EDITOR'S SUPPLEMENT TO REFERENCES

The Soviet journal Prikladnaia Matematika i Mekhanika (PMM) has been trans­lated into English on a cover-to-cover basis, by Pergamon Press, N. Y. and Oxford, beginning with Volume 22, 1958; therefore, all of the references to articles in PMM which are now available in English have been designated by *, and in some instances corrections in bibliographical data have been made and page numbers added. Two very important English language papers dealing with the problem of stability of bodies containing fluids are:

K. Stewartson, On the Stability of a Spinning Top Containing Liquid, J. Fluid Mechs., Vol. 5, part 4, 1959. E. H. Wedemeyer, The Unsteady Flow Within a Spinning Cylinder, J. Fluid Mechs., Vol. 20, part 3, 1964. A recently published monograph covering many of the practical aspects of liquid

behavior in moving containers is: H. N. Abramson (editor), The Dynamic Behavior 0/ Liquids in Moving Containers, SP-I06, NASA, Washington, D.C., 1966.

REFERENCES

1. Lagrange, J. L., Analytical Mechanics, Vol. 2 (French Transl.), Gostekhizdat, 1950.

2. Sommerfeld, A., Mechanics of Deformable Bodies, Vol. 3, Academic Press, New York, 1950.

3. Lur'ye, A. 1., Analiticheskaya mekhanika [Analytical Mechanies], Fizmatgiz, 1961.

4. Kochin, N. E., Vector and Tensor Analysis (in press), Chelsea Publishing Co., New York.

5. Rumyantsev, V. V., Uravneniya dvizheniya tverdogo tela, imeyushchego polosti, ne polnost'yu napolnennyye zhidkost'yu [Equations of Motion of a Rigid Body with Cavities not Completely Filled with a Fluid], PMM, Vol. 18, No. 6, 1954.

6. Kochin, N. E., I. A. Kibel and N. V. Rose, Theoretical Hydromechanics, Interscience Publishers, New York, 1964.

7. Gauss, K. F., Principia generalia theoriae figurae fluidorum in statu aequilibrii, Ges. Werke, Bd. 5, 1830 ..

8. Kirchhoff, G., Mechanics (German Transl.), Acad. Sci. SSSR, 1962. 9. Appel, P., Theoretical Mechanics (French Transl.), Vol. 2, Fizmatgiz, 1960.

10. Goryachev, D. N., Nekotoryye obshchiye integraly v zadache 0 dvizhenii tverdogo tela [Some General Solutions to the Problem of Motion of a Rigid Body], Warsaw, 1910.

11. Chetayev, N. G., Ob ustoychivosti vrashchatel'nykh dvizheniy tverdogo tela, polost' kotorogo napolnena ideal'noy zhidskot'yu [On the Stability of Rotational Motions of a Rigid Body with a Cavity Filled with an Ideal Fluid], PMM, Vol. 21, No. 2, 1957.

335

336 Re/erences

12. Beletskiy, V. V., Nekotoryye voprosy dvizheniya tverdogo tela v n'yuton­ovskom pole sil [Some Problems of the Motion of a Rigid Body in a New­tonian Force Field], PMM, Vol. 21, No. 6, 1957.

13. Poincare, H., Sur une forme nouvelle des equations de la mecanique, Compt. rend. Acad. Sei., Paris, Vol. 132, 1901.

14. Rumyantsev, V. V., Ob uravneniyakh dvizheniya tverdogo tela s polost'yu, napolnennoy zhidskot'yu [Concerning the Equation of Motion of a Rigid Body with a Fluid-Filled Cavity], PMM, Vol. 19, No. 1, 1955.

15. Zhukovskiy, N. Ye., 0 dvizhenii tverdogo tela imeyushchego polosti, napol­nennyye odnorodnoy kapel'noy zhidskost'yu [On the Motion of a Rigid Body with Cavities Filled with a Homogeneous Liquid Drop], Collected Works, Vol. 2, Gostekhizdat, 1948.

16. Landau, L. D. and Ye. M. Lifshitz, Mekhanika sploshnykh sred [Mechanics of Continuous Media], Gostekhizdat, 1953.

17. Lamb, Sir Horace, Hydrodynamics, sixth edition, Dover Publications, New York,1945.

18. Keldysh, M. V. and M. A Lavrent'yev, 0 yedinstvennosti zadachi Neymana [On the Uniqueness of Neumann's Problem], Doklady Akad. Nauk SSSR, Vol. 16, No. 3, 1937.

