) Lopes

) Le Mouël

) Courtillot

) Gibert

On the shoulders of Laplace

 can be considered as the central result of the paper. It shows that the sum of forces of the four Jovian planets matches in a striking way the polar motion reconstructed with SSA components (the Markowitz trend removed). All our results argue that significant parts of Earth's polar motion are a consequence (rather than a cause) of the evolution of planetary ephemerids. The Sun's activity and many geophysical indices show the same signatures, including many climate indices. Two different mechanisms (causal chains) are likely at work: a direct one from the Jovian Planets to Earth, another from planetary motions to the solar dynamo; variations in solar activity would in turn influence meteorological and climatic phenomena. Given the remarkable coincidence between the quasi-periods of many of these phenomena, it is reasonable to assume that both causal chains are simultaneously at work. In that sense, it is not surprising to find the signatures of the Schwabe, Hale and Gleissberg cycles in many "Laplace" -11/3/21 -2 nd Revision 3/45 terrestrial phenomena, reflecting the characteristic periods of the combined motions of the Jovian planets.

-Introduction

On July 5 1687, Isaac Newton published the three volumes of his Principia Mathematica, in which he put on a firm ground the law of universal attraction and the general laws of mutual attraction of masses. In the following two centuries, a corpus of laws that explained the motions of celestial bodies was established and vindicated by observations. Foremost among these works, Laplace published his Traité de Mécanique Céleste (Treatise of Celestial Mechanics) in 1799.

Based on Newton's law and the fundamental principle of dynamics, he established the general equations that govern the motions of material bodies (Laplace, 1799, book 1, chapter 7, page 74, system (D)). This system of differential equations of first order was later given the names of Liouville and Euler. It establishes both the rotation and translation of the rotation axis of any celestial body, and in particular the Earth. These same equations can be found in Guinot (in Coulomb and Jobert, 1977, p. 530) and more recently in the reference book of Lambeck (2005, p. 31). They are recalled in Appendices 1 and 2 in their most general form. When the forces and the moments that act on Earth are taken to be zero (i.e. the right hand side of the equations is zero), the

solution for the axis is a free oscillation with a Euler period 1/σ of 306 days (using the known values of the mean angular velocity and axial and equatorial moments of inertia (σ =((C-A)/A)Ω)).

Based on observations made between June 1884 and November 1885, Chandler (1891a,b) obtained a value of 427 days for 1/σ. Data provided by the International Earth Rotation and Reference System Service (IERS) yield a 1/σ that has varied between 431 days in 1846 and 434 days in 2020.

Newcomb (1892) verified Chandler's observations and concluded that the Earth should be viewed as an elastic body submitted to oceanic stresses. For this, Love numbers were introduced [START_REF] Love | The yielding of the Earth to disturbing forces[END_REF]. As a result, the Liouville-Euler system (D) of Laplace was made less general. Hough (1895) reinforced the idea that what made the Chandler period 121 days longer than the "theoretical" value was the fact that the Earth behaved as an elastic body. Based on Poincaré's (1885) work on the stability of rotating fluids with a free surface, Hough showed that the period should decrease rather than increase if one did not take elasticity into account. Works in the following decades strengthened the notion that the fluid envelopes of Earth (ocean, atmosphere and mantle) acted on Earth's rotation axis. An increasingly precise theory was thus proposed, whereas observations seemed to be increasingly remote from predictions.

"Laplace" -11/3/21 -2 nd Revision 4/45 Two papers [START_REF] Peltier | Glacial-isostatic adjustment-I", The forward problem[END_REF][START_REF] Nakiboglu | Deglaciation effects on the rotation of the Earth[END_REF] further strengthened the theory of an elastic Earth whose rotation axis was influenced by both its internal and external fluid envelopes. An important concept was that of Global Isostatic Adjustment (GIA), in which the Earth has a visco-elastic response to stress (load) variations, that originated at the onset of the last ice age. Melting ice would lead to sea level rise and a reorganization of surface masses that eventually modified the inclination of the rotation pole. Rather than writing in a physically explicit way the forces implied in system (D), as done by Laplace and Poincaré, more or less complex "excitation functions" were introduced (Appendix 1).

We return to the founding work of [START_REF] Laplace | Traité de mécanique céleste[END_REF] to see how these problems can be tackled further. In what follows, we refer to volumes, chapters, pages and equation numbers in the original edition of the Traité de Mécanique Céleste. Throughout the Treatise, Laplace (1799) rigorously shows that, whatever the nature of the oceans and atmosphere, the only thing that influences the rotation of celestial bodies is the action of other celestial bodies. On page 347 (chapter 1, volume 5) [START_REF] Laplace | Traité de mécanique céleste[END_REF] writes (this quotation is given in the original French in Appendix 4): "We have shown that the mean rotation movement of Earth is uniform, assuming that the planet is entirely solid and we have just seen that the fluidity of the sea and of the atmosphere should not alter this result. It would seem that the motions that are excited by the Sun's heat, and from which the easterly winds are born should diminish the Earth's rotation: these winds blow between the tropics from west to east and their continued action on the sea, on the continents and on the mountains they encounter, should seem to weaken imperceptibly that rotation movement. But the principle of conservation of areas, shows to us that the total effect of the atmosphere on this movement must be insensible; for the solar heat in dilating equally the air in all directions, should not alter the sum of areas covered by the vector radii of each molecule of the Earth and of the atmosphere, and when multiplied respectively by the corresponding molecules; which requires that the rotation motion be not diminished. We are therefore assured that as the winds analyzed diminish this motion, the other movements of the atmosphere that occur beyond the tropics, accelerate it by the same amount. One can apply the same reasoning to earthquakes, and in general, to all that can shake the Earth in its interior and at its surface. Only the displacement of these parts can alter this motion; if, for instance a body placed at the pole, was transported to the equator; since the sum of areas must always remain the same, the earth's motion would be slightly diminished; but for it to be noticeable, one should suppose the occurrence of great changes in the Earth's constitution."

These views are also shared by [START_REF] Poincaré | Les méthodes nouvelles de la mécanique céleste[END_REF]. They seem to be different from modern "Laplace" -11/3/21 -2 nd Revision 5/45 views as synthesized for instance by [START_REF] Lambeck | The Earth's variable rotation: geophysical causes and consequences[END_REF]. These authors agree on the Liouville-Euler system (D for Laplace) of differential equations, but the forces that act on the Earth are different (and interpreted in a different way, as shown below). In the present paper, we attempt to check Laplace's full system using the observations that have accumulated and improved since Laplace's time (time series starting in 1750 for the oldest and no later than 1850 for the shortest ones).

We first discuss some of the core ideas of the paper, based on Laplace's original developments (section 2). We then recall some concepts and tools that we use in the paper and introduce the data, i.e. the coordinates of the Earth's rotation pole from 1846 to 2020 (section 3). In section 4, we establish a striking result that is central to the paper: the detrended polar motion is highly correlated with the sum of the forces exerted by the four Jovian planets. We next submit the data to Singular Spectral Analysis (SSA) and discuss the first SSA components (section 4): the Markowitz, annual and Chandler rotations. Then, in section 5, we discuss the SSA components of the derivatives of the three components above. In section 6, we give several other examples, such as the excellent correlation of the 40yr SSA component of the derivative of the envelope of the Chandler oscillation with the 40yr SSA component of the combined forces of Uranus and Neptune.

We end with a discussion and concluding remarks (section 7).

-Forces, Moments and the Liouville-Euler System of Equations

In most classical applications of the mechanics of planetary rotation, one uses only the first two components of the trio of Euler angles, i.e. the coordinates of the rotation pole at the Earth's surface (Figure A1, Appendix 1). The Earth rotates about the Sun (and so do the other 7 planets) in the ecliptic plane that is almost perpendicular to the rotation axis. The Sun carries more than 99% of the mass of the solar system, and can be considered rather motionless (its center of gravity actually travels along a "small" variable "ellipse"). In addition to the gravitational attractions, one must consider the orbital kinetic moments of all planets (in others words the moment of the momentum, see eq. B1, Appendix 2), as emphasized by [START_REF] Laplace | Traité de mécanique céleste[END_REF]. Planets carry more than 99% of the total angular momentum of the system (19.3, 7.8, 1.7 and 1.7 x 10 42 kgm 2 s -1 respectively for Jupiter, Saturn, Uranus and Neptune). This can be compared to the Sun's attraction at the Earth's orbit, 3.5 10 22 kg.m.s -2 , that can be transformed to the dimension of a kinetic moment by multiplying it by the Sun-Earth distance and the orbital revolution period of Earth, yielding 1.7 10 41 kg.m 2 .s -1 : that is not negligible compared to the order of magnitude of the kinetic moments of the Jovian planets (to 1 or 2 orders of magnitude).

"Laplace" -11/3/21 -2 nd Revision 6/45

The central idea of this paper is to analyze variations in the Earth's rotation axis under the influence not only, as in many classic treatments (e.g. Dehant and Mathiews, 2015, ch. 2), of gravitational potentials, but also of kinetic moments. The classical system of differential equations that describe the pole's motion (Liouville-Euler) links the sum of simple physical entities with their time derivatives, hence a first order linear system (Appendix 1, part 1). See [START_REF] Bode | Network analysis and feedback amplifier design[END_REF] for more on the definition and consequences that can be drawn from such linear systems. One is that causes and consequences are similar, up to a constant factor, if the system is not too dissipative and is maintained: this implies that gravitational potentials, kinetic moments (of Jovian planets) and polar motions should share characteristic features.

