The Hamiltonian of a particle in cylindrical coordinates is given by \(H=\frac{1}{2m}\left(P_\rho^2+\frac{P_\varphi^2}{\rho^2}+P_z^2\right)-\frac{k}{\sqrt{\rho^2+z^2}},\) then its equivalent Hamilton-Jacobi equation is (Where, Pρ, Pφ and Pz are conjugate momenta and S(ρ, φ, z, t) is Hamilton’s principal function.)
\(\frac{1}{2m}\left\{\left(\frac{\partial S}{\partial\rho}\right)^2+\frac{1}{\rho^2}\left(\frac{\partial S}{\partial\varphi}\right)^2+\left(\frac{\partial S}{\partial z}\right)^2\right\}+\frac{\partial S}{\partial t}-\frac{k}{\sqrt{\rho^2+z^2}}=0\)
The Hamilton–Jacobi equation is obtained by replacing every momentum by the derivative of Hamilton's principal function, \(P_i=\frac{\partial S}{\partial q_i}\), and writing \(H\left(q_i,\frac{\partial S}{\partial q_i},t\right)+\frac{\partial S}{\partial t}=0\).
Here \(P_\rho\to\frac{\partial S}{\partial\rho}\), \(P_\varphi\to\frac{\partial S}{\partial\varphi}\) and \(P_z\to\frac{\partial S}{\partial z}\).
The angular term keeps its metric factor: \(\frac{P_\varphi^2}{\rho^2}\to\frac{1}{\rho^2}\left(\frac{\partial S}{\partial\varphi}\right)^2\) (not ρ² times it).
The potential term is unchanged, \(-\frac{k}{\sqrt{\rho^2+z^2}}\) – the Coulomb/Kepler potential depends on the distance \(r=\sqrt{\rho^2+z^2}\), not on its square.
The time derivative enters with a plus sign, since \(\frac{\partial S}{\partial t}=-H\).
Putting these together:
\(\frac{1}{2m}\left\{\left(\frac{\partial S}{\partial\rho}\right)^2+\frac{1}{\rho^2}\left(\frac{\partial S}{\partial\varphi}\right)^2+\left(\frac{\partial S}{\partial z}\right)^2\right\}+\frac{\partial S}{\partial t}-\frac{k}{\sqrt{\rho^2+z^2}}=0\)
Hence, the answer is \(\frac{1}{2m}\left\{\left(\frac{\partial S}{\partial\rho}\right)^2+\frac{1}{\rho^2}\left(\frac{\partial S}{\partial\varphi}\right)^2+\left(\frac{\partial S}{\partial z}\right)^2\right\}+\frac{\partial S}{\partial t}-\frac{k}{\sqrt{\rho^2+z^2}}=0\).
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.................... is a client server application that identifies each host on the internet with a unique user-friendly name.
Every OS has a ................ or .......... which primarily resides in the main memory of the computer to perform some of the basic functions of the OS.
(i) Kernel
(ii) Interface
(iii) Nucleus
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Codes:
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