19. Stokes, G., Mathematical and Physical Papers, Vol. 1, Cambridge, 1880. 20. Neumann, c., Hydrodynamische Untersuchungen, Leipzig, 1883. 21. Muskhelishvili, N. 1., Singulyarnyye integral'nyye uravneniya [Singular

Integral Equations], Stechert-Hafner, New York, 1961. 22. Rumyantsev, V. V., Ustoychivost' vrashcheniya tverdogo tela s ellipsoidal'noy

polost'yu, napolnennoy zhidskost'yu [The Stability of Rotation of a Rigid Body with an Ellipsoidal Cavity Filled with a Fluid], PMM, Vol. 21, No. 6, 1957.

23. Sludskiy, F. A, De la rotation de la terre supposee fluide a son interieur, Bulletin de la Societe des naturalistes de Moscou, Vol. 9, 1895.

24. Hough, The Oscillations of a Rotating Ellipsoidal Shell Containing Fluid, Phil. Transactions (A), Vol. 186, 1, 1895.

25. Poincare, H., Sur la precession des corps deformables, Bulletin astronomique, Vol. 27, 1910.

26. Dirichlet, Untersuchungen über ein Problem der Hydrodynamik, Crelle, Vol. 58, 181, 1860.

27. Zhak, S. V., 0 vozmozhnosti kvazitverdogo vrashcheniya zhidkosti [On the Possibility of Rotations of a Fluid as a Semirigid Body], PMM, Vol. 21, No. 4,1957.

28. Lyapunov, AM., Obshchaya zadacha ob ustoychivosti dvizheniya [The General Problem of Stability of Motion], Collected Works, Vol. 2, Acad. Sci. USSR Press, 1956.

29. Chetayev, N. G., Stability of Motion, Pergamon Press, New York, 1961. 30. Skimel', V. N., K zadacham ustoychivosti dvizheniya tyazhelogo tverdogo

tela vokrug nepodvizhnoy tochki [On the Problems of Stability of Motion ofa Solid Rigid Body About a Stationary Point], PMM, Vol. 20, No. 1, 1956.

Re/erences 337

31. Chetayev, N. G., Ob oelnom svoystve uravneniy Puankare [On One Property of Poincare Equations], PMM, Vol. 19, No. 5, 1955.

32. Rumyantsev, V. V., Metody Lyapunova v issledovanii ustoychivosti dvizheniy tverdykh tel s polostyami, napolnennymizhidkost'yu [Lyapunov's Methods in the Study of Stability of Motions of Rigid Bodies with Fluid-Filled Cavities], Izvestia Akad. Nauk SSSR, Mekhanika i Mashinostroyeniye, No. 6, 1953; see also Advances in Applied Mathematics, Vol. 8, Academic Press, N.Y.-London, 1964.

*33. Rumyantsev, V. V., Ob ustoychivosti dvizheniya girostatov [On the Stability of Motion of Gyrostats], PMM, Vol. 25, No. 1, pp. 9-19, 1961.

34. VoIterra, V., Sur la theorie des variations des latitudes, Acta math., Vol. 22, 1899. 35. Kelvin, Lord, Mathematical and Physical Papers, Vol. 4, Cambridge, 1882. 36. GreenhilI, A. G., On the general motion of a liquid ellipsoid, Proc. Camb.

Phil. Soc., Vol. 4, 1880. 37. Basset, AB., On the steady motion and stability of liquid in an ellipsoidal

vessel, The Quarterly Journal of Math., Vol. 45, 1914. 38. Sobolev, S. L., 0 dvizhenii simmetrichnogo voIchka s polost'yu napolnennoy

zhidkost'yu [On the Motion of a Symmetrical Top with a Fluid-Filled Cavity], Prikl. Mekhan. i Tekh. Fiz., No. 3, 1960.

