{
  "paper_id": "10.3176/phys.math.tech.1965.4.07",
  "source_pages": 3,
  "license_notice": "Translated from Russian. Original: Igor Keis, “Два решения задачи о движении гиростата, имеющего неподвижную точку”, Proceedings of the Academy of Sciences of the Estonian SSR. Physics, Mathematics and Technical Sciences Series, 1965, vol. 14 no. 4, pp. 552–554, https://doi.org/10.3176/phys.math.tech.1965.4.07. Published by Estonian Academy Publishers; license CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/) as deposited with Crossref by the publisher (the publisher's policy page names CC BY-NC 4.0). Changes: translated into English.",
  "title_en": "Two solutions of the problem of the motion of a gyrostat having a fixed point",
  "title_ru": "Два решения задачи о движении гиростата, имеющего неподвижную точку",
  "authors": [
    "I. Keis"
  ],
  "abstract_en": "The note gives two exact particular solutions for a gyrostat with a fixed point. For a heavy gyrostat whose mass and gyrostatic-moment vectors satisfy two quadratic relations, the Euler–Poisson equations admit a solution in which the angular velocity and the vertical rotate uniformly with a common phase; it corresponds to a regular nutation and generalizes the Grioli–Chikin motion. For a gyrostat in a Newtonian central field, under a stated relation between the parameters, the direction cosines are linear in the angular velocity (three Steklov-type partial integrals), and after a rescaling the equations reduce to Volterra's equations for the free motion of a gyrostat, whose general solution is then available in closed form.",
  "abstract_source": "generated",
  "translator_notes": [
    {
      "page": 1,
      "note": "The scan is of a two-sided sheet: the end of the preceding article shows through faintly (mirror-image) behind this page. Only this paper is translated. The journal masthead (Estonian and Russian) and the printed DOI line at the top are kept."
    },
    {
      "page": 1,
      "note": "Citations are printed as bracketed raised numbers, [¹], [²], …; reproduced as \\([^{1}]\\)."
    },
    {
      "page": 1,
      "note": "The cyclic-permutation tables printed in parentheses to the right of (1.1), (2.1) and the later displays are set here as small matrices in parentheses, as the source does."
    },
    {
      "page": 2,
      "note": "In the dimensionless variables the source prints \\tau = \\alpha^{+1/2} t with an explicit plus sign; kept."
    },
    {
      "page": 3,
      "note": "The Estonian summary is translated into English; the English summary is reproduced as printed (including its wording “Grioli-Chikin's one” and “Stekloff-type”). Reference 1's journal “ПММ” is Prikladnaya Matematika i Mekhanika."
    }
  ],
  "produced_with": "Claude Fable 5.1 (Claude Code), reading the scan rendered to 250 dpi PNG with macOS PDFKit; no OCR.",
