Euler and the dynamics of rigid bodies Sebastià Xambó Descamps Abstract
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Euler-RigidBody-x
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4.2. The inertia tensor. The inertia tensor of with respect to is the linear map defined with respect to any Cartesian basis by the matrix , where are the components of the position vector of with respect to . This does not depend on the Cartesian basis , because it is easy to check that the matrix in the expression is the matrix of the linear map such that . Notice, for example, that for we get , which is the first row of the matrix. In particular we have a coordinate-free description of , namely . The main reason for introducing the inertia tensor is that it relates the angular mo- mentum relative to , , and : . 10 Indeed, since is fixed with respect to (i.e., ), . In the last step we have used the formula for the double cross product. 4.3. Kinetic energy. The kinetic energy relative to is given by the formula . The proof is again a short computation: . But and hence , which establishes the claim because . Remark. If we let be the distance of to the rotation axis , then . Indeed, , with , and the claim follows from Pythagoras’ theorem, as is the or- thogonal projection of to the rotation axis. Note that this shows that is indeed the rotation kinetic energy of the solid. The kinetic energy with respect to an observer for which is, according to the second formula in §3.1 (with playing the role of ) , where , (the linear momentum of a point mass moving with ), and is the velocity of with respect to . This formula can also be established di- rectly, for the kinetic energy in question is , and the first term of last expression is the kinetic energy relative to and the third is . As a corollary we get, taking , that the kinetic energy with respect to is, setting , . 11 In other words, the kinetic energy of a rigid body is, for any observer, the sum of the kinetic energy of a point particle of mass moving as the braycenter of the body, and the rotation energy of the rotation about the axis with angular velocity . Download 1.26 Mb. Do'stlaringiz bilan baham: |
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