39. Ishlinskiy, A Yu. and M. Ye. Temchenko, 0 malykh kolebaniyakh verti­kal'noy osi voIchka, tselikom napolnennuyu ideal'noy neszhimayemoy zhidskost'yu [On the Small Vibrations of the Vertical Axis of a Top with a Cavity Completely Filled with an Ideal Incompressible Fluid], Prikl. Mekhan. i Tekh. Fiz., No. 3, 1960.

*40. Zhak, S. V., Ob ustoychivosti nekotorykh chastnykh sluchayev dvizheniya simmetricheskogo giroskopa, soderzhashchego zhidkiye massy [On the Sta­bility of Certain Particular Cases of Motion of a Symmetrical Gyroscope Containing Fluid Masses], PMM, Vol. 22, No. 2, pp. 330-337, 1958.

*41. Rumyantsev, V. V., Ob ustoychivosti vrashchatel'nykh dvizheniy tverdogo tela s zhidkim napolneniyem [On the Stability of Rotational Motions of a Rigid Body Filled with a Fluid], Coll.: Voprosy prikladnoy matematiki i mekhaniki [Problems of Applied Mathematics and Mechanics], No. 2, 1956; see also PMM, Vol. 23, No. 6, pp. 1512-1524, 1959.

42. Rumyantsev, V. V., Ob ustoychivosti dvizheniya po otnosheniyu k chasti pere­mennykh [On the Stability of Motion with Respect to aPart ofthe Variables], Vestnik Moscow State Univ., No. 4, 1957.

43. Gyunter, N. M., 0 dvizhenii zhidkosti, zaklyuchennoy v dannom peremesh­chayushchemsya sosude [On the Motion of a Fluid Contained in a Given Moving Vessel], Izvestia Akad. Nauk SSSR, Nos. 13-14, 15-17, 1926; Nos. 7-8, 9-11,18,1927; No. 1, 1928.

44. Ladyzhenskaya, O. A, Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1963.

*45. Movchan, A A, 0 pryamom metode Lyapunova v zadachakh ustoychivosti uprugikh sistem [On the Direct Lyapunov Method in Problems of Stability of Elastic Systems], PMM, Vol. 23, No. 3, pp. 686-700, 1959.

338 References

*46. Rumyantsev, V. V., Odna teoriya ob ustoyehivosti [One Theory on Stability], PMM, Vol. 24, No. 1, pp. 59-69, 1960.

*47. Rumyantsev, V. V., Ob ustoyehivosti dvizheniya volchka s polost'yu, napol­nennoy vyazkoy zhidskost'yu [On the Stability of Motion of a Top with a Cavity Filled with a Viseous Fluid], PMM, Vol. 24, No. 4, pp. 903-912, 1960.

48. Lyapunov, A. M., Ob ustoyehivosti ellipsoidal'nykh form ravnovesiya vrashehayusheheysya zhidkosti [On the Stability of Ellipsoidal Equilibrium Configurations of a Rotating Fluid], Colleeted Works, Vol. 3, Aead. Sei. USSR Press, 1959. .

49. Beletskiy, V. V., 0 libratsii sputnika [On the Libration of an Artifieial Earth Satellite], in Iskusstvennyye sputniki zemli [Artifieial Earth Satellites], No. 3, Aead. Sei. USSR Press, 1959.

*50. Kolesnikov, N. N., Ob ustoyehivosti svobodnogo tverdogo tela z polost'yu, zapolnennoy neszhimayemoy vyazkoy zhidkost'yu [On the Stability of an Uneonstrained Rigid Body with a Cavity Filled with an Ineompressible Viseous Fluid], PMM, Vol. 26, No. 4, pp. 914-923, 1962.

51. Moiseyev, N. N., 0 dvukh mayatnikah, napolnennykh zhidkost'yu [On Two Fluid-filled Pendulums], PMM, Vol. 26, No. 6, 1952.

52. Narimanov, G. S., Odvizhenii tverdogo tela, polost' kotorogo ehastiehno zapolnena zhidskost'yu [On the Motion of a Rigid Body Whose Cavity is Partially Filled with a Viscous Fluid], PMM, Vol. 20, No. 1, 1956.

53. Rumyantsev, V. V., Ob ustoyehivosti ravnomemykh vrasheheniy mekhani­eheskikh sistem [On the Stability of Uniform Rotations of Mechanieal Systems], Izvestia Akad. Nauk SSSR, Mekhanika i Mashinostroyeniye, No. 6, 1962.