Polar motion is described by three coordinates, usually labeled m1, m2 and m3 (Appendix 1). If one only wants to study the perturbations due to the gravitational potential of a planet in rotation about itself, two coordinates, m1 and m2, are sufficient to describe the motion. In the case of our Solar system, planets revolve about the Sun in (or close to) the ecliptic plane; the moments they generate are perpendicular to that plane (Appendix 2). They act on the inclinations of the rotation axes of all planets, including Earth's. This is the well-known phenomenon of interaction of spinning tops and is adequately described by the Liouville-Euler equations. This was known to Laplace who chose not to use the three Euler angles, but gave all the analytic formulas that allow one to compute the inclination θ of the rotation axis (Figure A1 and Appendix 3) as a function of time, under the influence of the Moon and Sun [START_REF] Laplace | Traité de mécanique céleste[END_REF]book 5, page 317, number 5), and the time derivative of the declination ψ of the rotation axis [START_REF] Laplace | Traité de mécanique céleste[END_REF]book 5, page 318, number 6). θ and ψ are defined in Laplace (1799; book 1, page 73, number 26). [START_REF] Laplace | Traité de mécanique céleste[END_REF]book 5, pages 352-355, number 14) deduces that, when neither the Moon nor the Sun act on Earth (conjunction nodes), the time derivative of the declination (which in modern terms is the Euler period 1/σ) has a value of 306 days (Appendix 3). This value is fully determined by the Earth's moments of inertia When the Moon and Sun act with maximum effect (conjunction bellies) 1/σ reaches a value of 578 days. 1/σ therefore oscillates between 306 and 578 days; Chandler (1891) found a value of 427 days and today one observes values of 432-434 days. Both inclination θ and declination ψ drift.

-The Toolbox: Rotation Pole Data, Ephemerids, Commensurability and Singular

Spectrum analysis

Some of the tools and data needed to pursue our goal are listed in this section. Of course, we require knowledge of planetary ephemerids, that are given by the IMCCE. Then we need/use:

3-1 Rotation pole data: Laplace did not have sufficient observations to demonstrate the influence of planets, though he certainly did not deny their possible role. We now have sufficiently long series of observations to test his full theory.

The rotation pole is defined by its components m1 and m2, respectively on the Greenwich (0°) and 90°E meridians (Figure A1). Two series of measurements of (m1, m2) are provided by IERS 1 under the codes EOP-C01-IAU1980 and EOP-14-C04. The first one runs from 1846 to July 1 st 2020 with a sampling rate of 18.26 days, and the second runs from 1962 to July 1 st 2020 with daily sampling (also giving access to the length of day). Figure 1 shows the components m1 and m2 of the longer series (data are given in milli arc second -mas -and converted here in radians per secondrad.s -1 ). Figure 2 integer numerator and denominator less than 9 [START_REF] Mörth | Planetary motion, sunspots and climate[END_REF][START_REF] Okhlopkov | The gravitational influence of Venus, the Earth, and Jupiter on the 11-year cycle "Laplace" -11/3/21 -2 nd Revision 43/45 of solar activity[END_REF][START_REF] Scafetta | Solar Oscillations and the Orbital Invariant Inequalities of the Solar System[END_REF]. Planets encounter a resonance and can be paired, and each pair can be considered as a single object (an egregor or aggregate). Jupiter/Saturn and Uranus/Neptune form two pairs.

Pairs of pairs can also be considered, thus the set (Jupiter/Saturn)/(Uranus/Neptune). Many analyses of sunspot series [START_REF] Lassen | Variability of the solar cycle length during the past five centuries and the apparent association with terrestrial climate[END_REF]Hataway, 2015;[START_REF] Usoskin | Solar activity during the Holocene: the Hallstatt cycle and its consequence for grand minima and maxima[END_REF][START_REF] Le Mouël | Identification of Gleissberg cycles and a rising trend in a 315-year-long series of sunspot numbers[END_REF][START_REF] Stefani | A Model of a Tidally Synchronized Solar Dynamo[END_REF][START_REF] Courtillot | On the prediction of solar cycles[END_REF]Le Mouël et al., 2020a;Stefani et al., 2020) and of a number of geophysical phenomena [START_REF] Courtillot | Multi-Decadal Trends of Global Surface Temperature: A Broken Line with Alternating~ 30 yr Linear Segments?[END_REF][START_REF] Scafetta | High resolution coherence analysis between planetary and climate oscillations[END_REF]Lopes et al., 2017;[START_REF] Scafetta | Multiscale Analysis of the Instantaneous Eccentricity Oscillations of the Planets of the Solar System from 13 000 BC to 17 000 AD[END_REF][START_REF] Bignami | Are normal fault earthquakes due to elastic rebound or gravitational collapse?[END_REF]Le Mouël et al., 2019a;Le Mouël et al., 2019b;[START_REF] Hilgen | Paleoclimate records reveal elusive~ 200-kyr eccentricity cycle for the first time[END_REF]Le Mouël et al., 2020b;[START_REF] Zaccagnino | Tidal modulation of plate motions[END_REF][START_REF] Courtillot | On the prediction of solar cycles[END_REF] contain components with periods that can be attributed to Jovian planets to first order, and all planets including the telluric ones to second order [START_REF] Courtillot | On the prediction of solar cycles[END_REF]. Table 1 lists planetary commensurabilities following [START_REF] Mörth | Planetary motion, sunspots and climate[END_REF]. The periods found in our analysis of the SSA components of polar motion (section 4) and of the derivatives of their envelopes (section 5) are labeled in red (there are 8, ranging from 1.2 to 165 years).

Note: Inspection of Table 1 may give the impression that there is a risk of "cherry picking".

But certain periods that could have been reconstructed are not present, such as 103 yr that could have been obtained with Neptune. Commensurabilities are built from two consecutive planets and once their effect has been aggregated, they can be used in the next step of aggregation/commensurablity. The concept of commensurability is used by astronomers in order to discriminate between planets and other objects. The corresponding periods are not random: they are directly related to the revolutions of these bodies, and result from calculating means or subtracting periods two by two. Thus what can be obtained is not random. Moreover, as already pointed out by [START_REF] Mörth | Planetary motion, sunspots and climate[END_REF] or more recently [START_REF] Scafetta | High resolution coherence analysis between planetary and climate oscillations[END_REF], uncovering a limited number of common periods in a number of geophysical observables including sunspots cannot be due to chance. The action of kinetic moments of Jovian planets on the Sun's surface is what has allowed us to predict the next solar cycle from the ephemerids in a previous paper [START_REF] Courtillot | On the prediction of solar cycles[END_REF].

3-3 Singular spectral analysis:

Finally we extract the relevant components of polar motion and ephemerids, and other long time series, with the help of Singular spectral analysis (SSA; [START_REF] Vautard | Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series[END_REF][START_REF] Vautard | Singular-spectrum analysis: A toolkit for short, noisy chaotic signals[END_REF]; for more up to date versions of the technique by the St Petersburg school of mathematics, see [START_REF] Golyandina | Singular Spectrum Analysis for time series[END_REF]. We have described and used our own version of SSA in a number of previous papers (e.g. [START_REF] Lopes | The mantle rotation pole position. A solar component[END_REF][START_REF] Courtillot | On the prediction of solar cycles[END_REF][START_REF] Courtillot | On the prediction of solar cycles[END_REF].

"Laplace" -11/3/21 -2 nd Revision 10/45

We discuss here a point that often comes up. An important factor in any time series analysis is the size of the window used in classical (Fourier) filters, to avoid erroneous interpretations [START_REF] Kay | Spectrum analysis-a modern perspective[END_REF]. In SSA, the lagged-vector analysis window L should be sufficiently large so that each eigen vector carries a large part of the information contained in the original time series. In more mathematical words, one should work in the frame of Structural Total Least Squares (STLS)

for a Hankel Matrix [START_REF] Lemmerling | Analysis of the structured total least squares problem for Hankel/Toeplitz matrices[END_REF]. A second issue is the separability of components. Many solutions are available, an exhaustive list being given by Golyandina and Zhigljavsky (2013, chap. 2.5.3, page 75). In this paper, we have used the sequential SSA. The window width L is variable, but remains close to 145 years.

-First Results

A striking reconstruction -Planets action on Earth's rotation:

We start to test the ideas and use the tools summarized above by comparing the sum of the forces exerted by the four Jovian planets (using the IMCCE ephemerids) and the m1 component of polar motion (1846 to 2020) as reconstructed from its SSA components, with the trend removed.

