By Armin Wachter
The Compendium of Theoretical Physics comprises the canonical curriculum of theoretical physics. From classical mechanics over electrodynamics, quantum mechanics and statistical physics/thermodynamics, all subject matters are taken care of axiomatic-deductively and confimed by means of workouts, options and brief summaries.
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2. According to Newton’s second axiom, the equation of motion is form-invariant in all inertial systems. This is clearly equivalent to the condition D 2 x0 = 0 , ω = 0 , since then all pseudo forces vanish. The condition can be satisﬁed most generally by letting x0 (t) = vt + q , R(t) = R , v, q, R = const . 16 1. Mechanics In other words: relative to K, the system K can only be rotated by a ﬁxed amount and move with constant velocity. If, additionally, the force F is not explicitly time-dependent, the time origin of K may also be shifted relative to that of K.
R, which can be chosen freely. In general, these are time-dependent functions that may also depend on the generalized coordinates and velocities. Using these multipliers, we rewrite the previous relation as n r δqj j=1 λl alj = 0 . 28) yields n δqj j=1 d ∂T ∂T − − Qj − dt ∂ q˙j ∂qj r λl alj =0. l=1 For the independent δqj , the corresponding coeﬃcients (bracket terms) can be set identically zero. Furthermore, we can choose the Lagrange multipliers such that the coeﬃcients of the dependent diﬀerentials also vanish.
If no external forces are present, the system is called closed. Given these deﬁnitions, Newton’s equations of motion for an N -particle system are F ij + F i , i = 1, . . , N . 14) j=i Before we study the corresponding kinematical quantities, such as momentum, angular momentum, and energy, it is useful to introduce the notion of center of mass: Deﬁnition: Center of mass xC of an N -particle system xC (t) = 1 M N N mi xi (t) , M = i=1 mi . i=1 In case of a continuous mass distribution ρ(x, t), these equations are replaced by xC (t) = 1 M xρ(x, t)d3 x , M = ρ(x, t)d3 x = const .