Abstract
The problem of equimomental systems of a rigid body is considered. It is shown that the system of four equal masses located at the vertices of an isosceles tetrahedron is equimomental to a given rigid body. A system of material points of equal masses located at the vertices of an isosceles tetrahedron, which is equimomental to the nucleus of the 67P Churyumov—Gerasimenko comet, is constructed.
REFERENCES
E. J. Routh, The Elementary Part of a Treatise on the Dynamics of a System of Rigid Bodies: Being Part I. Of a Treatise on the Whole Subject (Macmillan, London, 1882).
W. L. Loudon, An Elementary Treatise on Rigid Dynamics (Macmillan, New York, 1896).
Ph. Franklin, “Equimomental systems,” Stud. Appl. Math. 8 (1–4), 129–140 (1929).
D. M. Y. Sommerville, “Equimomental tetrads of a rigid body,” Math. Notes 26, 10–11 (1930). https://doi.org/10.1017/S1757748900002127
A. Talbot, “Equimomental systems,” Math. Gaz. 36, 95–110 (1952). https://doi.org/10.2307/3610326
N. C. Huang, “Equimomental system of rigidly connected equal particles,” J. Guid., Control, Dyn. 16, 1194–1196 (1993). https://doi.org/10.2514/3.21150
F. J. Gil Chica, M. Pérez Polo, and M. Pérez Molina, “Note on an apparently forgotten theorem about solid rigid dynamics,” Eur. J. Phys. 35, 045003 (2014). https://doi.org/10.1088/0143-0807/35/4/045003
L. P. Laus and J. M. Selig, “Rigid body dynamics using equimomental systems of point-masses,” Acta Mech. 231, 221–236 (2020). https://doi.org/10.1007/s00707-019-02543-3
J. M. Selig, Geometric Fundamentals of Robotics, 2nd ed. (Springer-Verlag, Berlin, 2005).
J. M. Selig, “Equimomental systems and robot dynamics,” in Proc. IMA Mathematics of Robotics, Sept. 9–11, Oxford, 2015 (2015).
L. P. Laus and J. M. Selig, “Rigid body dynamics using equimomental systems of point-masses,” Acta Mech. 231, 221–236 (2020). https://doi.org/10.1007/s00707-019-02543-3
I. F. Sharygin, Problems in Solid Geometry (Science for Everyone) (Nauka, Moscow, 1984; Mir, Moscow, 1986).
Ya. P. Ponarin, Elementary Geometry, Vol. 2: Stereometry, Space Transformations (MTsNMO, Moscow, 2006) [in Russian].
E. A. Nikonova, “On stationary motions of an isosceles tetrahedron with a fixed point in the central force field,” Prikl. Mat. Mekh. 86, 153–168 (2022). https://doi.org/10.31857/S0032823522020096
A. A. Burov and E. A. Nikonova, “The generating function for the components of the Euler–Poinsot tensor,” Dokl. Phys. 66, 139–142 (2021). https://doi.org/10.1134/S1028335821050037
A. R. Dobrovolskis, “Inertia of any polyhedron,” Icarus 124, 698–704 (1996). https://doi.org/10.1006/icar.1996.0243
G. N. Duboshin, Celestial Mechanics. Fundamental Problems and Methods (Nauka, Moscow, 1968) [in Russian].
L. Meirovitch, Methods of Analytical Dynamics (McGraw-Hill, New York, 1970).
Yu. A. Arkhangel’skii, Analytical Dynamics of a Rigid Body (Nauka, Moscow, 1977) [in Russian].
L. Meirovitch, “On the effects of higher-order inertia integrals on the attitude stability of Earth-pointing satellites,” J. Astronaut. Sci. 15 (1), 14–18 (1968).
