, 339 (1826).]
Fresnel	Polarization of light	1821	light polarization	transverse light wave	[⁹⁰90. A. Fresnel, Annales de Chimie et de Physique 17, 102 (1821).]
Fourier	Analytical Theory of Heat	1822	heat propagation	heat, temperature	[⁶⁹69. J. Fourier, Théorie Analytique de la Chaleur (Firmin Didot, Paris, 1822).]
Poisson	Theory of magnetism	1826	magnetism	magnetic fluid	[⁷⁶76. S.D. Poisson, Mémoires de l’Académie Royale de Sciences Mathématiques et Physiques de l’Institut de France annees 1821 et 1822, 5, 247 (1826)., ⁷⁷77. S.D. Poisson, Mémoires de l’Académie Royale de Sciences Mathématiques et Physiques de l’Institut de France annees 1821 et 1822, 5, 448 (1826).]
Ampère	Electrodynamic phenomena	1826	electrodynamics	electric current	[⁷⁹79. A.M. Ampère, Théorie des Phénomènes Électro-Dynamiques (Méquignon-Marvis, Paris, 1826).]
Navier	Laws of fluid motion	1827	fluid mechanics	pressure, density, viscosity	[⁹²92. C.L. Navier, Mémoires de l’Académie Royale des Sciences de l’Institut de France, année 1823, 6, 389 (1827).]

(\frac{d^{2} v}{d t^{2}}) = c^{2} (\frac{d^{2} v}{d x^{2}}) + c^{2} (\frac{d^{2} v}{d y^{2}}) + c^{2} (\frac{d^{2} v}{d z^{2}}) .

\sum m (\frac{d^{2} x}{d t^{2}} δ x + \frac{d^{2} y}{d t^{2}} δ y + \frac{d^{2} z}{d t^{2}} δ z) + \sum (P δ p + Q δ q + R δ r) = 0,

\sum (m \frac{d^{2} x}{d t^{2}} + X) δ x + \sum (m \frac{d^{2} y}{d t^{2}} + Y) δ y + \sum (m \frac{d^{2} z}{d t^{2}} + Z) δ z = 0 .

\sum m (x \frac{d^{2} y}{d t^{2}} - y \frac{d^{2} x}{d t^{2}}) + \sum (x Y - y X) = 0,

\sum m (z \frac{d^{2} x}{d t^{2}} - x \frac{d^{2} z}{d t^{2}}) + \sum (z X - x Z) = 0,

\sum m (y \frac{d^{2} z}{d t^{2}} - z \frac{d^{2} y}{d t^{2}}) + \sum (y Z - z Y) = 0,

\sum m (\frac{d^{2} x}{d t^{2}} d x + \frac{d^{2} y}{d t^{2}} d y + \frac{d^{2} z}{d t^{2}} d z) + d V = 0 .

d^{2} x δ x + d^{2} y δ y + d^{2} z δ z = d (d x δ x + d y δ y + d z δ z) - \frac{1}{2} δ (d x^{2} + d y^{2} + d z^{2}),

d \sum m (\frac{d x}{d t} δ x + \frac{d y}{d t} δ y + \frac{d z}{d t} δ z) - δ \sum m u d s = 0,

d x δ x + d y δ y + d z δ z = A d ξ δ ξ + B d ψ δ ψ + C d φ δ φ + D d ξ δ ψ + E d ξ δ φ + F d ψ δ φ + D d ψ δ ξ + E d φ δ ξ + F d φ δ ψ,

d^{2} x δ x + d^{2} y δ y + d^{2} z δ z = - \frac{δ α}{δ ξ} δ ξ - \frac{δ α}{δ ψ} δ ψ - \frac{δ α}{δ φ} δ φ + d (\frac{δ α}{δ d ξ}) δ ξ + d (\frac{δ α}{δ d ψ}) δ ψ + d (\frac{δ α}{δ d φ}) δ φ,

\sum m (\frac{d^{2} x}{d t^{2}} δ x + \frac{d^{2} y}{d t^{2}} δ y + \frac{d^{2} z}{d t^{2}} δ z) = (d \frac{δ T}{δ d ξ} - \frac{δ T}{δ ξ}) δ ξ + (d \frac{δ T}{δ d ψ} - \frac{δ T}{δ ψ}) δ ψ + (d \frac{δ T}{δ d φ} - \frac{δ T}{δ φ}) δ φ,

d \sum m (\frac{d x}{d t} δ x + \frac{d y}{d t} δ y + \frac{d z}{d t} δ z) - δ (T - V) d t = 0 .