The polar coordinates m1 and m2 are related to the forces acting on Earth (Appendix 1). To first order, we can consider that the total "force" is simply proportional to the sum of individual (Jovian) planetary kinetic moments, plus the Solar kinetic moment. We have computed these moments from the planetary ephemerids, revolution periods and masses; their sum is given as the top black curve in Figure 3a. The red curve below is the reconstructed m1 polar coordinate from

Figure 1, after it has been decomposed in its SSA components, then reconstructed from them, but with the first component (the trend, see sub-section 4.2) removed. Figure 3b shows an enlargement of the 1980-2019 part of Figure 3a. The correlation is quite striking. It is indeed expected, as already pointed out by Laplace (1799, book 5 in whole), that the Earth's rotation axis should undergo motions with components that carry the periods (and combinations of periods) of the Moon, Sun, and planets, particularly the Jovian planets as far as their kinetic moments are concerned (see also [START_REF] Mörth | Planetary motion, sunspots and climate[END_REF][START_REF] Mörth | Planetary motion, sunspots and climate[END_REF][START_REF] Courtillot | On the prediction of solar cycles[END_REF]. This first exercise demonstrates that one should indeed consider planetary kinetic moments when describing the motions of the Earth's rotation axis. Based on this remarkable result, the aim of the rest of this paper is to see whether characteristic components of the ephemerids are also found in Earth's polar motion and other related (or not obviously related) phenomena. We next analyze one by one the leading SSA components of the Earth's rotation pole coordinates.

First SSA Component (Markowitz)

The first SSA components, shown in Figure 4, correspond to the mean trend of polar motion called the Markowitz drift [START_REF] Markowitz | Concurrent astronomical observations for studying continental drift, polar motion, and the rotation of the Earth[END_REF]. The drift velocity is on the order of 13 cm/yr and is principally carried by the E-W component m2. As noted by a reviewer, these curves show changes in slope and inflection points that are reminiscent of the recent evolution of the Earth's global surface temperature (Le Mouël et al., 2019b, Figure 20). This important point is not discussed further in the present paper.

Second SSA Component (Annual)

The second SSA component is the forced annual oscillation (Figure 5). On that annual oscillation, Lambeck (2005, chapter 7, page 146) writes "The seasonal oscillation in the wobble is the annual term which has generally been attributed to a geographical distribution of mass associated with meteorological causes. Jeffreys in 1916 first attempted a detailed quantitative evaluation of this excitation function by considering the contributions from atmospheric and oceanic motion, of precipitation, of vegetation, and of a polar ice. Jeffreys concluded that these factors explain the observed annual polar motion, a conclusion that is still valid today".

Figure 5 shows that the annual components of m1 and m2 are significantly modulated, and in "Laplace" -11/3/21 -2 nd Revision 14/45 different ways (recall that the excitation functions are sums of sinuses and cosines with constant weights; Lambeck, 2005, page 153, equations 7.1.9). In the generally accepted theory, modulation is thought to be a response to reorganization of oceanic and atmospheric masses. We note in the modulation of m1 the suggestion of a periodicity on the order of 150 years or more that could correspond to the [START_REF] Jose | Sun's motion and sunspots[END_REF] 171.5 yr cycle. Note that, given uncertainties, the Jose cycle could actually be the Suess-de Vries ~200 yr cycle (Stefani et al. 2020). [START_REF] Hinderer | Geomagnetic secular variation, core motions and implications for the Earth's wobbles[END_REF][START_REF] Runcorn | The excitation of the Chandler wobble[END_REF][START_REF] Gibert | Wavelet analysis of the Chandler wobble[END_REF]Bellanger et al, 2001;[START_REF] Bellanger | A geomagnetic triggering of Chandler wobble phase jumps?[END_REF][START_REF] Gibert | Inversion of polar motion data: Chandler wobble, phase jumps, and geomagnetic jerks[END_REF]. The "Laplace" -11/3/21 -2 nd Revision 15/45 Component m1 is in red and m2 in blue.

Chandler oscillation extracted by SSA is similar to that obtained with wavelets by [START_REF] Gibert | Wavelet analysis of the Chandler wobble[END_REF]. It is also as regular as that obtained with SSA by [START_REF] Gorshkov | Manifestation of solar and geodynamic activity in the dynamics of the Earth's rotation[END_REF].

When the first three SSA components of m1 and m2 are added, they account for 73% of the original variance. The quality of that incomplete reconstruction is shown in Figure 7.

Pushing the SSA analysis further reveals an oscillation with period 1.22 yr with an 18.6 yr modulation (the nutation), one with period 1.15 yr with a symmetrical modulation as in the case of the Chandler term, one with period 1.10 yr. Some of these (quasi-) periods have already been found using SSA on time series of sunspots (Le Mouël et al, 2020a). These periods seem to be linked to the ephemerids of solar system planets, which has been used by [START_REF] Courtillot | On the prediction of solar cycles[END_REF] to predict the [START_REF] Usoskin | A history of solar activity over millennia[END_REF], but not all authors agree. Moreover, these components are found only in m2 "Laplace" -11/3/21 -2 nd Revision 16/45 and are much smaller in amplitude, on the order of 10 -13 to 10 -14 rad.s -1 vs 10 -10 to 10 -11 rad.s -1 for the first three [START_REF] Lopes | The mantle rotation pole position. A solar component[END_REF][START_REF] Japaridze | Study of the Periodicities of the Solar Differential Rotation[END_REF]. When all components from the trend to the Hale cycle are added, they account for 95% of the total variance of the original series. Except for the 1.10 and 1.15 yr components, all others are found in the table of planetary interactions (Table 1).

-On Some Derivatives of SSA Components of Polar Motion

System (D) expresses that there is a link between a force and the derivative of the resulting polar motion (Appendix 1, equation 2). In other words Earth acts as a natural integrator (Appendix 1, equation 2 implies that m is an integral of ξ; see Le Mouël et al, 2010). This leads us to analyze the derivatives of the first three (largest) SSA components identified in the previous section.

Markowitz Drift

We first calculate the derivative of the Markowitz drift (Figure 4), and analyze its major SSA components. They are a trend (Figure 8a), a 90 yr pseudo-cycle (Figure 8b), a 40 yr pseudo-cycle (Figure 8c), a 22 yr period and an 11 yr period (Figure 8d). 2017) have obtained a period of 90 ± 3 yr from sunspot series SN_m_tot_V2.02 . It corresponds to a characteristic period in the ephemerids of Uranus (Table 1). There is also a close correspondence of periods for the 11 yr oscillation (Figure 8d). The drift could be linked to the modulation and varying "periodicity" of sunspots (8 to 13 yr). This is close to a characteristic period of Jupiter's ephemeris. The trend (Figure 8a) could be linked to the [START_REF] Jose | Sun's motion and sunspots[END_REF] 171.5 yr cycle, attributed to Neptune (Table 1) or to the Suess-de Vries ~200 yr cycle (Stefani et al., 2020).

Envelope of the Forced Annual Oscillation

We next turn to the derivative of the envelope of the forced annual oscillation (Figure 5). Its first SSA component, the trend, is shown in Figure 9a. For m1 this trend is compatible with a little more than one period of a sine curve with a period close to 170 yr, that is the Jose (1965) solar cycle, corresponding to the ephemeris of Neptune (or again given uncertainties to the Suess-de Vries ~200 yr cycle). The next SSA component is a 70 yr cycle for m1 and a 60 yr cycle for m2 (Figure 9b). These periods, or pseudo-periods, are among those resulting from combinations of ephemerids of the Jovian planets [START_REF] Mörth | Planetary motion, sunspots and climate[END_REF][START_REF] Scafetta | Solar Oscillations and the Orbital Invariant Inequalities of the Solar System[END_REF]; Table 1). The 60 yr cycle had already been found in sunspot series by [START_REF] Scafetta | Empirical evidence for a celestial origin of the climate oscillations and its implications[END_REF] and Le Mouël et al (2020a).

We had also seen it as an important component of series of global temperature and PDO and AMO oceanic indices [START_REF] Courtillot | Multi-Decadal Trends of Global Surface Temperature: A Broken Line with Alternating~ 30 yr Linear Segments?[END_REF].

Free Chandler Oscillation

We now undertake the SSA analysis of the derivative of the envelope of the Chandler oscillation (Figure 6). We find components with periods 70, 40, 30 and 22 yr (Figures 10a to 10d).

It is remarkable that the components for m1 and m2 are quasi-identical and have a very regular behavior, close to sine functions but with some slower modulation: they could be described as "astronomical" (as opposed to "astrophysical", as defined by [START_REF] Mayaud | Derivation, meaning, and use of geomagnetic indices[END_REF].

-Further Examples

We can illustrate further how Jovian planets influence polar motion with the combined effects of the pair Uranus (84 yr) -Neptune (165 yr): this pair has revolution periods compatible with the envelopes in Figures 5, 6 and8b. Figure 11a shows the sum of the kinetic moments of these two Finally, in Figures 12a to 12c, we superimpose the signatures (components) of the ephemeris of Jovian planets on the components of polar motion. In Figure 12a, the 90 yr component of the envelope of m2 matches the ephemeris of Uranus offset by 32 years. In Figure 12b, the 165 yr component of the envelope of m1 matches the ephemeris of Neptune, also offset by 32 years. In 8d) has a variable phase drift with respect to the ephemeris. But, whereas "solar" components (periodicities) do appear at 22 and 11 yr (and 5.5 yr?) in polar motions, they are 3 to 4 orders of magnitude smaller than the leading components we discuss here.

