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By L. I. Sedov, J. R. M. Radok

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Additional info for A course in continuum mechanics, vol. 1: Basic equations and analytical techniques

Example text

Let Π ( X i , x j ) be a plane defined by the two vectors X i and x j ,  and the line Δij orthogonal to this plane, of unit vector nij . The angle   Q5 formed by the two vectors X i and x j is defined by the relation    X i ∧ x j = nij sin Q5 . The following diagram indicates the choice of angles. 16. – The indices i and j of various unit vectors of the two considered bases vary strictly according to a cyclic permutation of the numbers 1, 2 and 3 (for example, if j = 3, j + 1=1).

This means that: – to the extent that forces are inevitably acting, to account for the relative motion of the pseudo-Galilean system of reference relative to the Galilean system of reference, which is not strictly translation nor a rectilinear one, – and because rotations inevitably generate accelerations, corrective terms are added to the expressions of the fundamental law. But, if the orders of magnitude of these corrective terms generated by the motion of the working system of reference are negligible relative to those of the problem, and do not significantly influence the study of a motion, then the system of reference can be considered pseudo-Galilean and can serve as basic frame of reference for the application of the fundamental law.

Vector rotation R u,α  Let R u ,α be the rotation by angle α of a vector a about the axis   Δ with the direction vector u . Let b be the vector resulting from the  rotation of a .  Let Π be the plane perpendicular to vector u and passing through   the point A , the origin of vector a . The projection on the axis u and on plane Π can be written        a = ( u ⋅a )u + u ∧ ( a ∧ u )        . b = ( u ⋅b ) u + u ∧ ( b ∧ u )    Since b results from a by rotation about the axis ( A u ) , the two vectors have the same orthogonal projection on this axis, which gives       ( u ⋅ a ) u = ( u ⋅b ) u .

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