Question 9.4: Find a complete integral of the Hamilton-Jacobi equation for......

The Hamiltonian for this problem is given by (7.15) with V(r) replaced by V(r, θ) defined by (9.57):

H=\dot{r} p_{r}+\dot{\theta } p_{\theta }+\dot{\phi } p_{\phi }-L=\frac{1}{2m}\left(p^{2}_{r}+\frac{p^{2}_{\theta }}{r^{2}}+\frac{p^{2}_{\phi }}{r^{2}\sin^{2} \theta } \right) +V\left(r\right).  (7.15)

H = \frac{1}{2m}\left(p^{2}_{r}+\frac{p^{2}_{\theta}}{r^{2}}+ \frac{p^{2}_{\phi}}{r^{2} \sin^{2}\theta}\right)+a\left(r\right) + \frac{b\left(\theta \right) }{r^{2}}.  (9.58)

The Hamilton-Jacobi equation takes the form

\frac{1}{2m}\left[\left(\frac{\partial S}{\partial r} \right)^{2} +\frac{1}{r^{2}} \left(\frac{\partial S}{\partial \theta } \right)^{2}+\frac{1}{r^{2}\sin^{2}\theta }\left(\frac{\partial S}{\partial \phi } \right)^{2} \right] +a\left(r\right) +\frac{b\left(\theta \right) }{r^{2}} +\frac{\partial S}{\partial t} =0.  (9.59)

Since H does not explicitly depend on time and \phi is a cyclic coordinate, we can write

S =- \alpha _{1}t + \alpha _{ \phi} \phi+ W\left(r, \theta \right),   (9.60)

where W satisfies

\frac{1}{2m}\left[\left(\frac{\partial W}{\partial r}\right)^{2}+ \frac{1}{r^{2}}\left(\frac{\partial W}{\partial \theta}\right)^{2}+ \frac{\alpha ^{2}_{ \phi}}{r^{2} \sin^{2} \theta}\right]+ a\left(r\right) +\frac{b\left(\theta \right) }{r^{2}}= \alpha _{1}.   (9.61)

By trying a solution to this equation of the form

W(r, θ) = W_{1}(r) + W_{2}(θ), (9.62)

we are led to

\frac{1}{2m}\left[\left(\frac{\partial W_{1}}{d r}\right)^{2}+ \frac{1}{r^{2}}\left(\frac{\partial W_{2}}{d\theta}\right)^{2}+ \frac{\alpha ^{2}_{ \phi}}{r^{2} \sin^{2} \theta}\right]+ a\left(r\right) +\frac{b\left(\theta \right) }{r^{2}}= \alpha _{1}. (9.63)

Multiplying the above equation by r² we arrive at

r^{2}\left[\frac{1}{2m}\left(\frac{dW_{1}}{d r}\right)^{2}+a\left(r\right) -\alpha _{1}\right]=-\left[\frac{1}{2m}\left(\frac{dW_{2}}{d \theta}\right)^{2}+\frac{\alpha ^{2}_{ \phi}}{2m \sin^{2} \theta}+b\left(\theta \right)\right]= – \frac{\alpha ^{2}_{ \theta}}{2m},   (9.64)

with the variables r and θ separated and the separation constant conveniently denoted by −\alpha ^{2}_{ \theta} /2m. Equation (9.64) amounts to the two ordinary differential equations

\frac{dW_{1}}{d r}= \sqrt{2m\left[\alpha _{1}-a\left(r\right) \right]-\frac{\alpha ^{2}_{\theta }}{r^{2}} } ,  (9.65a)

\frac{dW_{2}}{d\theta}= \sqrt{\alpha ^{2}_{ \theta}-2mb\left(\theta \right) -\frac{\alpha ^{2}_{\phi }}{\sin^{2}\theta} } .  (9.65b)

Integrating these equations and inserting the results into (9.62) and (9.60) there results a complete integral of (9.59) in the form

S=-\alpha _{1}t+\alpha _{\phi }\phi +\int{\left[2m\left[\alpha _{1}-a\left(r\right) \right]-\frac{\alpha ^{2}_{\theta }}{r^{2}}\right]^{{1}/{2}}dr } +\int{\left[\alpha ^{2}_{ \theta}-2mb\left(\theta \right) -\frac{\alpha ^{2}_{\phi }}{\sin^{2}\theta}\right]^{{1}/{2}} d\theta } .  (9.66)

The solution of the equations of motion is given by

\beta _{1}=\frac{\partial S}{\partial \alpha _{1} }=-t+m\int{\frac{dr}{\left[2m\left[\alpha _{1}-a\left(r\right) \right]-{\alpha ^{2}_{\theta }}/{r^{2}} \right]^{{1}/{2}} } },  (9.67)

\beta _{\theta }=\frac{\partial S}{\partial \alpha _{\theta } }=-\int{\frac{\alpha _{\theta }dr}{r^{2}\left[2m\left[\alpha _{1}-a\left(r\right) \right]-{\alpha ^{2}_{\theta }}/{r^{2}} \right]^{{1}/{2}} } }+\int{\frac{\alpha _{\theta }d\theta }{\left[\alpha ^{2}_{\theta }-2mb\left(\theta \right) -{\alpha ^{2}_{\phi }}/{\sin ^{2}\theta } \right]^{{1}/{2}} } },  (9.68)

\beta _{\phi }=\frac{\partial S}{\partial \alpha _{\phi } }=\phi -\int{\frac{\alpha _{\phi }d\theta }{\sin ^{2}\theta\left[\alpha ^{2}_{\theta }-2mb\left(\theta \right) -{\alpha ^{2}_{\phi }}/{\sin ^{2}\theta } \right]^{{1}/{2}} } }.  (9.69)

It is hardly possible not to wonder at the impressive swiftness and simplicity with which the solution of this problem is reduced to quadratures by the Hamilton-Jacobi method. Equation (9.67) furnishes r(t) which, after substitution into (9.68), determines θ(t). Having found θ(t), by means of (9.69) we obtain \phi(t), completing the resolution of the equations of motion. If one is only interested in the shape of the geometric trajectory described by the particle, it suffices to use the two last equations to express the equation of the curve in the form r = r(θ), \phi=\phi(θ) or θ = θ(\phi), r = r(\phi).