*54. Pozharitskiy, G. K., Zadaeha minimuma v zadaehe ob ustoyehivosti ravno­vesiya tverdogo tela s ehastiehnym zhidkim napolneniyem [The Problem of a Minimum in the Problem of Stability of a Rigid Body Partially Filled with a Fluid], PMM, Vol. 26, No. 4, pp. 895-913, 1962.

55. Lyapunov, A. M., Zadaeha minimuma v odnom voprose ob ustoyehivosti figur ravnovesiya vrashehayusheheysya zhidkosti [The Problem of a Minimum in One Problem of Stability of Equilibrium Configurations of a Rotating Fluid], Colleeted Works, Vol. 3, Aead. Sei. USSR Press, 1959.

56. Rumyantsev, V. V., Ob ustoyehivosti ravnovesiya tverdogo tela, imeyush­ehego polosti, napolnennyye zhidskot'yu [On the Stability of Equilibrium of a Rigid Body Having Fluid-Filled Cavities], Doklady Akad. Nauk SSSR, Vol. 124, No. 2, 1959.

*57. Rumyantsev, V. V., Ob ustoyehivosti ustanovivshikhsya dvizheniy tverdykh tel s polostyami, napolnennymi zhidkost'yu [On the Stability of Steady Motions of Rigid Bodies with Fluid-Filled Cavities], PMM, Vol. 26, No. 6, 1962.

*58. Pozharitskiy, G. K. and V. V. Rumyantsev, Zadaeha minimuma v voprose ob ustoyehivosti dvizheniya tverdogo tela s polost'yu, zapolnennoy zhidkos'­tyu [The Problem of the Minimum in the Problem of Stability of Motion of a Rigid Body with a Fluid-Filled Cavity], PMM, Vol. 27, No. 1, pp. 18-32,1963.

References 339

59. Appel, P., Equilibrium Configurations of a Rotating Homogeneous Fluid, ONTI,1936.

60. Smirnov, V. I., Course of Higher Mathematics, Vol. 2, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1964.

61. Moisseyev, N. N., Sur certains problemes mathematiques du mouvement relatif des satellites. Dynamique des satellites, Symposium, Paris, 1962.

62. Moiseyev, N. N., Zadacha 0 dvizhenii tverdogo tela, soderzhayushchego zhidkiye massy, imeyushcheye svobodnuyu poverkhnost' [Problem of the Motion of a Rigid Body Containing Fluid Masses with a Free Surface], Matematicheskiy Sbornik, Vol. 32, No. 1, 1953.

63. Krasovskiy, N. N., Nekotoryye zadachi teorii ustoychivosti dvizheniya [Certain Problems of the Theory of Stability of Motion], Fizmatgiz, 1959.

*64. Pozharitskiy, G. K., 0 vliyanii vyazkosti na ustoychivost' ravnovesiya i statsionarnykh vrashcheniy tverdogo tela s polost'yu, chastichno zapolnennoy vyazkoy zhidkost'yu [On the Effect of Viscosity on the Stability of Equilib­rium and of Steady-State Rotations of a Rigid Body with a Cavity Partially Filled with a Viscous Fluid], PMM, Vol. 28, No. 1, pp. 67-76, 1964.

65. Mikhlin, S. G., Variational Methods in Mathematical Physics, second edition, Pergamon Press, New York, 1964.

66. Gunter, N. M., Potential Theory and Its Applications to Basic Problems of Mathematical Physics (in press), Frederick Ungar Publishing Co., New York.

67. Petrov, A. A., Priblizhennyy metod rascheta sobstvennykh kolebaniy zhid­kosti v sosudakh proizvol'noy formy i potentsialov Zhukovskogo dlya etikh sosudov [Approximate Method for Ca1culating the Natural Vibrations of a Fluid in Vessels of Arbitrary Shape and of Zhukovskiy's Potentials for These Vessels], Zh. Vychisl. Matem. i Matem. Fiz., No. 5, 1963.

68. Moiseyev, N. N. and A. A. Petrov, Chislennyye metody rascheta sobstvennykh chastot kolebaniy ogranichennogo ob"yema zhidkosti [Numerical Methods of Ca1culating the Natural Frequencies of Vibrations of a Bounded Volume of Fluid], Acad. Sei. USSR Press, Moscow, 1965.