  "body_md": "<!-- source-page: 001 -->\n\nEESTI NSV TEADUSTE AKADEEMIA TOIMETISED. XIV KÖIDE\nFÜÜSIKA-MATEMAATIKA- JA TEHNIKATEADUSTE SEERIA. 1965, NR. 4\n\nPROCEEDINGS OF THE ACADEMY OF SCIENCES OF THE ESTONIAN SSR. VOLUME XIV\nPHYSICS, MATHEMATICS AND TECHNICAL SCIENCES SERIES. 1965, No. 4\n\nhttps://doi.org/10.3176/phys.math.tech.1965.4.07\n\nI. KEIS\n\n# TWO SOLUTIONS OF THE PROBLEM OF THE MOTION OF A GYROSTAT HAVING A FIXED POINT\n\n**1.** The equations of motion of a heavy gyrostat fixed at a point have the form \\([^{1}]\\)\n\n\\[\nA\\frac{dp}{dt} + (C - B)\\,qr + k_3 q - k_2 r = Mg\\,(e_3\\gamma_2 - e_2\\gamma_3), \\qquad\n\\frac{d\\gamma_1}{dt} = r\\gamma_2 - q\\gamma_3\n\\quad \\begin{pmatrix} 1 & 2 & 3 \\\\ A & B & C \\\\ p & q & r \\end{pmatrix}.\n\\tag{1.1}\n\\]\n\nHere \\((e_1, e_2, e_3)\\) is the radius vector of the center of mass of the gyrostat; \\((k_1, k_2, k_3)\\) is the vector of the relative angular momentum, assumed constant in the body of the gyrostat; the remaining notation has the generally accepted meaning, and the equations of the system (1.1) that are not written out are obtained from the line given by the cyclic permutation indicated in the parentheses.\n\nWhen \\(e_2 = 0\\), \\(k_2 = 0\\), \\((B - C)e_1^2 = (A - B)e_3^2\\), \\((B - C)k_1^2 = (A - B)k_3^2\\), the equations (1.1) admit the particular solution\n\n\\[\n\\begin{gathered}\np = \\omega_0(\\lambda + \\mu\\cos\\varphi), \\qquad q = -\\omega_0\\sin\\varphi, \\qquad r = \\omega_0(\\mu - \\lambda\\cos\\varphi),\\\\[4pt]\n\\gamma_1 = \\frac{\\delta_1}{2}(1 - \\cos 2\\varphi) + \\delta_0\\cos\\varphi, \\qquad \\gamma_2 = -\\delta_0\\sin\\varphi + \\frac{\\delta_1}{2}\\sin 2\\varphi, \\qquad \\gamma_3 = \\delta_1\\cos\\varphi,\\\\[4pt]\n\\varphi = \\varphi_0 + \\omega_0(t - t_0).\n\\end{gathered}\n\\tag{1.2}\n\\]\n\nHere the following notation has been introduced:\n\n\\[\n\\begin{gathered}\n\\omega_0^2 D = Mg\\delta_1\\sqrt{e_1^2 + e_3^2}, \\qquad D = \\sqrt{(A - B)(B - C)},\\\\[4pt]\n\\lambda\\sqrt{A - C} = \\sqrt{A - B}, \\qquad \\mu\\sqrt{A - C} = \\sqrt{B - C},\\\\[4pt]\n\\delta_0 = K^{-1}\\bigl[(A + C - B)\\omega_0 + m\\bigr], \\qquad \\delta_1 = \\omega_0 D K^{-1}, \\qquad m^2 = k_1^2 + k_3^2,\\\\[4pt]\nK^2 = \\bigl[A^2 + AC + C^2 - B(A + C)\\bigr]\\omega_0^2 + 2(A + C - B)\\omega_0 m + m^2.\n\\end{gathered}\n\\]\n\nTo the solution (1.2) there corresponds a regular nutation \\([^{2}]\\) of the gyrostat\n\n<!-- source-page: 002 -->\n\n\\[\n\\psi = \\psi_0 = \\text{const}, \\qquad \\sin\\psi_0 = 0, \\qquad \\Theta = \\varphi = \\varphi_0 + \\omega_0(t - t_0),\n\\]\n\nwhich coincides with the motion established by Grioli \\([^{3}]\\) for a rigid body, but exists under initial conditions differing from those given in \\([^{4}]\\).\n\n**2.** If one considers the equations of motion of a gyrostat about a fixed point in a Newtonian field of forces \\([^{5}]\\)\n\n\\[\nA\\frac{dp}{dt} + (C - B)\\,qr + k_3 q - k_2 r = Mg\\,(e_3\\gamma_2 - e_2\\gamma_3) + \\frac{3g}{R}(C - B)\\gamma_2\\gamma_3\n\\quad \\begin{pmatrix} 1 & 2 & 3 \\\\ A & B & C \\\\ p & q & r \\end{pmatrix},\n\\tag{2.1}\n\\]\n\n\\[\n\\frac{d\\gamma_1}{dt} = r\\gamma_2 - q\\gamma_3,\n\\tag{2.2}\n\\]\n\nfor the case when the relations\n\n\\[\nA'(1 - B'C')(D - A)\\,\\alpha^{\\frac12} k_1 = A(1 + B'C')\\,Mge_1\n\\quad \\begin{pmatrix} 1 & 2 & 3 \\\\ A & B & C \\\\ A' & B' & C' \\end{pmatrix}\n\\]\n\nhold, then analogously to \\([^{6}]\\) one can show that the equations (2.1) and (2.2) admit the particular solution\n\n\\[\n\\gamma_1 = A'\\alpha^{-\\frac12}p + \\frac{MRe_1}{3(D - A)}, \\qquad\n\\gamma_2 = B'\\alpha^{-\\frac12}q + \\frac{MRe_2}{3(D - B)}, \\qquad\n\\gamma_3 = C'\\alpha^{-\\frac12}r + \\frac{MRe_3}{3(D - C)}.