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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Extra resources for Compendium of Theoretical Physics
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.