69. Moiseyev, N. N., Variatsionnyye zadachi teorii kolebaniy zhidkosti i tela s zhidkost'yu [Variational Problems of the Theory of Vibration of a Fluid and of a Body with a Fluid], in the collection Variatsionnyye metody v zadachakh o kolebanii zhidkosti i tela s zhidkost'yu [Variational Methods in Problems of Vibration of a Fluid and of a Body with a Fluid], Acad. Sei. USSR Press, Moscow, 1962.

70. Moiseyev, N. N., 0 kolebaniyakh tyazheloy ideal'noy i neszhimayemoy zhidkosti v sosude [On the Vibrations of a Heavy, Ideal and Incompressible Fluid in aVesseI], Doklady Akad. Nauk SSSR, Vol. 85, No. 5, 1952.

71. Moiseev, N. N., Introduction to the Theory of OsciHations of Liquid-Con­taining Bodies. Advances in applied mechanics, Vol. 8, Academic Press, New York and London, 1964.

72. Stoker, J., Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publishers, New York, 1950.

340 References

73. Moiseyev, N. N., Dinamika korablya, imeyushehego zhidkiye gruzy [Dynam­ies of a Ship with Fluid Cargo], Izvestia Akad. Nauk SSSR, Otd. Tekh. Nauk, No. 7,' 1952.

74. Bolotin, V. V., 0 dvizhenii zhidkosti v kolebluyushehemsya sosude [On the Motion of a Fluid in an Oseillating Vessel], PMM, Vol. 20, No. 2, 1956.

75. Krushinskaya, S. I., Kolebaniya tyazheloy vyazkoy zhidkosti v podvizhnom sosude [Vibrations of a Heavy Viseous Fluid in a Moving Vessel], Zh. Vyehisl. Matern. i Matern. Fiz., No. 3, 1965.

76. Stokes, On some eases of Fluid Motion, Camb. Trans. 8, 1843. 77. Okhotsimskiy, D. Ye., K teorii dvizheniya tela s polostyami, ehastiehno

zapolnennymi zhidskost'yu [On the Theory of Motion of a Body with Cavities Partially Filled with a Fluid], PMM, Vol. 20, No. 1, 1956; available in English as NASA TT-F-33 (May 1960).

78. Kreyn, S. G. and N. N. Moiseyev, 0 kolebaniyakh tverdogo tela, soderzha­shehego zhidkost' so svobodnoy poverkhnost'yu [On the Vibrations of a Rigid Body Containing a Fluid with a Free Surfaee], PMM, Vol. 21, No. 2, 1957.

79. Pontryagin, L. S., Ermitovy operatory v prostranstvakh s indefinitivnoy metrikoy [Hermitian Operators in Indefinite Metrie Spaees], Izvestia Akad. Nauk SSSR, Sero Maternat., No. 8, 1944.

80. Myshkis, A. D., M. A. Belyayeva and A. D. Tyuptsov, Gidrostatika v slabykh gravitatsionnykh polyakh [Hydrostaties in Weak Gravitational Fields], Izvestia Akad. Nauk SSSR, Mekhanika, No. 5, 1964.

81. Hobson, E. W., Spherieal and Ellipsoidal Harmonies, Chelsea Publishing Co., NewYork,1955.

82. Moiseyev, N. N., Asimptotieheskiye metody tipa uzkikh polos [Asymptotie methods of the Narrow-Strip Type], in the eolleetion Nekotoryye problemy matematiki i mekhaniki [Some Problems of Mathematies and Meehanies], Siberian Division Aead. Sei. USSR Press, Novosibirsk, 1961.

83. Moiseyev, N. N., 0 krayevykh zadaehakh dlya linearizovannykh uravneniy Nav'ye-Stoksa v sluehaye, kogda vyazkost' malaya [On Boundary-Value Problems for Linearized Navier-Stokes Equations for the Case of Low Viseosity], Zh. Vyehisl. Matern. i Matern. Fiz., Vol. 1, No. 3, 1961.

