By Walter Greiner
Intended for complex undergraduates and starting graduate scholars, this article relies at the hugely winning path given through Walter Greiner on the college of Frankfurt, Germany. the 2 volumes on classical mechanics offer not just a whole survey of the subject but in addition a huge variety of labored examples and difficulties to teach scholars truly tips to follow the summary ideas to sensible problems.
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Additional info for Classical Mechanics: Point Particles and Relativity (Classical Theoretical Physics)
F| (b) The torque of a force F p acting at point P(x, y, z), that is at the position r = (x, y, z), is deﬁned with respect to the coordinate origin (center of rotation) as the vector M = r × Fp . ϑ r F′ F Center of rotation Here r is the position vector from the cen- The absolute value of the torque results from ter of rotation to the action point of the force the force component perpendicular to the distance vector. F p . The magnitude of M is obviously given by 35 APPLICATION IN PHYSICS M = r · F , where F = F sin ϑ is the force component perpendicular to the position vector (compare in the ﬁgure).
In order to become independent of the coordinate frame, a set of orthogonal unit vectors is put at the point of the trajectory of the mass point given by s. ” As unit vectors one uses 49 50 THE MOVING TRIHEDRAL (ACCOMPANYING DREIBEIN)—THE FRENET FORMULAS T tangent vector, N principal normal vector, B binormal vector. 8 Because the vectors form an orthonormalized set, it holds that (N × B) = T, cyclically permutable. In the following we give the precise deﬁnition of these three base vectors of the moving trihedral and show how they are calculated for a given space curve r(t).
The triple scalar product is therefore also denoted as a mixed product. The triple scalar product is a scalar. Triple scalar product in component representation: a · (b × c) = (a1 , a2 , a3 ) · [(b1 , b2 , b3 ) × (c1 , c2 , c3 )] e1 e2 e3 = (a1 , a2 , a3 ) · b1 b2 b3 c1 c2 c3 = (a1 , a2 , a3 ) · (b2 c3 − b3 c2 , −b1 c3 + b3 c1 , b1 c2 − b2 c1 ) = a1 (b2 c3 − b3 c2 ) − a2 (b1 c3 − b3 c1 ) + a3 (b1 c2 − b2 c1 ). The three terms may again be combined to a determinant, such that a1 a2 a3 a · (b × c) = b1 b2 b3 = (a × b) · c.