δ x (d \frac{d x}{d t} - P d t) + δ y (d \frac{d y}{d t} - Q d t) + δ z (d \frac{d z}{d t} - R d t) = 0,

\frac{x_{0}}{r_{0}^{3}} = \frac{X}{R^{3}} - \frac{m_{1} x}{m R^{3}} + \frac{3 m_{1} X (X x + Y y + Z z)}{m R^{5}},

\frac{x_{1}}{r_{1}^{3}} = \frac{X}{R^{3}} + \frac{m_{0} x}{m R^{3}} - \frac{3 m_{0} X (X x + Y y + Z z)}{m R^{5}} .

(\frac{d^{2} V}{d θ^{2}}) + \frac{\cos θ}{\sin θ} (\frac{d V}{d θ}) + \frac{1}{\sin^{2} θ} (\frac{d^{2} V}{d ω^{2}}) + r (\frac{d^{2} r V}{d r^{2}}) = 0 .

\frac{1}{\sin ω} \frac{\partial}{\partial ω} (\sin ω \frac{\partial U_{n}}{\partial ω}) + \frac{1}{\sin^{2} ω} \frac{\partial^{2} U_{n}}{\partial φ^{2}} + n (n + 1) U_{n} = 0,

I = - \int (\frac{d^{2}}{d X^{2}} \frac{1}{R} + \frac{d^{2}}{d Y^{2}} \frac{1}{R} + \frac{d^{2}}{d Z^{2}} \frac{1}{R}) d X d Y d Z,

Q = \frac{1}{R^{3}} (α α^{'} + β β^{'} + γ γ^{'}) - \frac{3}{2 R^{5}} (X α + Y β + Z γ) (X α^{'} + Y β^{'} + Z γ^{'}) .

C = \frac{1}{r^{3}} \int (y^{'} d x^{'} - x^{'} d y^{'}) + \frac{3}{r^{5}} \int (x^{2} x^{'} d y^{'} - y^{2} y^{'} d x^{'}) .

r δ r = \frac{d δ x}{d x} α^{2} + \frac{d δ y}{d y} β^{2} + \frac{d δ z}{d z} γ^{2} + \frac{d δ x}{d y} α β + \frac{d δ y}{d x} α β + \frac{d δ x}{d z} α γ + \frac{d δ z}{d x} α γ + \frac{d δ y}{d z} β γ + \frac{d δ z}{d y} β γ .

ρ (\frac{d u}{d t} + u \frac{d u}{d x} + v \frac{d u}{d y} + w \frac{d u}{d z}) = P - \frac{d p}{d x} + ε (\frac{d^{2} u}{d x^{2}} + \frac{d^{2} u}{d y^{2}} + \frac{d^{2} u}{d z^{2}}),

ρ (\frac{d v}{d t} + u \frac{d v}{d x} + v \frac{d v}{d y} + w \frac{d v}{d z}) = Q - \frac{d p}{d y} + ε (\frac{d^{2} v}{d x^{2}} + \frac{d^{2} v}{d y^{2}} + \frac{d^{2} v}{d z^{2}}),

ρ (\frac{d w}{d t} + u \frac{d w}{d x} + v \frac{d w}{d y} + w \frac{d w}{d z}) = R - \frac{d p}{d z} + ε (\frac{d^{2} w}{d x^{2}} + \frac{d^{2} w}{d y^{2}} + \frac{d^{2} w}{d z^{2}}) .

[1] * Correspondence email address: oliveira@if.usp.br