84. Bagayeva, N. Ya. and N. N. Moiseyev, Tri zadaehi 0 kolebanii vyazkoy zhidkosti [Three Problems of the Vibrations of a Viseous Fluid], Zh. Vyehisl. Matern. i Matern. Fiz., Vol. 4, No. 2, 1964.

85. Moiseyev, N. N., 0 matematieheskikh metodakh issledovaniya nelineynykh kolebaniy [On Mathematieal Methods in the Study of Nonlinear Vibrations]. Proeeedings of the International Symposium on Nonlinear Vibrations, Vol. 2, Kiev,1963.

*86. Krasnoshehekov, P. S., 0 kolebanii fizieheskogo mayatnika, imeyushehego polosti, zapolnennyye vyazkoy zhidkost'yu [On the Oseillations of a Physieal Pendulum Containing Cavities Filled with a Viseous Fluid], PMM, Vol. 27, No. 2, pp. 289-302, 1963.

87. Krasnoshehekov, P. S., 0 vliyanii vyazkoy zhidkosti, nakhodyasheheysya

Re/erences 341

vnutri sputnika, na yego dinarniku [On the Effect Exerted on the Dynarnics of an Artificial Earth Satellite by a Viscous Fluid Contained in It], Doklady Akad. Nauk SSSR.

*88. Rurnyantsev, B. N., 0 dvizhenii tverdogo tela, soderzhayushchego polosti, zapolnennyye vyazkoy zhidkost'yu [On the Motion of a Rigid Body Con­taining Cavities Filled with a Viscous Fluid], PMM, Vol. 28, No. 6, pp. 1353-1358,1964.

*89. Chemous'ko, F. L., Dvizheniye tverdogo tela s polostyarni, zapolnennyrni vyazkoy zhidkost'yu pri rnalykh chislakh Renol'dsa [Motion of a Rigid Body with Cavities Filled with a Viscous Fluid for Low Reynolds Nurnbers], Zh. Vychisl. Matern. i Matern. Fiz., Vol. 5, No. 6, 1965.

90. Sretenskiy, L. N., 0 volnakh na poverkhnosti zhidkosti [On Waves on the Surface of a Fluid], Trudy TsAGI, No. 541, 1941.

91. Vishik, M. I. and L. A. Lyustemik, Regulyarnoye vyrozhdeniye i pogranichnyy sloy dlya differentsialnykh uravneniy s rnalyrn parametrom [Regular Degen­eration and the Boundary Layer for Differential Equations with a Small Parameter], Uspekhi Matern. Nauk, Vol. 12, No. 5, 1957.

Index

Absolute motion, 6 Algebraic equations, nonhomogeneous,

286 system of, 281, 282

Angular momentum, theorem of, 18 theory of, 34

Arbitrary cylinders, 290 Asymptotic solutions, theory of, 309

Beam-fluid system, 214, 232, 233 Bernoulli's integral, 37 Bessel function, nth-order, 60 Bifurcation points, 118, 153 Body-fluid system, 204, 238 Bond numbers, 261, 262, 264-266, 281,

291,292 large, 267, 283 perturbation theory for, 282, 283 smalI, 263, 264

Boundary layer, equation of, 269 Boundary-Iayer theory, 272-274

methods of, 269, 314 Boundary-value problem, 207

formulation of, 178

Capillary forees, action of, 252 Cartesian coordinates, 21, 61, 276, 277 Cauchy-Lagrange equation, analog of,

322 Cauchy-Lagrange integral, 169, 241, 242,

259,260 Cauchy-Poisson problems, 335 Cauchy problem, 248, 249, 335 Cauchy's conditions, 253, 254

integral, 37 Central impact theorem, 243, 244 Characteristic directions, definition of,

246,247 Chetayev's functions, 70 Chetayev's instability theorem, 69, 70, 76,

80 Chetayev's method, 71, 82, 103 Congruencetheorem,164

Conservation of energy, law of, 72 Constraints, steady-state, 32, 33 Contact angle condition, 262 Continuity condition, 158 Continuity equation, 253. Curviiinear system of coordinates, 256

d'Alembert-Lagrange principle, 2, 4,38, 116

Degenerate equation, 269 Displaeement thickness, 268-270 Dissipative forces, addition of, 192

external, 192 Disturbed motion, equations of, 69 Dynamic parameters, 10 Dynamics, general theorems of, 29 Dynamics of a bubble, 296

problems of, 295 Dynamics problems, variational

formulation of, 3

Elastic oscillations of a beam, 223 Energy integral, 29 Equations of motion, solutions of, 3 Equilibrium configurations, theory of, 274 Equivalent body, 49 Euler's angles, 23, 100,241 Euler's equations, 19, 26, 28, 65, 108,