We have seen that the sum of the Markowitz drift, annual oscillation and Chandler oscillation explain some 70% of polar motion. The same is true for the leading components of sunspots, i.e. the sum of the trend (Jose ~171.5 yr cycle), Schwabe cycle (~11 yr) and Gleissberg cycle (~90 yr) (on the same time range). These periods correspond to those of Neptune (~165 yr), Uranus (84 yr ) and Jupiter (11.8 yr ).

Many if not most of the (quasi-)periods found in the SSA components of polar motion, of their modulations, of their derivatives can be associated with the Jovian planets. Only one, the 432-434 day period is due to the Earth's mass and moments of inertia and not to the Jovian planets, as predicted by Laplace (1799).

-Summary, Discussion and Conclusion

The general laws that govern the motions of celestial bodies have been derived and discussed by [START_REF] Laplace | Traité de mécanique céleste[END_REF] in his remarkable Traité de Mécanique Céleste. Laplace established the system of linear differential equations now known as the Liouville-Euler equations. He provided the full set of equations for the three Euler angles that specify the motions of a body's axis of rotation. Laplace differs from most later authors in the way he uses the Liouville-Euler system. Laplace makes full use of the system (D) for a rotating body that undergoes both rotation and translation, and solves the algebraic transcendant equations of Appendix 3, given all astronomical parameters. Most others use a simplified version with the formalism of excitation functions (Appendix 1, equation 2; a second order system) in which the possibility of a translation of the body's rotation axis is denied.

When Laplace obtains system (D) on page 74 of Chapter 7 of Book I, after 7 chapters that led him to these equations, he recognizes the fact that the system accounts for rotation as well as translation of a rotating body's polar axis. When Lambeck (for instance) follows the same route, his Chapter 3 (entitled «Rotational Dynamic») on page 30 begins with the following sentence : «The fundamental equations governing the rotation of a body are Euler's dynamical equations».

Lambeck links the angular momentum to the torque that generated it. One means only rotation: that would be valid if the Earth's inclination were zero or a constant. The equations are the same, but one soon forgets that the momentum that is the source of the torque (Lambeck's system (3.1), page 30) is a 3D vector (with no reason to be restricted to 2D, since the Earth is neither flat, nor is its inclination constant; its rotation axis revolves about the Sun and is therefore subjected at least to our star's kinetic momentum). This oversight has some consequences. Since one only considers rotations, not translations, then the (Chandler) free rotation is obtained by zero-ing all torques and disregarding the third equation for the m3 polar coordinate (that is assumed constant). Then, the forced annual oscillation cannot be due to the revolution of Earth about the Sun and one must find causes for these forced oscillations (the excitation functions). Laplace of course knew that polar coordinates m1 and m2 were connected to m3. Therefore, Laplace did not constrain polar motion to the two surface components (m1, m2) but represented it by two meaningful components, the axis' inclination θ and the time derivative of its declination ψ depends on the inclination "Laplace" -11/3/21 -2 nd Revision 28/45

(previously calculated as a solution of the first Liouville-Euler equation). Laplace showed that there existed a free oscillation that would drift with a period between 306 (conjunction nodes) and 578 days (conjunction bellies), fully determined by the Earth's moments of inertia. This free oscillation, the Chandler oscillation, has a current value of 432-434 days. We now have long time series, up to a couple of centuries long, available and we use series of coordinates of the rotation pole m1 and m2

(Figure 1) to extend some of Laplace's (1799) results. A simple Fourier transform (Figure 2) shows the dominant spectral lines at 1 yr (forced annual oscillation) and 1.19 yr (free Chandler oscillation).

Singular spectral analysis (SSA) allows to better characterize the three leading components, the trend (~13cm/yr) called the (free) Markowitz drift (Figure 4), then the (forced) annual oscillation (showing different modulations for m1 and m2, Figure 5) and the Chandler oscillation (with a very large modulation and a phase change in 1930, similar for m1 and m2, Figure 6). Under the current theory, modulation is thought to be a response to reorganization of oceanic and atmospheric masses (e.g. Lambeck, 2005, chapter 7). Taken together, the first three SSA components explain 73% of the signal's total variance (Figure 7). The smaller components that follow have (pseudo-) periods of 1.22 (with an 18.6 yr modulation), 1.15 and 1.10 yr. Some of these periods have been encountered in sunspot series and in the ephemerids of Jovian planets (Le Mouël et al., 2020a;[START_REF] Courtillot | On the prediction of solar cycles[END_REF].

We have next analyzed in the same way the envelopes of the derivatives of the first three SSA components of polar motion (Figure 8). We find a trend in the derivative of the Markowitz drift, that could also correspond to the 171. One can think in Laplace's terms that the kinetic moments of planets act directly on Earth, or that these moments act on the external layers of the Sun (which is a fluid mass) and perturb its rotation, hence its revolution and kinetic moment M (Appendix 1), eventually affecting the Earth's 1799) has shown that one should consider the orbital kinetic moments of all planets and that the Earth's rotation axis should undergo motions with the signatures/periods of the Sun and planets: the moments of the Jovian planets range from 1.7 to 19.3 10 42 kg.m 2 .s -1 , and for the Sun, an equivalent is 1.7 10 41 kg.m 2 .s -1 . To first order, the total kinetic moment applied to the Earth's rotation axis is simply the sum of individual (Jovian) planetary kinetic moments plus the Solar kinetic moment.

We have shown (Figure 3) that the m1 component of polar motion reconstructed with SSA, with the Markowitz trend removed, matches remarkably well the sum of kinetic moments of the four Jovian planets. We have also computed these kinetic moments from the planetary ephemerids of Uranus and Neptune (Figure 11a). They "predict" remarkably well (Figure 11b) the 40 yr SSA component of the derivative of the envelope of the Chandler oscillation (Figure 11d).

We have previously determined the characteristic SSA components of solar activity, using sunspot numbers as a proxy (Le Mouël et al., 2019b). The sum of the Markowitz drift, annual oscillation and Chandler oscillation explain over 70% of polar motion. The same is true for sunspots, on the same time range, regarding the sum of the trend (Jose ~171.5 yr), Schwabe (~11 yr) and Gleissberg (~90 yr) cycles. These periods correspond to those of Neptune (~165 yr), Uranus (~90 yr) and Jupiter (~11 yr). We have superimposed the signatures (components) of the ephemerids of Jovian planets on the components of polar motion. The 90 yr component of the envelope of m2 matches the ephemerids of Uranus, offset by 32 years (Figure 12a). The 165 yr component of the envelope of m1 matches the ephemerids of Neptune, also offset by 32 years (Figure 12b). And the 30 yr component of the envelope of m1 of the Chandler oscillation matches the ephemerids of Saturn, offset by 15 years (Figure 12c).

We have followed [START_REF] Mörth | Planetary motion, sunspots and climate[END_REF], who determined the commensurable periods of pairs and pairs of pairs of Jovian planets ( 2019d). This has led to attempts to increase the complexity of the model, such as the forcing by climate or the visco-elastic response to glacial isostatic rebound. We have seen that this theory uses only 2 of the 3 Euler angles. By using the full system of equations in the Liouville-Euler system (D for Laplace), Laplace (1799) was able to go beyond the synthetic treatments of (for instance) Guinot (1977) or [START_REF] Lambeck | The Earth's variable rotation: geophysical causes and consequences[END_REF]. We have seen in this paper numerous applications of this theory that explain many pseudo-periodic components of a number of geophysical (and solar) phenomena, making the leading role of planetary ephemeris clear.

The shorter periods (months to a few decades) often show as modulations of even shorter variations. And trends, with about 200 years of data, are possibly due to periods in the ephemeris comparable to or longer than the range of available observations. Still, these 200 years allow us to test Laplace's work further than he himself could. We have for instance been able to use this formalism to predict the future evolution of solar Cycle 25 [START_REF] Courtillot | On the prediction of solar cycles[END_REF].

It is widely assumed that both forced and free oscillations of Earth can, at least in part, be associated with climate forcings. Such has been the case from Jeffreys (1916) to [START_REF] Lambeck | The Earth's variable rotation: geophysical causes and consequences[END_REF], and recently to [START_REF] Zotov | On modulations of the Chandler wobble excitation[END_REF] and [START_REF] Zotov | A possible interrelation between Earth rotation and climatic variability at decadal time-scale[END_REF]. In all these works, causality is absent, be it from a time perspective or based on the orders of magnitude of the forces required to perturb the Earth's rotation. The periods that for instance [START_REF] Zotov | A possible interrelation between Earth rotation and climatic variability at decadal time-scale[END_REF] associate with an interaction between Earth's fluid and rigid envelopes are found in other geophysical phenomena such as the Earth's magnetic field or sunspots (Le Mouël et al., 2019a,b,c,d ;Le Mouël et al., 2020a,b ;[START_REF] Courtillot | On the prediction of solar cycles[END_REF][START_REF] Courtillot | On the prediction of solar cycles[END_REF]and references therein). We have come to the same conclusion regarding many climatic indices (Le Mouël et al., 2019d). If there is a good correlation of many characteristic periods, pseudo-periods and components extracted with SSA, for instance between Earth's rotation and many features of climate, it is reasonable to assume that this "Laplace" -11/3/21 -2 nd Revision 31/45 is because they are subject to some common forcings. This is not an overly speculative hypothesis:

with the views of Laplace on tides, we know that the fluid envelopes react on short time scales (to changes in the Moon's declination for 2/3rds and the Sun for 1/3 rd ). On longer time scales, the whole (including solid) Earth responds (e.g. Dehant et Mathiews, 2015), all being governed by the Liouville-Euler equations.