P. C. Sulikashvili, “The effect of third- and fourth-order moments of inertia on the motion of a solid,” J. Appl. Math. Mech. 51, 208–212 (1987). https://doi.org/10.1016/0021-8928(87)90066-9
R. S. Sulikashvili, “On the stationary motions in a Newtonian field of force of a body that admits of regular polyhedron symmetry groups,” J. Appl. Math. Mech. 53, 452–456 (1989). https://doi.org/10.1016/0021-8928(89)90051-8
K. R. Koch and F. A. Morrison, “Simple layer model of the geopotential from a combination of satellite and gravity data,” J. Geophys. Res. 75, 1483–1492 (1970). https://doi.org/10.1029/JB075i008p01483
H. J. Melosh, “Mascons and the Moon’s orientation,” Earth Planet. Sci. Lett. 25, 322–326 (1975). https://doi.org/10.1016/0012-821X(75)90248-4
T. G. G. Chanut, S. Aljbaae, and V. Carruba, “Mascon gravitation model using a shaped polyhedral source,” Mon. Not. R. Astron. Soc. 450, 3742–3749 (2015). https://doi.org/10.1093/mnras/stv845
P. T. Wittick and R. P. Russell, “Mascon models for small body gravity fields,” in Astrodynamics 2017: Proc. AAS/AIAA Astrodynamics Specialist Conf., Stevenson, Wash., Aug. 20–24, 2017 (Univelt, San Diego, Calif., 2018), in Ser.: Advances in Astronautical Sciences, Vol. 162, pp. 2003–2020.
R. Gaskell, L. Jorda, C. Capanna, S. Hviid, and P. Gutierrez, SPC SHAP5 Cartesian Plate Model for Comet 67P/C-G 6K PLATES, RO-C-MULTI-5-67P-SHAPEV2.0:CG_SPC_SHAP5_006K_CART, NASA Planetary Data System and ESA Planetary Science Archive (2017).
A. A. Burov and V. I. Nikonov, “Computation of attraction potential of asteroid (433) Eros with an accuracy up to the terms of the fourth order,” Dokl. Phys. 65, 164–168 (2020). https://doi.org/10.1134/S1028335820050080
A. A. Burov and V. I. Nikonov, “Sensitivity of the Euler–Poinsot tensor values to the choice of the body surface triangulation mesh,” Comput. Math. Math. Phys. 60, 1708–1720 (2020). https://doi.org/10.1134/S0965542520100061
A. A. Burov and V. I. Nikonov, “Inertial characteristics of higher orders and dynamics in a proximity of a small celestial body,” Nelineinaya Din. 16, 259–273 (2020). https://doi.org/10.20537/nd200203
A. A. Burov, A. D. Guerman, E. A. Nikonova, and V. I. Nikonov, “Approximation for attraction field of irregular celestial bodies using four massive points,” Acta Astronaut. 157, 225–232 (2019). https://doi.org/10.1016/j.actaastro.2018.11.030
A. A. Burov, A. D. German, and V. I. Nikonov, “Using the K-means method for aggregating the masses of elongated celestial bodies,” Cosmic Res. 57, 266– 271 (2019). https://doi.org/10.1134/S0010952519040026
A. A. Burov, A. D. German, E. A. Raspopova, and V. I. Nikonov, “On the use of the K-means algorithm for determination of mass distributions in dumbbell-like celestial bodies,” Nelineinaya Din. 14, 45–52 (2018). https://doi.org/10.20537/nd1801004
J. Lages, I. I. Shevchenko, and G. Rollin, “Chaotic dynamics around cometary nuclei,” Icarus 307, 391–399 (2018). https://doi.org/10.1016/j.icarus.2017.10.035
J. Lages, D. L. Shepelyansky, and I. I. Shevchenko, “Chaotic zones around rotating small bodies,” Astron. J. 153, 272 (2017). https://doi.org/10.3847/1538-3881/aa7203
