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
### (i) Canonical Transformation A canonical transformation is a transformation in classical mechanics that preserves the form of Hamilton's equations. It transforms one set of canonical coordinates \((q, p)\) into another set \((Q, P)\) while maintaining the Hamiltonian structure of the equations of motion. #### Why do we use Canonical Transformations? Canonical transformations are useful because they allow us to simplify complex problems in Hamiltonian mechanics. They help: 1. Solve equations of motion more easily by choosing a new set of coordinates where the system becomes separable or _____ _______ _____ ____ __________ _______ ________ _____ _________ _______ ____ _______.
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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.
Consider a uniform square plate of length x ya and mass m. Obtain the moment of inertia along z-axis
Consider a damped harmonic oscillator whose Lagrangian is given by:
(i) Write the equation of motion and hence obtain the corresponding Hamiltonian.
(ii) Using Hamilton-Jacobi equation associated with the Hamiltonian, solve the equations of motion.
Using symplectic condition for canonical transformation, show that the transformation is canonical
Obtain the generating function for the transformation.
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
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.
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?