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 issues are handled axiomatic-deductively and confimed by way of workouts, strategies and brief summaries.
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We consider only the case of all constraints being given by s = 3N − n holonomic conditions. To perform the transformation of variables, 48 1. Mechanics L(qi , q˙i , t) −→ H qi , ∂L ,t ∂ q˙i , we use the Legendre transformation n pi q˙i − L H = H(q1 , . . , qn , p1 , . . 38), evaluate the derivatives n n ∂ q˙i ∂L ∂ q˙i ∂H = q˙j + pi − = q˙j ∂pj ∂p ∂ q˙i ∂pj j i=1 i=1 n ∂H = ∂qj n pi i=1 ∂ q˙i ∂L ∂ q˙i ∂L ∂L − − =− = −p˙j . 21: Hamilton equations for s holonomic constraints The Hamilton function of an N -particle system with n generalized coordinates and momenta is given by n pi q˙i − L(q1 , .
We have d ∂L ∂L ˙ t) . − = 0 , qi = qi (Q, t) , q˙i = q˙i (Q, Q, dt ∂ q˙i ∂qi Because q˙i = j ∂qi (Q, t) ˙ ∂qi Qj + , ∂Qj ∂t it follows that ∂qi ∂ q˙i = . ∂Qj ∂ Q˙ j ˙ t), we now calcuStarting from the transformed Lagrange function L (Q, Q, late the Lagrange equations in the new coordinates. To this end, we need Applications ∂L = ∂Qi j d ∂L = dt ∂ Q˙ i =⇒ ∂L ∂qj ∂L ∂ q˙j + ∂qj ∂Qi ∂ q˙j ∂Qi j ∂L = ∂ Q˙ i j 41 ∂L ∂ q˙j = ∂ q˙j ∂ Q˙ i ∂L ∂ q˙j d dt d ∂L ∂L − = dt ∂ Q˙ i ∂Qi j ∂L ∂qj ∂ q˙j ∂Qi ∂qj + ∂Qi j d ∂L ∂L − dt ∂ q˙j ∂qj j + j ∂qj ∂Qi ∂L d ∂ q˙j dt ∂qj ∂Qi d ∂qj ∂ q˙j − dt ∂Qi ∂Qi ∂L .
19). Thus, for t − t < 0, the integration does not contribute at all. This is in accordance with the causality principle, which states that a system, at time t, can only be inﬂuenced by the past (t < t). For t − t > 0, we obtain • Weak damping: ω02 > γ 2 . G(t, t ) = 1 −γ(t−t ) e sin[ω (t − t )] , ω = mω ω02 − γ 2 . • Strong damping: ω02 < γ 2 . G(t, t ) = 1 −γ(t−t ) e sinh[ω (t − t )] , ω = mω γ 2 − ω02 . • Critical damping: ω02 = γ 2 . G(t, t ) = t − t −γ(t−t ) . 2 Lagrangian Mechanics In the problems we have encountered so far, our starting point has been the Newtonian equations of motion.