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On the minimization properties of the Tisserand systems
Alberto
1

1 Escapa

Д , Tomas B

1 aenas

Д , Jose Manuel Ferr

1 Д andiz

, & Juan Get

2 ino

Dept. Applied Mathematics, University of Alicante, P. O. 99, Alicante 03080, Spain 2 Dept. Applied Mathematics, University of Valladolid, Valladolid 47011, Spain Alberto.Escapa@ua.es

1. Introduction
a collection of material par ticles S experience relative displacements, it is no possible to define unambiguously a rotational motion of the set. In these situations it is assigned to S a reference system Oxy z (the "body axes") with origin in its bar ycenter O and connected with it in a prescribed way.

W
Y

HEN

From Eqs. (1) and (3), the deformation kinetic energy can be written as (Escapa 2011) T
d ef

1 ( ) = T - L + I , 2

(5)

where T is the kinetic energy of S . Hence, for an arbitrar y vector different from 0, we have 1 (6) Tdef + = Tdef ( ) - L + I + I . 2 If we consider condition (a), defining the angular momentum of the system L, in Eq. (6), we get 1 Tdef T + - Tdef (T ) = I . 2 Since the matrix of iner tia is definite positive, we have that (7)

doing so, the rotation of the par ticles is identified with the rotation of the body axes with respect to some iner tial, or quasi- iner tial, reference system OX Y Z . This rotation admits a precise definition in mathematical terms.

B

T

are different possibilities to connect the body axes Oxy z with the considered set of par ticles (Munk & McDonald 1960). From the point of view of simplifying the equations of motion, one convenient method is to employ the so-called Tisserand systems (Tisserand 1891).
HERE

2. Tisserand systems

1 I > 0, R3, = 0. (8) 2 Therefore, Eq. (7) implies that Tdef ( ) takes its minimum at T , i.e., condition (b).

T

O

introduce Tisserand systems, let us write the velocity, relative to OX Y Z, of a par ticle of S with position xi and mass mi as Vi = з xi + vi( ). (1)

4. Tisserand systems evolution
angular velocity T , considered as a known function of time, determines the rotational kinematics of the body axes, but not its orientation in an univocal manner (Tisserand 1891).

The vector is, at this stage, arbitrar y and common for the set S . In contrast vi( ), the deformation velocity (Moritz & Mueller 1987), depends on the material par ticle i and the par ticular choice of . Tisserand systems can be defined by any of the following conditions that fix to a par ticular value T : (a) The angular momentum of S L=
iS

T

HE

m

i

xi з V

i

(2)

It allows defining a rotation matrix R(t) that brings the OX Y Z system to the body axes through (Wintner 1941) dRT T (t) = R, (10) dt where the superscript T denotes the transpose of a matrix. The solution of this linear differential equation is given by
t

Specifically, from the components of T in the OX Y Z system, we can construct the skew-symmetric matrix 0 -T Z (t) T Y (t) 0 -T X (t) . T (t) = T Z (t) (9) -T Y (t) T X (t) 0

is L = I T (Tisserand 1891), where I is the matrix of iner tia of S (b) The kinetic energy of S associated to the deformation velocity 1 Tdef(T ) = 2 mivi(T )
i S 2

(3)

is minimum (Jeffreys 1976) (c) The angular momentum of S related with the deformation velocity h(T ) =
i S

R(t) = R (t0) exp -
t
0

T (s)ds ,

(11)

R (t0) providing the numerical value of R(t) at the epoch t0. In this way, besides any of the conditions (a), (b), or (c), the specification of a par ticular Tisserand system requires providing explicitly the initial orientation of the body axes relative to OX Y Z .

mi [xi з vi(T )]

(4)

is the null vector (Tisserand 1891)

References 3. Equivalence of the conditions
former characterizations turn out to say, (a) (b), (b) (c), and (c) third implications are detailed in the exist & Mueller 1987). Let us focus on the first З З З З Escapa, A.: Celest. Mech. Dyn. Astr., Vol. 110, 99-142, 2011 Jeffreys, H.: The Ear th. Cambridge University Press, 1976 Moritz, H. & Mueller, I.: Ear th Rotation. Frederic Ungar, 1987 Д Д Д Tisserand, F. F.: Traite de Mecanique Celeste, Vol. 2. Gauthier- Villars, 1891 З Wintner, A.: The Analytical Foundations of Celestial Mechanics. Princeton University Press, 1941

T

HE

to be equivalent, that is (a). The second and ing literature (e.g., Moritz one.

Session 4. Ear th's rotation and geodynamics

Д ` ДД Journees "Systemes de reference spatio-temporels", St. Petersburg, Russia, 22-24 September 2014