By Armin Wachter
The Compendium of Theoretical Physics includes the canonical curriculum of theoretical physics. From classical mechanics over electrodynamics, quantum mechanics and statistical physics/thermodynamics, all subject matters are handled axiomatic-deductively and confimed by way of routines, recommendations and brief summaries.
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23) where the q˙j = dqj /dt are called generalized velocities. Virtual displacements and d’Alembert principle. An inﬁnitesimal displacement δxi of the ith particle is called virtual if it occurs at ﬁxed time t (dt = 0) and is consistent with the given holonomic and nonholonomic constraints: n δxi = j=1 ∂xi δqj , ∂qj n alj δqj = 0 , l = 1, . . , r . 24) j=1 We call such a displacement “virtual”, to distinguish it from a real displacement that occurs during a time interval dt, and for which the forces and constraints may change.
The translational force F T points in the direction of the z3 -axis and thus diminishes the strength of the gravitational force on the earth’s surface, depending on the geographical latitude. F T can therefore be absorbed in the gravitational constant g, and we are left with the equation m¨ z = F G + F P (z) − 2mω × z˙ or 18 1. Mechanics z1 S − 2mω(z˙3 cos λ − z˙2 sin λ) L z2 m¨ z2 = − S − 2mω z˙1 sin λ L m¨ z3 = −mg + S3 + 2mω z˙1 cos λ . m¨ z1 = − If we consider only small deviations from the vertical z3 -axis, we have S ≈ mg , z˙3 z˙2 , so that the pendulum’s z1 z2 -motion becomes g z¨1 = − z1 + 2ω z˙2 , ω = ω sin λ L g z¨2 = − z2 − 2ω z˙1 .
As an example for the use of Newton’s equations of motion in accelerated coordinate systems, we study the eﬀect of the earth’s rotation on the oscillation plane of a pendulum. 2 determines the axes of the inertial system3 K and that of the relative to K accelerated system K (earth-ﬁxed system). We consider a mathematical pendulum consisting of 3 Strictly speaking, this system is not an inertial system, in particular due to its accelerated motion around the sun. However, this can be neglected for the argument that follows.