ON THE FUNDAMENTL FORMULÆ OF DYNAMICS
15
If the variations
, etc. are capable both of positive and of negative values, we must have
![{\displaystyle {\frac {dU}{d{\ddot {\omega }}_{1}}}=\Omega _{1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22325d5c9b38567905dcf97f1a26259c050fd9da) etc.,
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(29)
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or![{\displaystyle {\frac {dU}{d{\ddot {\omega }}_{1}}}={\frac {dV}{\omega _{1}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69b6342b0130eb8f785a278e6b16dbd7d82658fe) etc.
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(29)
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To illustrate the use of these equations in a case in which
, etc. are not exact differentials, we may apply them to the problem of the rotation of a rigid body of which one point is fixed. If
denote infinitesimal rotations about the principal axes which pass through the fixed point,
, will denote the moments of the impressed forces about these axes, and the value of
will be given by the formula
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where
, and
are constants,
being the moments of inertia about the three axes. Hence,
![{\displaystyle {\frac {dU}{d{\ddot {\omega }}_{1}}}=(b-c){\dot {\omega }}_{2}{\dot {\omega }}_{3}+(b+c){\ddot {\omega }}_{1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d079d2dfee95f0d61935e57dde51d1a2b882cb) |
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and the equations of motion are
![{\displaystyle {\ddot {\omega }}_{1}={\frac {(c-b){\dot {\omega }}_{2}{\dot {\omega }}_{3}+\Omega _{1}}{c+b}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf7536c6c80814ac3d83801cb2ae76e82051562)
![{\displaystyle {\ddot {\omega }}_{2}={\frac {(a-c){\dot {\omega }}_{3}{\dot {\omega }}_{1}+\Omega _{2}}{a+c}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51067512049b397936bc537b91f789110173d5e0)
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