Use the Hamilton-Jacobi method to find Hamilton's principal function W for a particle in the three dimensional isotropic oscillator well with a potential Hence obtain the corresponding momentum
and the corresponding action variables
To solve this problem using the Hamilton-Jacobi method, we follow these steps: --- ### Step 1: Hamilton-Jacobi Equation (HJE) The Hamiltonian for a three-dimensional isotropic harmonic oscillator is given by: \[ H = \frac{1}{2m} (p_x^2 + p_y^2 + p_z^2) + \frac{1}{2} k (x^2 + y^2 + z^2) \] Using the Hamilton-Jacobi equation (HJE), we replace \( p_x, p_y, p_z \) with derivatives of Hamilton’s principal function \( W(x,y,z) \): \[ H \left( \frac{\partial W}{\partial x}, \frac{\partial W}{\partial y}, \frac{\partial W}{\partial z}, x, y, z \right) = E \] Substituting the expressions for momentum: \[ \frac{1}{2m} \left( \left( \frac{\partial W}{\partial x} \right)^2 + \left( \frac{\partial W}{\partial y} \right)^2 + \left( \frac{\partial W}{\partial z} \right)^2 \right) + \frac{1}{2} k (x^2 + y^2 + z^2) = E \] Multiplying by \( 2m \): \[ \left( \frac{\partial W}{\partial x} \right)^2 + ____ ____ _________ ________ ______ _________ _____.
_______ _______ ____ _________ _____ _______.
___ _________ ______ ______ _________ _________ ________ ___ ________ __________.
______ _______ ______ _______ ____ _____ _________.
______ ____ ________ ________ _________ _________ _________ ______ __________ __________.
__________ __________ ___ _________ _______ ______ __________ __________ ___ ____.
_________ _____ ________ ________ ______.
________ ____ ________ _________ ______ _______ ______ ______.
___ __________ ____ ___ ______.
______ ______ _____ ____ ________ ______ ________ _______ _______ _______ ________ ____.
_____ ____ _______ ___ _______ ____ ___ _________ ___ _________ ________ ________.
____ _______ ______ ____ __________ ______.
________ ________ _____ ________ ______ ____ _________ ___ _______ __________ __________.
_________ _____ __________ ______ ___ ________ _________.
______ __________ ____ __________ ________ _________ _____ __________.
______ ___ _______ _______ _________ _____ _____ ________ _____ _________.
___ ______ ___ _______ _______ ______ __________ _____.
________ __________ _________ _______ _______ _______.
______ _________ _____ __________ ________ _________ _____ ________ ________ ________ ________ ___.
____ _________ ____ _________ ____ _____.
________ __________ ___ ____ _______ _______ ________ _________ ________ ______ _____ __________.
____ ____ ________ ______ _______ _______ _______ ______ ________ ______.
__________ ________ ______ _____ ________ ________ _____ _________ ___.
__________ ________ ________ ________ _______ ____ ____ _____ _______ __________ _________.
_____ ___ ________ ____ ____ _________ ______ ________ __________ ______.
_____ _________ ______ _______ _________ ___ _______ _______.
____ _________ ____ ______ ________ _______ ___ _____.
________ ________ ______ _______ ______ __________ _______ ________ ___.
________ ___ _______ ___ ______ ___ _____ _______ ____ ____ _________.
_________ _____ __________ _________ _________ ___ ________ _________.
________ ______ _______ ______ ______ _________.
____ ______ ____ ______ _____ ____ ____ ____ __________ ___.
_____ ________ _____ ______ ________ _________ _____ _________ ___.
___ ________ _________ _________ __________.
__________ ____ ____ __________ _______ _________ __________ ___ ________.
__________ _______ ________ _______ ____ _______ ________.
__________ ______ _____ ________ ____ _________ ____ _____ _________ ______ ________ __________.
_______ _______ ____ ______ _________ ______ _______ ____ ________.
______ ____ ____ ___ _______ _________ ____ ______ ________ ____.
_________ _______ ______ ____ __________ ____.
______ ________ _______ __________ ________ _________ ______.
______ _____ _________ ______ ________ _______ ____ __________ _______ ____ ____.
______ ___ _________ ____ __________ ____ __________ ___.
_____ ______ ________ _____ __________ _________ ______ _____ _____ ___.
______ __________ ________ ___ _______ __________ ____ ____ ______ _____ ____.
_______ ____ __________ _______ _____ _______ ______ __________ _______ _____ _______ ________.
____ _________ __________ ___ _______ ____ ___.
Get Full Answer on WhatsApp
Consider a Lagrangian L where q is cyclic. Show that the momentum conjugate to q is conserved under the translation q→q+a.
For a particle moving in central force field, the Hamiltonian of the system is given by
Assuming the motion to be elliptical, obtain action variables Jr and Jo.
What is Legendre transformation of a function? Express the Legendre transformation of a Lagrangian to obtain the Hamilton's function H.
(ii) The Hamiltonian of a simple harmonic oscillator is given as:
Show that the velocity vector is tangential to the curve defined by the Hamiltonian. Draw a labeled diagram of the velocity vector for the phase trajectory for energy E.
What is a canonical transformation? Why do we use canonical transformation?
Give an example where a canonical transformation becomes useful.
(ii) The Hamiltonian of a harmonic oscillator if given by
Solve the problem of the harmonic oscillator using canonical transformation with the generator
Use the Hamilton-Jacobi method to find Hamilton's principal function W for a particle in the three dimensional isotropic oscillator well with a potential Hence obtain the corresponding momentum
and the corresponding action variables
State Liouville's theorem and write the mathematical expression. Show that if pdepends on q.p through the Hamiltonian
Consider a particle of mass m moving in a plane attracted to the origin due to the potential k/r:
(i) Choose an appropriate coordinate
(ii) Write down the Lagrangian, the momenta conjugate to your choice of coordinates, the Hamiltonian and the action for the system.
(ii) Show that the action is invariant under translation of time. What conser vation law does this yield?