\n\\tag{2.3}\n\\]\n\nHere the notation introduced is\n\n\\[\n\\begin{gathered}\nA' = \\left|\\frac{u}{vw}\\right|^{\\frac12}\\operatorname{sgn} u, \\qquad\nB' = \\left|\\frac{v}{uw}\\right|^{\\frac12}\\operatorname{sgn} v, \\qquad\nC' = \\left|\\frac{w}{uv}\\right|^{\\frac12}\\operatorname{sgn} w, \\qquad\nD = A + B + C + \\frac{2C}{w - 1},\\\\[4pt]\n\\alpha = 3gR^{-1}, \\qquad Cu = C - A + Aw, \\qquad Cv = C - B + Bw,\n\\end{gathered}\n\\]\n\n\\(w\\) is an arbitrary constant.\n\nPassing in (2.2) to the dimensionless variables\n\n\\[\np' = \\alpha^{-\\frac12}p, \\qquad q' = \\alpha^{-\\frac12}q, \\qquad r' = \\alpha^{-\\frac12}r, \\qquad \\tau = \\alpha^{+\\frac12}t,\n\\]\n\nwe obtain, after the substitution (2.3), the equations\n\n\\[\nA'\\frac{dp'}{d\\tau} + (C' - B')\\,q'r' + \\frac{MR}{3}\\left(\\frac{e_3}{D - C}\\,q' - \\frac{e_2}{D - B}\\,r'\\right) = 0\n\\quad \\begin{pmatrix} 1 & 2 & 3 \\\\ A & B & C \\\\ A' & B' & C' \\\\ p' & q' & r' \\end{pmatrix}.\n\\]\n\nThe general solution of these equations is easily obtained from the formulas \\([^{7}]\\) for \\(p, q, r\\), if one replaces in them \\(A, B, C\\) by \\(A', B', C'\\); \\(m_1, m_2, m_3\\) by\n\n\\[\n3^{-1}MR(D - A)^{-1}e_1, \\qquad 3^{-1}MR(D - B)^{-1}e_2, \\qquad 3^{-1}MR(D - C)^{-1}e_3\n\\]\n\nand \\(p, q, r, t\\) by \\(p', q', r', \\tau\\).\n\n<!-- source-page: 003 -->\n\n## References\n\n1. V. V. Rumyantsev, PMM [Prikl. Mat. Mekh.], 25, No. 1, 9–16 (1961) (Russian).\n2. V. A. Chikin, Tr. Ryazansk. Radiotekhnich. In-ta, 4, 83–94 (1958) (Russian).\n3. G. Grioli, Ann. Mat., 24, No. 3–4, Ser. IV, 271–281 (1947).\n4. I. A. Keis, Vestn. MGU, Ser. I, Matematika, Mekhanika, No. 1, 76–79 (1964) (Russian).\n5. V. V. Beletskii, Dokl. AN SSSR, 113, No. 2, 287–290 (1957) (Russian).\n6. V. A. Stekloff, Comptes Rendus, 135, 526–528 (1902).\n7. V. Volterra, Acta Mathematica, 22, 201–358 (1898–1899).\n\nInstitute of Cybernetics,\nAcademy of Sciences of the Estonian SSR\n\nReceived May 7, 1965\n\n## Summary (Estonian)\n\nI. KEIS\n\n**Two particular solutions of the equations of motion of a gyrostat with a fixed point**\n\nThe first particular solution is a generalization of the Grioli–Chikin solution; the second particular solution, which contains the Steklov integrals, corresponds to Volterra's solution for the free motion of a gyrostat with a fixed point.\n\n## Summary (English, as printed)\n\nI. KEIS\n\n**TWO SOLUTIONS OF THE EQUATIONS CORRESPONDING TO THE MOTION OF A GYROSTAT WITH A SINGLE POINT FIXED**\n\nThe first precise partial solution presented in the paper is the generalization of the Grioli-Chikin's one, while the second, obtaining three partial integrals of the Stekloff-type, comes to the modification of Volterra's solution for the spontaneous motion of a gyrostat.\n"
}