252, 253, 259 linearized, 169 study of, 19

Euler's hydrodynamic equation, 18,22 Euler's kinematic· equations, 23

Flexural-torsional osci1lations, 205, 206 Fluid links, 202, 204 Foree-defining parameters, 66 Forees, gravitational, 251

342

hydrostatic, 251 inertial, 251, 252 mass, 251 reaction, 21 surfaee tension, 251, 252, 276, 284, 286

Index 343

Fouriercoefficients, 285, 286, 293, 320, 327 Fourier method, 281, 282 Fourier series, 55 Frechet derivative, 256, 257 Fredholm's alternative, 297, 298 Fredholm's integral equation, 177 Friedrich's metric, 221, 222 Friedrich's norm, 229, 275, 276 Froude number, 309, 310, 313

Galerkin method, 291, 292,320,327,328 Gauss' concepts, 14 Gauss-Ostrogradskiy formula, 26, 30,

113, 114 Gauss-Ostrogradskiy theorem, 9, 24, 25 Gravitational forces, 252 Gravity field, homogeneous, 20 Gravity forces, 32 Green's formula, 158, 168, 180, 185,209,

210,217,241,244,245,275,276,294, 295, 325

Green's function, 162,227,228 of Neumann's problem, 159, 211, 217

Green's theorem, 9 Group theory, methods of, 21

Hamilton-Ostrogradskiy equation, 5 Hamilton-Ostrogradskiy principle, 1, 3,

4,21 Hamilton's operator, 9 Hamilton's principle, 157, 159,209,210,

214, 252, 304 formulation of, 253

Harmonie functions, theory of, 249 Helmholtz's equation, 18, 36, 64, 67, 331 Helmholtz's theorem, 36, 65 Hermitian operators, 221, 222

theory of, 222 Hilbert space, 217 Holonomous systems, mechanics of, 1,3 Huygens-Steiner theorem, 50, 54, 142, 143 Hydrodynamic press ure, 17

boundary conditions for, 16, 17 Hydrodynamies, problem of, 19 Hydrostat, defined, 51 Hydrostatic forces, 251

Impact theory, 249 Incompressibility conditions, 5 Incompressibilityequation, 16 Indices, cyclical transposition of, 53 Inertia, centrifugal moments of, 11 Inertia products, 11 Inertia tensor, diagonal components of, 11

nondiagonal components of, 11

Inertial forces, 252 Infinitesimal transformations, sequence of,

21 Infinitesimals, second-order, 7 Integrals, angular momentum, 33

linear, 33 Integro-differential equation, system of,

250 Isoperimetrie condition, 262

Kelvin's discovery, 81 Kelvin's theorem, 36, 192 Kinematic conditions, 17, 158 Kinematic viscosity coefficient, 309 Kinematics, formulas of, 5 Kinetic energy, partial derivatives of, 12

theorem of, 31 König's theorem, 31, 51 Kronecker delta, 166, 232, 233, 292, 293

Lagrange function, 116 Lagrange multiplier, 304, 305

indeterminate, 5 Lagrange's equations, 21, 22, 25, 32,

204-206 Lagrange's theorem, 36, 127, 221, 222 Lagrangian coordinates, 21, 68, 112 Lagrangian multiplier, 5 Lamb's formula, 316, 321 Lamb's problem, 313

generalization of, 332 Lamb's representation, 312 Laplace formula, 17 Laplace transform, 335 Laplace's equation, 62, 298, 299, 302 Laplacian operator, 9, 309, 310 Legendre equation, 266 Legendre functions, 267, 297

first kind, 267 second kind, 267

Legendre operator, 264, 265 Legendre polynomials, theory of, 297 Legendre's equations, 267 Limited dimensions, 250 Limits of integration, conditions at, 16 Linear momentum, theory of, 33 Linear momentum integrals, 33 Linear operators, theory of, 220, 225 Linear problems, solvability of, 273 Linear theory, 168 Linearization, 253 Lyapunov-Chetayev theorems, 67 Lyapunov's definition, 68 Lyapunov's direct method, 78 Lyapunov's functions, 7(}-73, 75, 77, 82,

