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 through workouts, suggestions and brief summaries.
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3N − s = n , dt ∂ q˙j ∂qj it follows that the Lagrange equations also hold for ˙ t) = L[q(Q, t), q(Q, ˙ t), t] . ˙ L (Q, Q, Q, Equivalence of Lagrange and Newtonian formalism. In the absence of constraining forces, the Lagrange formalism for an N -particle system must yield Newtons equations of motion. If we choose Cartesian coordinates as the generalized ones, q = x, we can see this (in vector notation) as follows: N L=T −V = i=1 mi 2 x˙ − V (x1 , . . , xN , t) 2 i =⇒ ∇xi L = −∇xi V = F i , d ¨i .
For a system of N particles with n = 3N − s degrees of freedom, they describe the motion of the system in an abstract 2n-dimensional space, called phase space, which is spanned by the generalized coordinates and momenta. The following table shows the main results of Lagrangian and Hamiltonian mechanics. 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.
We consider a mathematical pendulum consisting of 3 Strictly speaking, this system is not an inertial system, in particular due to its accelerated motion around the sun. However, this can be neglected for the argument that follows. 1 Newtonian Mechanics y ω 17 z3 z2 L body-fixed system K z3 z1 q ω x λ FP space-fixed system K z2 m z FG z1 Fig. 2. Determination of space-fixed and body-fixed systems on the earth’s surface. a massless rope of length L, fixed at the top, so that it can rotate freely in all directions, and to which a particle of mass m is attached at the other end.