Equivalence of Euler equations and torque-angular momentum relation
DOI:
https://doi.org/10.31349/RevMexFisE.18.136Keywords:
Rigid body, rotational motion, Euler equations, torque-angular momentum relationAbstract
Euler derived equations for rigid body rotations in the body reference frame and in the stationary reference frame by considering an infinitesimal part of the rigid body.Another derivation is possible, and it is widely used: transforming torque-angular momentum relation to the body reference frame.
However, their equivalence is not shown explicitly.
In this work, for a rigid body with different moments of inertia, we calculated Euler equations explicitly in the body reference frame and in the stationary reference frame and torque-angular momentum relation.
We also calculated equations of motion from Lagrangian.
These calculations show that all four of them are equivalent.
References
L. Euler, Decouverte d’un noveau principe de mecanique, The Euler Archive E-177 (1752), http://eulerarchive.maa.org
L. Euler, Du mouvement de rotation des corps solides autour d’un axe variable, The Euler Archive E-292 (1765), http://eulerarchive.maa.org
J. E. Marquina, M. L. Marquina, V. Marquina and J. J. Hern{'{a}}ndez-G{'{o}}mez, Leonhard Euler and the mechanics of rigid bodies, Eur. J. Phys., 38 (2017) 015001,
https://doi.org/10.1088/0143-0807/38/1/015001
V. Tanr{i}verdi, Comment on 'Leonhard Euler and the mechanics of rigid bodies', Submitted to Eur. J. Phys, (2020)
H. Goldstein, Classical Mechanics, 2nd ed. (Massachusetts: Addison-Wesley, Massachusetts, 1980)
J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems, 5th ed. (Brooks/Cole, Belmont, 2004)
A. Sommerfeld, Mechanics. Lectures on Theoretical Physics. Volume 1 (Academic Press, New York, 1952)
J. R. Taylor, Classical Mechanics, (Dulles: University Science Books, Dulles, 2005)
G. R. Fowles and G. L. Cassiday, Analytical Mechanics, 7th ed. (Brooks/Cole, Belmont, 2005)
V. Barger and M. Olson, Classical Mechanics: A Modern Perspective, (McGraw-Hill, New York, 1994)
V. J. Jorge and J. S. Eugene, Classical dynamics. A contemporary approach, (Cambridge University Press, Cambridge, 1998)
J. L. McCauley, Classical mechanics: Transformations, flows, integrable, and chaotic dynamics, (Cambridge: Cambridge University Press, Cambridge, 1997)
E. Corinaldesi, Classical Mechanics for Physics Graduate Students, (World Scientific Publishing Company, Singapore, 1999)
E. A. Desloge, Classical Mechanics. Volume 1, (New York: John Wiley and Sons, New York, 1982)
K. R. Symon, Mechanics, 3rd ed. (Addison-Wesley, Massachusetts, 1971)
R. D. Gregory, Classical Mechanics, (Cambridge University Press, New York, 2006)
V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer-Verlag, New York, 1989)
L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed. (Butterworth-Heinenann, New Delhi, 2000)
W. D. MacMillan, Dynamics Of Rigid Bodies, (Dover, New York, 1960)
W. Greiner, Classical Mechanics, Systems of particles and Hamiltonian dynamics, (Springer, New York, 2003)
J. B. Scarborough, The Gyroscope Theory and Applications, (Interscience Publishers, London, 1958)
E. Leimanis, The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point, (Springer-Verlag, Berlin, 1965)
L. D. Mott, Another Derivation of Euler's Equations of Rigid-Body Rotation, Am. J. Phys. 34 (1966) 1197,
https://doi.org/10.1119/1.1972684
H. Lamb, Higher Mechanics, (Cambridge University Press, Cambridge, 1920)
H. Crabtree, An elementary treatment of the theory of spinning tops and gyroscopic motion, (Longmans, Green and Co., London, 1909)
R. B. Hayward, On a direct method of estimating velocities, accelerations, and all similar quantities with respect to axes moveable in any manner in space, with applications, Cambridge Philosophical Society. Transactions of the Cambridge Philosophical Society (Cambridge: University Press), 10 (1864) 1-20,
https://catalog.hathitrust.org/Record/000526741
F. Klein and A. Sommerfeld, The theory of the Top, Volume II, (Birkhauser, Boston MA, 2010)
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