Download Compendium of Theoretical Physics by Armin Wachter PDF

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 via workouts, strategies and brief summaries.

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22: Momentum conservation If H does not depend explicitly on the generalized coordinate qj , the corresponding momentum pj is conserved: ∂H = 0 =⇒ pj = const . ∂qj Such a coordinate is called cyclic. Differentiating the Hamilton function with respect to time, dH = dt n j=1 ∂H ∂H q˙j + p˙j ∂qj ∂pj + ∂H , ∂t and using the Hamilton equations yields dH ∂H = . 23: Conservation of the Hamilton function If H (or L) does not depend explicitly on t, then H (or the corresponding Lagrange equations) is constant along the solutions of Hamilton’s equations: ∂H = 0 =⇒ H = const .

Qn , q˙1 , . . , q˙n , t) − V (q1 , . . , qn ) . 2 i From this, one obtains the equations of motion (Lagrange equations) in terms of generalized coordinates for the case of s holonomic and r nonholonomic constraints: ∂L d ∂L − − dt ∂ q˙j ∂qj r λl alj = 0 , j = 1, . . 30) l=1 n alj q˙j + alt = 0 , l = 1, . . , r . j=1 The Lagrange equations constitute a system of coupled ordinary differential equations of second order for the time dependence of the generalized coordinates. 22), which are now to be regarded as differential equations, we have a total of n + r equations for the n generalized coordinates and the r Lagrange multipliers.

2, which characterize inertial systems. 10: Inertial systems and Galilei invariance Inertial systems are connected via coordinate transformations of the form xi −→ xi = Rij xj + vi t + qi , t −→ t = t + t0 , with R, v, q, t0 = const , RRT = 1 , det R = 1 . These transformations form a proper orthochronous Lie group with 10 parameters. It is called the group of Galilei transformations. Newton’s laws are form-invariant in all inertial systems. We say: Newtonian mechanics is Galilei-invariant. In inertial systems, force-free particles move linearly uniformly.

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