2.2. (a) Verify explicitly the invariance of the volume element do of the phase space of a single particle under transformation from the Cartesian coordinates (x,y, z, px, Py, Pz) to the spherical polar coordinates (r,e,4,Pr.Pe,Po). (b) The foregoing result seems to contradict the intuitive notion of "equal weights for equal solid angles," because the factor sine is invisible in the expression for do. Show that if we average out any physical quantity, whose dependence on po and po comes only through the kinetic energy of the particle, then as a result of integration over these variables we do indeed recover the factor sin0 to appear with the subelement (de do).

2.2. (a) Verify explicitly the invariance of the volume element do of the phase space of a single particle under transformation from the Cartesian coordinates (x,y, z, px, Py, Pz) to the spherical polar coordinates (r,e,4,Pr.Pe,Po). (b) The foregoing result seems to contradict the intuitive notion of "equal weights for equal solid angles," because the factor sine is invisible in the expression for do. Show that if we average out any physical quantity, whose dependence on po and po comes only through the kinetic energy of the particle, then as a result of integration over these variables we do indeed recover the factor sin0 to appear with the subelement (de do).

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter7: Hamilton's Principle-lagrangian And Hamiltonian Dynamics

Section: Chapter Questions

Problem 7.8P

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