In the present study, we have been able to find planetary signatures in polar motions, strictly based on observational data and using only classical mechanics. A possible causal chain thus emerges that has gravity potential and kinetic moments of planets acting directly or modulating motions of the fluid parts of celestial bodies, i.e. the Sun's outer layers (sunspots) and the Earth's atmosphere and ocean. These effects are in general not yet modeled: this is a domain where climate modeling warrants significant research advances.

In summary and conclusion of this work, two different mechanisms (causal chains) are likely at work. One is illustrated by the spectacular and direct effect of the kinetic moments of the (Jovian) planets on the Chandler wobble, whose intrinsic period (somewhere between 306 and 578 days) is synchronized to 433 days (a value that depends on Earth properties). The causal chain is directly from the Jovian Planets to Earth. Another causal chain would be an effect of planetary motions on the solar dynamo; variations in solar activity would in turn influence meteorological and climatic phenomena, such as mass transport between the equator and the poles, length of day, sea-level,...

Given the remarkable coincidence between the quasi-periods of many of these phenomena, it is reasonable to assume that both causal chains are simultaneously at work. In that sense, it is not surprising to find the signatures of the Schwabe, Hale and Gleissberg cycles in many terrestrial phenomena, reflecting the characteristic periods of the combined motions of the Jovian planets.

  (i.e. the internal distribution of masses). The equations derived by Laplace are: "Laplace" -11/3/21 -2 nd Revision 7/45 All celestial and terrestrial parameters in these equations are defined in Appendix 2. The time variation of declination of the Earth's rotation pole is a function of inclination. Since the (θ ,ψ) and (m1, m2) couples represent the same physics, the pattern of the sum of planetary kinetic moments that "force" part of the Earth's polar motions should be found in m1 and m2 (see below). Laplace obtained these equations taking into account "only" the Moon and Sun.

45 FFigure 1 :Figure 2 :

 4512 Figure 1: Components (m1, m2) of polar motion since 1846 (time series EOP-C01-IAU1980)

"

  Figure 3a: Upper curve (in black) the sum of the forces of the four Jovian planets affecting Earth. 289 Ephemerids from the IMCCE. Lower curve (in red) the m1 component of polar motion (1846-2020) 290 reconstructed with SSA and with the trend (Markowitz) removed. 291

Figure 5 :

 5 Figure 5: 2nd SSA comp. of polar motion (annual oscillation) since 1846. (m1 red, m2 blue).4.4 Third SSA Component (Chandler) Figure 6 shows the third SSA component, that is the Chandler component. Its amplitude is twice that of the annual component and its behavior is very different. The modulations are very large, similar for m1 and m2, and undergo a sharp and simultaneous change in phase and amplitude in 1930. Many scientists have studied this phase change[START_REF] Hinderer | Geomagnetic secular variation, core motions and implications for the Earth's wobbles[END_REF][START_REF] Runcorn | The excitation of the Chandler wobble[END_REF]

Figure 6 :

 6 Figure 6: Third SSA components of polar motion (Chandler oscillation) since 1846.

Figure 7 :

 7 Figure 7: Reconstruction of polar motion since 1846 using only its first three SSA components. Top: observed component m1 in black and reconstructed in red; bottom: observed component m2 in black and reconstructed in blue.

  Figure 8a: First SSA component (trend) of the derivative of the Markowitz drift (first SSA component of polar motion). Component m1 in red and m2 in blue.

Finally

  , the 40 yr component has been shown byMörth and Shlamminger (1979; see also[START_REF] Courtillot | On the prediction of solar cycles[END_REF] to correspond to a commensurable revolution period of the four Jovian planets. It is interesting to point out that in both terrestrial polar motion and solar activity (as studied through the proxy of sunspots) the first 3 components that emerge from SSA are a trend, then the Gleissberg and Schwabe quasi-cycles.

Figure 8b :

 8b Figure 8b: Second SSA component (90 yr period)) of the derivative of the Markowitz drift (first SSA component of polar motion). Component m1 in red and m2 in blue. In black: Gleissberg cycle extracted from sunspots (sign reversed).

Figure 8c :

 8c Figure 8c: Third SSA component (40 yr period)) of the derivative of the Markowitz drift (first SSA component of polar motion). Component m1 in red and m2 in blue.

Figure 8d :

 8d Figure 8d: 22 yr SSA component (top, component m2 in blue) and 11 yr SSA component (bottom, component m2 in blue) of the derivative of the Markowitz drift (first SSA component of polar motion). Bottom, black curve: the 11 yr Schwabe cycle extracted by SSA from the sunspot series (sign reversed).

"

  Figure 9a: First SSA component (trend) of the derivative of the envelope of SSA component 2 496 (annual oscillation) of polar motion. 497 498

  Figure 10a: First SSA component (70 yr quasi-period) of the derivative of the envelope of the Chandler oscillation.

Figure 10c :

 10c Figure 10c: Third SSA component (30 yr quasi-period) of the derivative of the envelope of the Chandler oscillation

Figure 10d :

 10d Figure 10d: Fourth SSA component (22 yr quasi-period) of the derivative of the envelope of the Chandler oscillation

Figure 11c :

 11c Figure 11c: Second component of the Uranus-Neptune pair (top) and forced annual oscillation of the polar motion m1 (bottom).

Figure 12c ,

 12c Figure 12c, the 30 yr component of the envelope of m1 of the Chandler oscillation matches the ephemeris of Saturn offset by 15 years. The 11 yr component detected in the m2 component of the derivative of the Markowitz drift (Figure 8d) has a variable phase drift with respect to the

"

  Figure 12a: Superimposition of the ~90 yr SSA component of the envelope of m2 (blue curve; see

  5 yr Jose cycle (associated with the ephemeris of Neptune) or the ~200 yr Suess-de Vries cycle). Next, a 90 yr component, reminiscent of the Gleissberg solar cycle (associated with the ephemeris of Uranus), a 40 yr component, corresponding to a commensurable revolution period of the four Jovian planets, a 22 yr and an 11 yr component, that can be associated with Jupiter and/or the Sun. For the modulation of the annual component of polar motion, SSA finds periods of 165, 70 and 60 years (Figure 9). The 60 yr component has been found in sunspots, global temperature of Earth's surface, and the oceanic oscillation patterns PDO and AMO (and Saturn). Finally, for the Chandler component, excellent matches are found for m1 and m2 with periods of 70, 40, 30 and 22 years (Figure 10).

Table 1

 1 that is to calculate the inclination θ and declination ψ following Laplace's (1799) full treatment of the equations (Appendix 2). Since the Liouville-Euler equations are linear differential equations of first order, we have been able to use the frame of small perturbations and we have considered that the influence of planets can be taken as the sum of individual influences. When one works within this theoretical frame, there remain unexplained observations such as the 434 day value of the current period of the Chandler wobble or the 6 month component of oceanic indices(Le Mouël et al.,

	): we find that 8 of them, ranging from

that starts with the Sun and Moon, and continues with the Jovian planets. It would be satisfying to undertake a rigorous demonstration of the influence of all planets on the Sun and on the Earth's

Observatoire royal de Belgique, http://www.sidc.be/silso/datafiles

Acknowledgements: We thank two anonymous reviewers for very useful comments on the original draft of this paper. V.C. acknowledges input from Georges Consolo. IPGP Contribution no

Appendix 1: Polar Coordinates and Excitation Functions

Figure A1 The reference system for the pole (m1 and m2).

Figure A1 gives the notations for the reference system that we use. The rotation pole is defined by its components m1 and m2, respectively on the Greenwich (0°) and 90°E meridians. We follow Lambeck's (2005, chapter 3) formalism. The rotation of the pole ω can be decomposed into three Euler angles (ω1, ω2, ω3) associated with the axes (X1, X2, X3) of the fixed reference frame.

These Euler angles are a function of the Earth's mean angular velocity Ω and of the apparent position of the pole at the Earth's surface m1, m2, m3 :

Noting that the Earth rotates about its axis and that its radius is constant, the Liouville-Euler system of equations (D for Laplace, 1799) becomes (1/σ is the Euler period): One sees the rotation as the sum of two oscillations, one intrinsic to the Earth linked to the constant 3m/4n, that varies like (1+λ).m.cos θ for all the nodes of luni-solar orbits, and is therefore a function of inclination θ, and another one forced by the Moon and Sun, linked to longitudes (ft+ζ) and (f't+ζ') and to right ascensions ν and ν' of these two orbs. Laplace therefore can estimate that the rotation period varies from 306 and 578 days.


Jean-Charles David¹
email: jean-charles.david@etudiant.univ-rennes2.fr.

Arnaud Buchet²

Jean-Noël Sialelli³

Sylvain Delouvée
email: sylvain.delouvee@univ-rennes2.fr.

