Chapter 6.1, Problem 6.4P

Verify the Jacobi identity for Poisson brackets, {A, [Expect to use a few lines to answer this question.] {B,C'}}+{B, {C, A}} + {C, {A, B}} =

A. at pattern. Let's ssuming that the e angular spatial poral frequencies w, correspond to e is then W, 1) ir) [7.33] avelength of the quals the group eing amplitude- e waves of fre- of modulating and sum over is called the wer sideband. 7.19 Given the dispersion relation w = ak', compute both the phase and group velocities. -7.20* Using the relation 1/v = dk/dv. prove that 1 Vg I V₂ 7.21* In the case of lightwaves, show that V = 1 Ve v dv v² dv V 7.22 The speed of propagation of a surface wave in a liquid of depth much greater than λ is given by v dn c dv 11 -+- C solve 7.25 only where g = acceleration of gravity, λ = wavelength, p = density. Y = surface tension. Compute the group velocity of a pulse in the long wavelength limit (these are called gravity waves). 7.23* Show that the group velocity can be written as dv dλ 7.24 Show that the group velocity can be written as gλ 2πΥ + 2πT pλ Vg = V-A ( n + w(dn/dw) 7.25 With the previous problem in mind prove that dn (v) dv n₂ = n(v) + v.…

A point particle moves in space under the influence of a force derivablefrom a generalized potential of the formU(r, v) = V (r) + σ · L,where r is the radius vector from a fixed point, L is the angular momentumabout that point, and σ is the fixed vector in space. Find the components of the force on the particle in spherical polar  coordinates, on the basis of the equation for the components of the generalized force Qj: Qj = −∂U/∂qj + d/dt (∂U/∂q˙j)