S. A. Stern, H. A. Weaver, J. R. Spencer, C. B. Olkin, G. R. Gladstone, W. M. Grundy, J. M. Moore, D. P. Cruikshank, H. A. Elliott, W. B. McKinnon, J. W. Parker, A. J. Verbiscer, L. A. Young, D. A. Aguilar, J. M. Albers, T. Andert, J. P. Andrews, F. Bagenal, M. E. Banks, B. A. Bauer, J. A. Bauman, K. E. Bechtold, C. B. Beddingfield, N. Behrooz, K. B. Beisser, S. D. Benecchi, E. Bernardoni, R. A. Beyer, S. Bhaskaran, C. J. Bierson, R. P. Binzel, E. M. Birath, M. K. Bird, D. R. Boone, A. F. Bowman, V. J. Bray, D. T. Britt, E. L. Brown, M. R. Buckley, M. W. Buie, B. J. Buratti, L. M. Burke, S. S. Bushman, B. S. Carcich, A. L. Chaikin, C. L. Chavez, A. F. Cheng, E. J. Colwell, S. J. Conard, M. P. Conner, C. A. Conrad, J. C. Cook, S. B. Cooper, O. S. Custodio, C. M. Dalle Ore, C. C. Deboy, P. Dharmavaram, R. D. Dhingra, G. F. Dunn, A. M. Earle, A. F. Egan, J. Eisig, M. R. El-Maarry, C. Engelbrecht, Enke1 B. L., C. J. Ercol, E. D. Fattig, C. L. Ferrell, T. J. Finley, J. Firer, J. Fischetti, W. M. Folkner, M. N. Fosbury, G. H. Fountain, J. M. Freeze, L. Gabasova, L. S. Glaze, J. L. Green, G. A. Griffith, Y. Guo, M. Hahn, D. W. Hals, D. P. Hamilton, S. A. Hamilton, J. J. Hanley, A. Harch, K. A. Harmon, H. M. Hart, J. Hayes, C. B. Hersman, M. E. Hill, T. A. Hill, J. D. Hofgartner, M. E. Holdridge, M. Horányi, A. Hosadurga, A. D. Howard, C. J. A. Howett, S. E. Jaskulek, D. E. Jennings, J. R. Jensen, M. R. Jones, H. K. Kang, D. J. Katz, D. E. Kaufmann, J. J. Kavelaars, J. T. Keane, G. P. Keleher, M. Kinczyk, M. C. Kochte, P. Kollmann, S. M. Krimigis, G. L. Kruizinga, D. Y. Kusnierkiewicz, M. S. Lahr, T. R. Lauer, G. B. Lawrence, J. E. Lee, E. J. Lessac-Chenen, I. R. Linscott, C. M. Lisse, A. W. Lunsford, D. M. Mages, V. A. Mallder, N. P. Martin, B. H. May, D. J. McComas, R. L. McNutt, Jr., D. S. Mehoke, T. S. Mehoke, D. S. Nelson, H. D. Nguyen, J. I. N´u˜nez, A. C. Ocampo, W. M. Owen, G. K. Oxton, A. H. Parker, M. Pätzold, J. Y. Pelgrift, F. J. Pelletier, J. P. Pineau, M. R. Piquette, S. B. Porter, S. Protopapa, E. Quirico, J. A. Redfern, A. L. Regiec, H. J. Reitsema, D. C. Reuter, D. C. Richardson, J. E. Riedel, M. A. Ritterbush, S. J. Robbins, D. J. Rodgers, G. D. Rogers, D. M. Rose, P. E. Rosendall, K. D. Runyon, M. G. Ryschkewitsch, M. M. Saina, M. J. Salinas, P. M. Schenk, J. R. Scherrer, W. R. Schlei, B. Schmitt, D. J. Schultz, D. C. Schurr, F. Scipioni, R. L. Sepan, R. G. Shelton, M. R. Showalter, M. Simon, K. N. Singer, E. W. Stahlheber, D. R. Stanbridge, J. A. Stansberry, A. J. Steffl, D. F. Strobel, M. M. Stothoff, T. Stryk, J. R. Stuart, M. E. Summers, M. B. Tapley, A. Taylor, H. W. Taylor, R. M. Tedford, H. B. Throop, L. S. Turner, O. M. Umurhan, J. Van Eck, D. Velez, M. H. Versteeg, M. A. Vincent, R. W. Webbert, S. E. Weidner, G. E. Weigle, J. R. Wendel, O. L. White, K. E. Whittenburg, B. G. Williams, K. E. Williams, S. P. Williams, H. L. Winters, A. M. Zangari, and T. H. Zurbuchen, “Initial results from the New Horizons exploration of 2014 MU69, a small Kuiper Belt object,” Science 364, eaaw9771 (2019). https://doi.org/10.1126/science.aaw9771
G. Rollin, I. I. Shevchenko, and J. Lages, “Dynamical environments of MU69 and similar objects,” Icarus 357, 114178 (2021). https://doi.org/10.1016/j.icarus.2020.114178
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Nikonova, E.A. Isosceles Tetrahedron and an Equimomental System of a Rigid Body. Vestnik St.Petersb. Univ.Math. 56, 119–124 (2023). https://doi.org/10.1134/S1063454123010107
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DOI: https://doi.org/10.1134/S1063454123010107