Compendium of Theoretical Physics by Armin Wachter

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.

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