344 Dynamic Stability 0/ Bodies Containing Fluid

84,85,90 method of, 89

Lyapunov's method, 69, 86 Lyapunov's stability theorem, 69,70,71,

73,76, 77, 79, 83, 90, 127

Mass force vector, 13 Mathieu equations, 175 Mayyevskiy's condition, 77 Mechanical system, motion of, 21 Molecular theory, 13

fine points of, 14 Motion, absolute, 6

relative, 6 Moving coordinate axes, 19

Navier-Stokes equations, 94, 108, 307, 308, 331 linearized, 309, 332

Neumann operator, 159,250, 273, 274 Neumann-Zhukovskiy problems, 179 Neumann's problem, 43, 52, 168,209

Green's function of, 159, 217 Newtonian force field, 20, 104 Newtonian incompressible fluid, 38 Newton's law, 127 Nondiagonal components, 11 Normal stresses, condition of absence of,

326

Parameters, force-defining, 66 Pendulum problems, 182 Permissible displacements, principle of,

113 Perturbation theory, 165,292

method of, 320, 329 Petrov's method of perturbation theory,

276,277 Petrov's outline of perturbation method,

165 Poincare equations, 21, 67 Poincare method, 263, 264 POlsson's differential equations, 7 Possible deviation, 121 Potential condition, 158 Potential energy, expression for, 233 Pressure-constancy condition, 241, 242

Quasi-velocities, variations of, 8

Reaction forces, moments of, 21 Relative angular momentum, theory of,

28 Relative motion, 6 Resonance values; 323

Reynolds numbers, 307-310, 313, 315, 316, ~29, 330, 335

Rigid body, angular velocities of, 7 equations of motion of, 21 fluid-containing, 14 kinetic energy of, 21

Ritz method, 162, 163, 178, 180, 209, 210, 212, 213, 215, 216, 218, 220, 221, 228,229,265,275,276

Rotating fluid, theory of stability of equilibrium shapes of, 112

Routh's equation, 116 Routh's function, 116 Routh's method, 115 Routh's theorem, 127

Saint-Venant method of solution, 56 Schwarz's inequality, 236, 237, 244, 245 Segments, smooth-surfaced, 5 Shallow water, asymptotic behavior for,

300 Shell impermeability condition, 158 Small vibrations, problem of, 274

theory of, 133, 273, 275 Spherical coordinates, problems in, 265,

266 Spherical functions, 264, 265 Spring plus mass system, 326 Stability, concept of, 222

theory of, 68 Stability conditions, 251 Stability of motion, definition of, 88 Standing gravity waves, problem of

damping, 313 Stokes equation, 36 Stokes-Zhukovskiy potentials, 178, 180,

181,242,243,245,246 Stokes-Zhukovskiy problem, 178 Stokes' equivalent body, 49 Stokes' theorem, 10 Stream function, 330 Strouhal number, 281, 282 Surface tension coefficients, 17, 253, 276 Surface tension forces, 13, 253

potential energy of, 30 Sylvester's inequalities, 82, 234, 235 Sylvester's sign-definiteness criterion, 70,

71, 73, 97, 100, 103, 107, 111 System momentum, theorem of, 18

Tangential stresses, condition of absence of,326

Thermodynamic equilibrium, condition of,254

Torsion of prisms, problem of, 56

Index

Torsional oscillations, problem of, 205, 206 Torsional oscillations of a beam, 235 Transformed body, 51

Variation of constants, method of, 267 Variational equation, 158 Variational problem, formulation of, 252 Variational quasi-coordinates, derivatives

of,8 Vector analysis, 5 Velocity addition theorem, 6, 115 Velocity potential, 37

Vibrations, free, 222 superposition of, 22 theory of, 252, 253

Viscosity, phenomenological theory of, 178

Viscous fluids, hydrodynamics of, 308 Vortex lines, 36 Vortex tube, intensity of, 36

Weightlessness, conditioJ?s of, 14

Zhukovskiy's theorems, 51,98

345