76 lines
2.8 KiB
Markdown
76 lines
2.8 KiB
Markdown
---
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title: 36-eulers-equations
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---
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precesses about the direction of g (i.e. the vertical) with a mean angular velocity
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Spr (ul/M)g cos a
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(2)
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which is small compared with Senu
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Spr
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in
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no
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a
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FIG. 50
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In this approximation the quantities M and cos a in formulae (1) and (2) are constants,
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although they are not exact integrals of the motion. To the same accuracy they are related
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to the strictly conserved quantities E and M3 by M3 = M cos a,
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§36. Euler's equations
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The equations of motion given in §34 relate to the fixed system of co-
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ordinates: the derivatives dP/dt and dM/dt in equations (34.1) and (34.3)
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are the rates of change of the vectors P and M with respect to that system.
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The simplest relation between the components of the rotational angular
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momentum M of a rigid body and the components of the angular velocity
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occurs, however, in the moving system of co-ordinates whose axes are the
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principal axes of inertia. In order to use this relation, we must first transform
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the equations of motion to the moving co-ordinates X1, X2, X3.
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Let dA/dt be the rate of change of any vector A with respect to the fixed
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system of co-ordinates. If the vector A does not change in the moving system,
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its rate of change in the fixed system is due only to the rotation, so that
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dA/dt = SxA; see §9, where it has been pointed out that formulae such as
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(9.1) and (9.2) are valid for any vector. In the general case, the right-hand
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side includes also the rate of change of the vector A with respect to the moving
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system. Denoting this rate of change by d'A/dt, we obtain
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dAdd
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(36.1)
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§36
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Euler's equations
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115
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Using this general formula, we can immediately write equations (34.1) and
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(34.3) in the form
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=
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K.
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(36.2)
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Since the differentiation with respect to time is here performed in the moving
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system of co-ordinates, we can take the components of equations (36.2) along
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the axes of that system, putting (d'P/dt)1 = dP1/dt, ..., (d'M/dt)1 = dM1/dt,
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..., where the suffixes 1, 2, 3 denote the components along the axes x1, x2, X3.
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In the first equation we replace P by V, obtaining
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(36.3)
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=
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If the axes X1, X2, X3 are the principal axes of inertia, we can put M1 = I,
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etc., in the second equation (36.2), obtaining
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=
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I2 = K2,
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}
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(36.4)
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I3 = K3.
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These are Euler's equations.
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In free rotation, K = 0, so that Euler's equations become
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= 0,
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}
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(36.5)
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= 0.
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As an example, let us apply these equations to the free rotation of a sym-
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metrical top, which has already been discussed. Putting I1 = I2, we find from
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the third equation SQ3 = 0, i.e. S3 = constant. We then write the first two
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equations as O = -wS2, Q2 = wS1, where
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=
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(36.6)
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is a constant. Multiplying the second equation by i and adding, we have
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= so that S1+iD2 = A exp(iwt), where A is a
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constant, which may be made real by a suitable choice of the origin of time.
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Thus
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S1 = A cos wt
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Q2 = A sin wt.
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(36.7)
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116
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Motion of a Rigid Body
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