Exact solution of Euler's equations in the case of non-zero applied torques (rigid body dynamics): reduction to Abel ordinary differential equation.
Sergey V. Ershkov Institute for Time Nature Explorations, M.V. Lomonosov's Moscow State University, Leninskie gory, 1-12, Moscow 119991, Russia.
Keywords: Euler's equations (rigid body dynamics), applied torques, the principal moments of inertia, components of the angular velocity, principal axes, Abel ordinary differential equation, Riccati’s ordinary differential type equation.
Here is presented a new exact solution of Euler's equations (rigid body dynamics, non-zero applied torques) in the case of reduction one of Euler's equations to a proper Abel ordinary differential equation. One component of angular velocity of rigid body for such a solution - is a proper classical solution for the case of symmetric rigid rotor (which has two equal moments of inertia), but applied torques is not zero. The second component is proved to be appropriate solution of Abel ordinary differential equation. Due to a very special character of Riccati’s type equation (such as Abel ODE), it’s general solution is known to have a proper gap of above component of angular velocity of rigid body. It means a possibility of sudden acceleration of rigid body rotation around appropriate principal axe at definite moment of parametrical time. The 3-d could be obtained from a simple algebraic equality. It is a multiplying all the components of angular velocity of rigid body (which is to be a constant for such a solution).
1 In accordance with [1-3], Euler's equations describe the rotation of a rigid body in a frame of reference fixed in the rotating body:
d 1 I I( I ) K , 1 d t 3 2 2 3 1
d 2 I I( I ) K , 1.1 2 d t 1 3 1 3 2
d 3 I I( I ) K , 3 d t 2 1 1 2 3
- where K i are the applied torques, I i ≠ 0 are the principal moments of inertia and i are the components of the angular velocity vector along the principal axes (i = 1, 2, 3).
Let’s multiply all the equations of (1.1) onto i properly, then divide them by I i:
d1 I3 I2 K1 , 1 1 2 3 1 td I1 I1
d2 I1 I3 K2 , 2 1 2 3 2 td I2 I2
d3 I2 I1 K3 , 3 1 2 3 3 td I3 I3
- we obtain [4] (I 1 ≠ I 2 , I 2 ≠ I 3):
I1 K1 d1 I 2 K2 d2 1 1 1 2 3 2 2 I3 I2 I1 td I1 I3 I 2 td 2.1
I2 K 2 d2 I3 K3 d3 2 2 1 2 3 3 3 I1 I3 I 2 td I 2 I1 I3 td
Thus, in above system (1.2) all variables i are proved to be separated.
2 Besides, 1-st equation of (1.2) determines non-linear character of dependence the function 1 in respect to 2, as well as 2-nd equation - 2 in respect to 3.
Let’s note:
1 2 3 f (t) ,
- then we could represent system (1.2) in a form below:
I1 K1 d1 ,)t(f 1 1 I3 I 2 I1 td
I 2 K 2 d 2 ,)t(f 3.1 2 2 I1 I 3 I 2 td
I3 K 3 d 3 )t(f . 3 3 I 2 I1 I3 td
Above system let’s make a conclusion that existence of exact solution of equations (1.3) is possible only if f (t) = const, then it leads to below (i = 1, 2, 3):
di Ki C ),t(f i i i td Ii 4.1
I3 I2 I1 I3 I2 I1 C ,)t(f C ,)t(f C .)t(f 1 2 3 I1 I2 I3
- Abel ordinary differential equation, which is known to be of Riccati’s type [5].
Due to a very special character of such an equation, it’s general solution is known to
have a proper gap of above component of angular velocity i.
3 It means a possibility of sudden acceleration of rigid body rotation around appropriate
principal axe at definite moment of parametrical time t 0, or existence of such a continuous solution only at some definite, restricted range of parameter t [6].
According to [5], an exact solutions of Abel ordinary differential equation are known to exist in very rare cases. For example, exact solutions of (1.4) could be found under conditions below (t ≥ 0, i = 1, 2, 3):
- torque-free precession of the rotation axis of rigid rotor [3]: K i = 0,
- the rigid rotor is symmetric (has two equal moments of inertia):
I 1 = I 2, or I 2 = I 3 C i = 0 .
The last condition means an existence of a proper exact solution of system (1.1), namely:
- one component of angular velocity 1 (t) for such a solution - is a proper classical solution for the case of symmetric rigid rotor (which has two equal moments of inertia), but applied torques is not zero, see (1.4) under condition C i = 0.
- the 2-nd component 2 (t) is proved to be appropriate solution of Abel ordinary differential equation (1.4). Due to a very special character of Riccati’s type equation (such as Abel ODE), it’s general solution is known to have a proper gap of angular velocity i at some definite moment of parametrical time t 0. It means a possibility of sudden acceleration of rigid body rotation around appropriate principal axe.
- the 3-rd component 3 (t) one could obtain from the equality below:
3 = f(t) / ( 1 · 2) ,
- where f(t) = const (which is given by initial conditions).
4 References:
1. Landau L.D. and Lifshitz E.M. (1976) Mechanics, 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover). 2. Goldstein H. (1980) Classical Mechanics, 2nd. ed., Addison-Wesley. ISBN 0-201-02918-9. 3. Symon KR. (1971) Mechanics, 3rd. ed., Addison-Wesley. ISBN 0-201-07392-7. See also: http://en.wikipedia.org/wiki/Euler's_equations_(rigid_body_dynamics) . 4. Ershkov S.V. Self-similar solutions of Euler’s equations of rigid body rotation // Moscow State University, proceedings of seminar on Temporology exploring (2005) (in Russian): http://www.chronos.msu.ru/RREPORTS/yershkov_kontseptsia.pdf. 5. Dr. E.Kamke. Hand-book for ordinary differential equations // Moscow: “Science” (1971). 6. Ershkov S.V., Schennikov V.V. Self-Similar Solutions to the Complete System of Navier-Stokes Equations for Axially Symmetric Swirling Viscous Compressible Gas Flow // Comput. Math. and Math. Phys. J. (2001) Vol.41, № 7. P.1117-1124.
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