The use of antibiotics in veterinary medicine: Representations of antibiotics and biosecurity by pig farmers

par les éleveurs de porcs

Introduction

The use of antibiotics in livestock farming: context and issues

According to the WHO (World Health Organisation), more than half of the antibiotics produced in the world are intended for animals and a large number of these medicines are used in industrial livestock farming. In fact, the rearing conditions characteristic of these facilities (a large number of animals in close proximity) lead to a high consumption of antibiotics and, at the same time, increase the risk of bacterial resistance [START_REF] Rushton | Antimicrobial Resistance: The Use of Antimicrobials in the Livestock Sector[END_REF][START_REF] Schwarz | Use of antimicrobial agents in veterinary medicine and food animal production[END_REF].

Moreover, multi-resistant bacteria originating from farming can be transmitted to humans as the same classes of antibiotics are used in human and veterinary medicine [START_REF] Bonnet | Thèse : Utilisation raisonnée des antibiotiques en élevage porcin. Démarche d'accompagnement dans sept élevages[END_REF]. The challenge is thus to preserve the effectiveness of antibiotics through a One Health approach [START_REF] Parodi | Une seule santé «One world, one health»: la place des vétérinaires[END_REF]. With this objective, the WHO and other international institutions are encouraging countries to implement strategies aimed at reducing the use of antibiotics in veterinary medicine.

In France, in line with these recommendations, the National Agency of Veterinary Medicines (Anses-ANMV) carries out an annual survey of sales of veterinary medicines containing antibiotics. The data collected are used to measure the level of exposure of animals to antibiotics (ALEA: Animal Level of Exposure to Antimicrobials). In November 2011, Ecoantibio (the national plan to reduce the risks of antimicrobial resistance in veterinary medicine) predicted a drop of 25% in this exposure over 5 years from 2012 to 2016. The Anses report of 2018 shows that the overall objective of the Ecoantibio plan was reached with a fall in the ALEA of 36.6% (41.5 % for pigs). Nevertheless, pig farms remain one of the sectors most exposed to antibiotics with an average ALEA of 0.623 (Anses, 2018). The objective of the new 2017-2021 Ecoantibio plan is to improve on these results by developing, for example, communication campaigns about antimicrobial resistance and training in biosecurity.

In addition, an Ifop study (2017) revealed that 49% of French people have little or no knowledge of the term "antimicrobial resistance" while 71% of them think that the public authorities do not communicate enough about this phenomenon. The report also indicated that although the French people are only partially familiar with antimicrobial resistance and its causes, their opinions on this topic are becoming fixed. For example, 76% are in favour of eating meat from animals reared without antibiotics. The expectations of consumers are thus carrying more and more weight and are encouraging farmers to change their practices, especially those enabling them to reduce their reliance on antibiotics. As a result, a large body of research in animal epidemiology is trying to identify the technical, health and structural factors influencing the use of antibiotics [START_REF] Van Der Fels-Klerx | Farm factors associated with the use of antibiotics in pig production[END_REF][START_REF] Chauvin | Usage des antibiotiques en filières porcine, avicole et cunicole en France[END_REF]. Among them, biosecurity figures as one of the most promising strategies in terms of reducing this use (Laanen et al., 2013;[START_REF] Postma | The biosecurity status and its associations with production and management characteristics in farrow-to-finish pig herds[END_REF]. Biosecurity includes actions such as cleaning/disinfecting farm buildings and changing clothing. These types of measures are now the subject of many training courses, particularly in the poultry and pig sectors.

However, in order to communicate effectively about the risks linked to antimicrobial resistance and provide training in biosecurity suitable for farmers, it is essential to understand the different perceptions those involved have of antibiotics and the actions required to reduce their use. Their opinions, attitudes and beliefs are all factors that should be taken into account in order to support farmers appropriately in these changes in practices.

The contribution of social representations to explaining and preventing the use of antibiotics

The concept of social representation was introduced by Serge Moscovici in 1961 in his book entitled Psychoanalysis, its image and its public. In this work, Moscovici defined representations as "forms of naive knowledge meant to organise behaviours and guide communications". These representations are composed of opinions, attitudes, beliefs and information related to an object or a situation (Lo Monaco, Delouvée & Rateau, 2016;[START_REF] Rateau | Social representation theory[END_REF]. The content of a representation as well as its organisation are determined by the individual and the social and ideological environment to which he/she belongs and are modulated by the nature and intensity of the links the individual has with this social system. Thus, a social representation enables individuals in the same group to understand, explain, and take a stand regarding a phenomenon in agreement with the values and ideas of the group in question. The representations have at least four functions in the social environment of a person: a communication function; an identity function in the determination, construction and conservation of the group identity as well as in intergroup relationships and the maintenance of social distance (Jodelet, 1989); a justification function in that they allow the positions taken and the attitudes toward the object of the representation to be justified; and lastly, a behavioural guidance function, they are "prescriptive of behaviours and practices. They define what is lawful, tolerable or inacceptable in a given situation" (Abric, 1994, p. 17).

Today, there are several theoretical orientations in the way of understanding social representations [START_REF] Rateau | Social representation theory[END_REF]. Among them is the sociodynamic approach of [START_REF] Doise | Les représentations sociales[END_REF], one of the principles of which is based on the anchoring process proposed by Moscovici (e.g. [START_REF] Clémence | Social positioning and social representations[END_REF]Doise et al., 1992 ;[START_REF] Palmonari | Le modèle socio-dynamique[END_REF]. Doise considers that "Social representations are principles that generate positions linked to specific insertions in a set of social relationships and organize the symbolic processes involved in these relationships" [START_REF] Doise | Les représentations sociales[END_REF]p.125). This model suggests examining the organization of relations between the social meta-system (i.e., the common positions taken about an object of representation), the cognitive system that corresponds to individual differences, and the social relations in which the positions are produced (Palomari & Emiliani, 2016).

This research therefore sought to study the anchoring points of farmers in relation to antibiotics and biosecurity at these three levels. The first step was to analyse the farmers' common positions on these issues. In a second step, the organizing principles of the individual or sub-group positions were studied. Finally, these position papers were discussed in the light of the context and issues currently surrounding the use of antibiotics in veterinary medicine.

Consequently, the main objective of this study was to highlight the social representations of antibiotics and biosecurity by analysing the associations of a group of farmers. Several factors were taken into account in this analysis: the geographical location of the farmers and their way of farming, namely their reliance on antibiotics.

Research Questions

Moliner (1993) and Bonetto & Lo Monaco (2018) consider that the epistemic and affiliation needs of individuals underlie the processes of construction of social representations. Given the context and the issues associated with the use of antibiotics in the livestock farming sectors (emergence of antimicrobial resistance, regulatory standards, expectations of consumers, societal pressure), we consider that antibiotics and biosecurity have all the necessary characteristics to be objects of social representation among pig farmers.

For these two objects, we hypothesised that their organization and their content would be determined by both the geographical location (French department) and the ALEA of the farms.

The geographical hypothesis refers to the idea of the greater or lesser proximity of these farms to the historic home of the cooperative: the Côtes-d'Armor. It was expected that the sociorepresentational universes of the farms situated within this territory would converge as regards the values, standards and attitudes asserted by the cooperative. The hypothesis taking into account the ALEA of farms is based on the link between the social representations and the practices of individuals in relation to the object considered.

Method

Participants

This research compared the social representations of farmers who belong to the same agricultural cooperative: Cooperl Arc Atlantique. This group, founded in 1966, specialises in the production of pigs and the processing of pork. The cooperative currently has 2,700 farmers and produces 5,800,000 pigs each year. Although over the years the group has become an agroindustrial complex of international stature, Cooperl Arc Atlantique was created on the initiative of a few farmers in the region of Lamballe (Côtes-d'Armor) seeking sustainability for the agricultural activities within their region. By founding this cooperative, these farmers hoped to improve their living conditions, put their production on a long-term footing, change the image of the farming world and curb the exodus of young people from the region. Cooperl Arc Atlantique was thus founded on regional values. Since then, in order to adapt to the changes taking place in agriculture (globalisation, environmental impact of activities, emergence of antimicrobial resistance), the cooperative has diversified its activities and extended its geographical location over all of western France. Gradually, farms in other departments have joined Cooperl Arc Atlantique. As a result, it is now a very mixed group of farms, in terms of livestock units, rearing practices and cultural affiliations, which must tackle antimicrobial resistance. The pig farms in our sample are representative of this diversity. They are situated in different French departments (Côtes-d'Armor, Ille-et-Vilaine, Morbihan, Loire-Atlantique, Mayenne, Manche; see Figure 1) and present variable ALEA. In views of these specific features, we believed that these farmers could highlight differences in the perception of antibiotics and biosecurity.

[INSERT FIGURE 1]

We administered the questionnaire from February to July of 2018; the participants completed the questionnaire in their own offices. Initially, we had at our disposal a sample composed of about a hundred pig farmers. However, some farmers did not consent to participate in the study, and others did not complete the questionnaire in full. Our sample consisted of 87 farmers from the Cooperl Arc Atlantique group: 8 women and 79 men (M = 48.82; SD = 9.55, min = 27; max = 67). The "gender" variable was not taken into account in the analysis because we met very few women. The women we met worked on farms based on a family model. Most of the time, they indicated that they were primarily involved in maternity care. They were selected by the farm veterinary surgeons depending on their ALEA and geographical location in order to provide a representative sample of farms from Cooperl. A large number of them were situated in Brittany.

