Question 9.6: Construct Hamilton’s principal function for a particle subje......
Construct Hamilton’s principal function for a particle subject to a constant force F.
The physical motion with x(0) = x_{0} is
x(\tau ) =x_{0} + v_{0}\tau +\frac{F}{2m}\tau^{2}. (9.107)
Imposing the condition x(t) = x we obtain
v_{0}= \frac{x-x_{0}}{t}-\frac{F}{2m}t, (9.108)
whence
x(\tau ) =x_{0}+ \left(x-x_{0}\right)\frac{\tau}{t}+\frac{F\tau}{2m}\left(\tau − t\right). (9.109)
Therefore, with t_{0} = 0,
S\left(x,x_{0},t\right)=\int_{0}^{t}{\left[\frac{m}{2}\left(\frac{dx}{d\tau } \right)^{2}+ Fx\left(\tau \right )\right]d\tau }=\frac{m\left(x-x_{0}\right)^{2} }{2t} +\frac{x+x_{0}}{2}Ft-\frac{F^{2}t^{3}}{24m}. (9.110)
The reader is invited to check that the Hamilton-Jacobi equation
\frac{1}{2m}\left(\frac{\partial S}{\partial x}\right)^{2}-Fx+\frac{\partial S}{\partial t}=0 (9.111)
is satisfied by Hamilton’s principal function (9.110).