The average ALEA of the sample was 0.51, i.e. lower than the average ALEA of the pig sector in France. The farms were divided into two classes: the first, called "low" ALEA, included farms with an ALEA lower than 0.50; the second, called "high" ALEA, included farms where the animals were exposed to an ALEA equal to or higher than 0.50. The ALEA cut-off of 0.50 was chosen according to the criteria usually followed by veterinary studies and reports of the National Agency for Food, Environmental and Occupational Health Safety (ANSES in French).

This leads, in our study, to a slightly unbalanced sample (there were 55 farms with a "low" ALEA and 32 with a "high" ALEA) but this allows a comparison with other studies in the field (a high ALEA is then identical).

[INSERT TABLE 1]

Materials

Procedure

After giving their consent to participate in the study, the farmers completed two free association tasks (Lo Monaco, Piermattéo, Rateau, & Tavani, 2017;Moliner & Lo Monaco, 2017): they had to produce the 4 words or expressions that came to mind when they thought of the stimulus "antibiotic" and then again for the stimulus "biosecurity". Thus, we were able to understand how the farmers constructed meanings about these two objects, as well as their discourse associated with their use of antibiotics (real, future or envisaged) and the actions that they carried out in terms of biosecurity. The data collected were then lemmatised. Lemmatisation refers to the lexical analysis of a corpus comprised of words of the same family but taking different forms (noun, plural, infinitive verb, etc.); for example, the words "disinfectant" and "to disinfect" found in our corpus were transformed into "disinfection".

Results

The reason we chose to ask farmers about "antibiotics" and "biosecurity" was, as mentioned in the introduction, because we assumed that the two terms related to different realities and perceptions, [START_REF] Abric | L'approche structurale des représentations sociales : développements récents[END_REF]Flament & Rouquette, 2003) although they should both be linked in the fight against antibiotic resistance. In order to identify the relations of proximities and distances maintained by our two objects of representations questioned on the basis of the examination of the associative answers relative to each of them, we have calculated a corpus similarity index called Ellegård's index (Ellegård, 1959). This index consists of dividing the number of themes common to two corpuses ("antibiotics" and "biosecurity"; numerator) by the square root of the total number of themes evoked for the first corpus ("antibiotics") multiplied by the total number of themes evoked for the second corpus ("biosecurity").

In line with the literature (e.g., Brunel et al., 2017;[START_REF] Di | Intergroup alliances and rejections within a protest movement (analysis of the social representations)[END_REF][START_REF] Di Giacomo | Alliance et rejets intergroupes au sein d'un mouvement de revendication[END_REF]Doise et al., 1992;Moliner & Lo Monaco, 2017;[START_REF] Robieux | L'espoir dans la maladie chronique: représentations sociales de l'espoir chez les patients et soignants[END_REF], we thus calculated a similarity index between our two corpuses. The Ellegård index revealed that farmers share only 25% of common themes between the two corpuses. In other words, they share a very weakly similar vocabulary in their social representations of antibiotics and biosecurity. Farmers with a high ALEA shared a greater similarity with farmers with a low ALEA regarding the term "antibiotic" only (r n =.53). This result was also found with biosecurity (r n =.51). As the common vocabulary seems higher, we will now see, in detail, the organization of the two objects of social representations.

Results for "antibiotics"

The corpus was composed of productions from 87 participants. These were categorised by the authors, independently, and using classical rules of content analysis [START_REF] Di | Intergroup alliances and rejections within a protest movement (analysis of the social representations)[END_REF]Lambert, Graham, & Fincham, 2009;[START_REF] Rosenberg | A method for investigating and representing a person's implicit theory of personality: Theodore Dreiser's view of people[END_REF]. In total, for all the participants, 337 verbal associations were collected as some subjects did not provide the 4 words expected. Once the corpus had been cleaned, 113 different semantic units were obtained (See Appendix 1 for the frequencies of the different categories).

In the same way as the work that is part of the socio-dynamic approach, all the associations were submitted to a Correspondence Factor Analysis (CORR. F. A., [START_REF] Benzécri | L'analyse des correspondances[END_REF][START_REF] Deschamps | Analyse des correspondances et variations des contenus des représentations sociales [Correspondence analysis and variations in the content of social representations[END_REF]Doise, Clémence & Lorenzi-Cioldi, 1992). This factorial analysis highlights the differences in terms of frequencies of behaviours relative to the independent variables. It gives access to a summary of the data by revealing a particular structure. It also enables identification of the most significant factorial axes. The axes are composed of the different modalities of the independent variables. Lastly, it emphasises the correspondences between the modalities of the independent variables and the behaviours associated by the participants. This analysis was performed on associations whose frequency was greater than 5 (N = 18, 76.93% of the corpus without unique terms) and using the departments and the ALEA as independent variables. The two first factors represented 60.33% of total inertia (i.e. Factor 1 = 32.41%; Factor 2 = 27.92%). Only the modalities of variables and the types of responses contributing to the construction of factors were retained. We retained the modalities or the types whose contribution per factor was higher than the average contribution [START_REF] Deschamps | Analyse des correspondances et variations des contenus des représentations sociales [Correspondence analysis and variations in the content of social representations[END_REF].

Thus, the modalities of variables that contributed to the construction of Factor 1 were the departments of Côtes-d'Armor (D22), Loire-Atlantique (D44), Manche (D50) and Morbihan (D56): CPF (D22) = .13 + CPF (D44) = .37 + CPF (D50) = .14 + CPF (D56) = .28. The total contribution to the definition of Factor 1 was .92 (i.e. 92%). Factor 2 was also constructed on the basis of different departments: CPF (D44) = .41 + CPF (D50) = .25 + CPF (D56) = .20, i.e. a contribution of 86% to the formation of Factor 2. Figure 2 illustrates this organisation.

[INSERT FIGURE 2]

Figure 2 shows that the geographical location of the farm (i.e. the department) determines the socio-representational field of antibiotics for the farmers. The Côtes-d'Armor centralises the socio-representational content of farmers for the object antibiotics with evocations such as animal and medicine in a very descriptive register. The graphical representation of the results of the CORR. F. A. shows that the departments of Loire-Atlantique, Manche and Morbihan are opposed on the two axes. The farmers of Morbihan share a socio-representational universe related to antimicrobial resistance, animal well-being and the label pwa (pork without antibiotics). In contrast, the departments of Loire-Atlantique and Manche give rise to evocations related to animals, the cost of antibiotics and alternative solutions enabling their use to be reduced. The results also show that the productions of participants, in contrast to what was expected, are not significantly associated with the consumption of antibiotics of farmers.

Results for "biosecurity"

The corpus was composed of the same participants as for the first phase of the study of productions from 87 participants. In total, for all the participants, 333 verbal associations were collected. Once the corpus had been cleaned, 111 different semantic units were obtained (See Appendix 2 for the frequencies of the different categories according to the variables taken into account).

All the associations were submitted to a Correspondence Factor Analysis (CORR. F. A., [START_REF] Benzécri | L'analyse des correspondances[END_REF][START_REF] Deschamps | Analyse des correspondances et variations des contenus des représentations sociales [Correspondence analysis and variations in the content of social representations[END_REF]. This analysis was performed on associations whose frequency was greater than 5 (N = 22, 76.93% of the corpus without hapax), and using the departments and the ALEA as independent variables. The two first factors represented 57.36% of total inertia (i.e. Factor 1 = 33.17%; Factor 2 = 24.19%). Only the modalities of variables and the types of responses contributing to the construction of factors were retained. We retained the modalities or the types whose contribution per factor was higher than the average contribution [START_REF] Deschamps | Analyse des correspondances et variations des contenus des représentations sociales [Correspondence analysis and variations in the content of social representations[END_REF]. Thus, the modalities of variables that contributed to the construction of Factor 1 were the departments of Manche (D50) and Morbihan (D56) as well as the ALEA: CPF (D50) = .31 + CPF (D56) = .24 + CPF (low ALEA) = .10* + CPF (high ALEA) = .16. The total contribution to the definition of Factor 1 was .81 (i.e. 81%). Factor 2 was constructed only from the variable "department": CPF (D22) = .14 + CPF (D35) = .44, i.e. a contribution of 58% to the formation of Factor 2. Figure 3 illustrates this organisation.

[ INSERT FIGURE 3] The departments of farms determine the productions of participants and the sociorepresentational field of biosecurity. For example, the department of Ille-et-Vilaine features evocations such as disinfection and footbath, whereas the farmers of Côtes-d'Armor (historical site of the creation of the cooperative) use ideas more associated with the health aspects of their farm, well-being at work and consumers. The ALEA of the farm also characterises the productions of farmers. Those who use large quantities of antibiotics mention words such as the constraints, vigilance, performance or standards surrounding this farming practice.

Discussion and conclusion

Our objective was to compare the discourse of farmers depending on their use of antibiotics (low ALEA vs. high ALEA) and the geographical location of their farm. Thus, we were firstly interested in the potential relationships between practices and social representations [START_REF] Abric | Pratiques sociales et représentations [Social practices and representations[END_REF]Jodelet, 1989;Moscovici, 1961). In fact, we investigated this link between the different farming practices related to the use of antibiotics and the social representations associated with them. Our results show that the ALEA of farms is not a determining factor in the representations of farmers regarding this object. Indeed, regardless of the level of consumption of antibiotics, the farmers refer to ideas opposing their use (e.g. alternative solutions, antimicrobial resistance, pwa). In other words, the large consumers of antibiotics do not seem to have particular cognitions within their socio-representational field that would subsequently lead to greater use.

On the contrary, the vast majority of farmers participating in this study produced discourse that demonstrates an awareness of the need to change certain farming practices (e.g. stopping the preventive use of antibiotics). Arguments of different types (economic, health, quality of life at work) appeared in the evocations of farmers. This diversity seems to be the sign of a successful socio-representational structure for this object. In fact, for several years, these farmers have been the target of awareness campaigns to convince them of the benefits of demedication. They have been informed of the risks linked to the "irrational" use of antibiotics, made aware of the economic and social issues inherent in demedication, told about alternative solutions and behaviours to implement in order to reduce this use, etc. The farmers are therefore in constant contact with many arguments inviting them to change some of their behaviours. Nevertheless, although these strategies have led to concrete action in terms of livestock management for some farmers, for others the shift from attitudes to behaviours has been less obvious. This is supported by the social representations of farmers regarding biosecurity. In our study, the ALEA of farms characterises the perception of biosecurity by farmers. When the ALEA is high, these farmers prioritise ideas against implementing biosecurity and their representations are characterised by constraints. They mention the regulatory standards and the attention required for this practice.

They perceive, at the same time, a pressure exerted by some institutions (European Union, consumers, associations), and the ever more demanding expectations concerning how they should organise their activity. An evaluative dimension runs through their socio-representational field in that biosecurity figures in their perceptions as a set of procedures that complicates the exercise of their activity, generating both financial and attentional costs. During discussions at the end of the questionnaire, the farmers mentioned this practice as "extra work" going against one of their main objectives: "their well-being". In fact, these representations may act as obstacles to implementing biosecurity measures on their farm. The temporal dimension, which also plays an important role in their discourse, gives weight to this hypothesis. Biosecurity is seen by these farmers as a future practice whose content remains vague at present as do the benefits (technical, economic, health) it is supposed to bring. Actually, for them, biosecurity is a farming issue for future generations. The farmers with a low ALEA are ideologically opposed to this type of representation. They produced more descriptive and functional evocations, using elements directly from their behaviour in terms of biosecurity. For example, they mentioned words such as "disinfection", "clothing", "footbath", etc. For these farmers, biosecurity consists of concrete measures, applicable as of now and generating appreciable effects in different areas of farming. Firstly, concerning health, these farmers perceive a gain in control over the occurrence of some events (e.g. bacterial proliferation, presence of a virus). Secondly, the feeling of working in a "hygienic" work environment increases their quality of life at work. In contrast to the large consumers of antibiotics, biosecurity for these farmers is currently a meaningful and beneficial practice. Their representational universe is thus focused on the reasons for observing the rules of biosecurity in farming.

Another objective of our research was to explore the potential link between the department of the farm and the representations of antibiotics. Certain results confirm our hypothesis that the perception of antibiotics by the farmer is partly determined by his/her regional roots. The territorial network of Cooperl Arc Atlantique extends over several departments and some are far from the historical home of the cooperative responsible for its history, values, farming standards, etc. In fact, we thought that some territories could give rise to variability in the perception of certain important social objects in farming. The farm's department seems to be a more important factor than its ALEA in predicting farmers' representations of antibiotics. Thus, and regardless of the consumption of antibiotics, the farmers situated in the Côtes-d'Armor produce similar discourse regarding antibiotics. They preferentially mention ideas related to the therapeutic dimension of antibiotics in veterinary medicine. Their representations may be described as "classic" in the sense that their associations are limited to the basic functions of antibiotics. They perceive them as a farming tool among others. The other departments differ in their discourse by stating the consequences of antibiotic use (e.g. "antimicrobial resistance", "cost") and solutions to remedy it. Thus, two types of territory can be contrasted for this object of representation: the Côtes-d'Armor farms that mention global and non-contextualised ideas vs. the farms situated, for example, in the Morbihan that produce discourse taking account of health (i.e. antimicrobial resistance) and economic (e.g. pwa label) issues linked to antibiotic consumption in the livestock farming sectors. The representations for the other departments recall the need to reduce this use and show a certain adherence to the demedication policy of the cooperative. These results seem paradoxical with regard to the relatively late integration of these farms into the cooperative. Our initial hypothesis therefore ran counter to this by thinking that the farmers located in the historic home of the cooperative would act as messengers advocating demedication. Although these farmers are not particularly averse to demedication, it is rather the departments like Morbihan (department in our sample characterised by a low ALEA) that develop a "positive" and supportive discourse on this topic. A historical explanation can be provided here. During its lifetime, the cooperative has faced many challenges, particularly related to the impact of its activity on the environment. The group's original farms (i.e. in Côtes-d'Armor) have therefore become accustomed during their professional experience to changing some of their practices in order to adapt to new contextual data. A phenomenon of habituation could have appeared for these farmers concerning the dynamic aspect of modern agricultural activities. The "more recent" farms of the cooperative have proportionally had to cope less with types of upheaval such as the emergence of antimicrobial resistance. Thus, the seriousness and the exceptional character of this phenomenon are undoubtedly more striking for these pig farms. This variability in the perception of the phenomenon could encourage them to become more actively involved in the field of ideologies.

This study thus reveals several elements. First, the discourse concerning antibiotics does not seem to determine the reliance on their use on farms. Overall, the farmers are all aware of the need to reduce the consumption of antibiotics in livestock farming. The campaigns of persuasion (carried out at national and organisational levels) have thus had real effects on farmers' cognitions. However, these effects are limited in terms of behaviour on many farms. This finding is supported by the representations of farmers regarding biosecurity. Biosecurity requires the implementation of very concrete behaviours in order to lower the consumption of antibiotics. We note that for this object, the ALEA characterises the discourse of farmers. For the large consumers of antibiotics, the representations are abstract and extremely evaluative. Although an ideological opposition to this practice is emerging for these farmers, they are unable to connect this attitude to direct experiences linked to biosecurity measures. These results encourage us to think that supporting farmers in the process of demedication should not only consist of persuasive communication. We believe that measures including communications based on these representations and preceded by the implementation of concrete measures related to demedication (e.g. commitment to carry out some biosecurity procedures on farms) would reduce the consumption of antibiotics within the cooperative more effectively. Thus, this persuasive communication would bridge the gap between the representations of farmersalready in favour of demedication -and the actions needed to achieve it.

Finally, these results have implications for prevention. Several studies show that farmers or slaughterhouse staff are occupationally exposed to resistant bacteria such as resistant Staphylococcus aureus methicillinus aureus (MRSA) [START_REF] Parisi | MRSA in swine, farmers and abattoir workers in Southern Italy[END_REF][START_REF] Van Gompel | Occupational Exposure and Carriage of Antimicrobial Resistance Genes (tetW, ermB) in Pig Slaughterhouse Workers[END_REF]. Resistant bacteria pose a risk if transmitted to these workers. Indeed, they can cause illnesses that are difficult to cure if effective antibiotic treatment is not available (Khachatourians, 1998;Michael et al., 2014). Biosecurity, on the other hand, makes it possible to limit the dissemination and transmission of these pathogens (Moss et al., 2012;[START_REF] Bragg | Bacterial resistance to quaternary ammonium compounds (QAC) disinfectants[END_REF]. In view of the representations collected during this study, it would be appropriate, in a One Health approach, to raise awareness and train farmers in biosecurity. Note: Grayed blocks refer to the experimental conditions. "Experimental conditions" contribute to the formation of Factor 1; "Experimental conditions" refer to the experimental conditions which contribute to the formation of Factor 2; "Experimental condition" refer to the experimental conditions which contribute to the formation of both Factors 1 and 2. "Observations" refer to the observations which contribute to the formation of Factor 1; "Observations" refer to the observations which contribute to the formation of Factor 2; "Observations" refer to the observations which contribute to the formation of both Factors 1 and 2.

* « psa » refers to a label Cooperl. This label concerns pigs without antibiotics.

Figure 1 .Figure 2 .

 12 Figure 1. Location of the pig farms in our sample

Figure 3 .

 3 Figure 3. Graphical representation of the results obtained by means of the CORR. F. A. concerning Factors 1 and 2.

Table 1 .

 1 Distribution of farmers based on the department and the ALEA class

		Number of farmers « Low » ALEA	« High » ALEA Average ALEA
	Côtes-d'Armor	45	29	16	0.58 (0.38)
	Ille-et-Vilaine	20	13	7	0.42 (0.39)
	Morbihan	15	9	6	0.34 (0.35)
	Loire-Atlantique	1	1	0	0.15 (*)
	Manche	3	2	1	0.37 (0.28)
	Mayenne	3	1	2	1.01 (0.89)

Appendix 2 -Farmers' evocations of the word "biosecurity" as a function of variables.

487

Evocations

Freq.

Côted'Armor