# Hamiltonian Dynamics. Cylindrical Constraint Problem  A particle of mass m is attracted to the origin by a force proportional to the distance.  The.

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Hamiltonian Dynamics

Cylindrical Constraint Problem  A particle of mass m is attracted to the origin by a force proportional to the distance.  The particle is constrained to move on the surface of a cylinder of radius a.  Use Hamilton’s equations to find the equations of motion.

Lagrangian Form  The generalized coordinates are cylindrical coordinates. Radial term constant for the constraintRadial term constant for the constraint  Hamilton’s method starts with a Lagrangian.

Conjugate Replacement  The conjugate momenta are found and used in place of generalized velocities. Replace in the LagrangianReplace in the Lagrangian Replace in the HamiltonianReplace in the Hamiltonian

Hamilton’s Equations  Hamilton’s equations give the equations of motion. Constant angular momentum Simple harmonic oscillations in z

Electromagnetic Hamiltonian  The Lagrangian can be written in terms of a generalized potential. Both E and B derive from potentials , A. Cartesian coordinates for momentum and velocity  The Hamiltonian is a function of momentum and potentials. , A depend on position and time

Electromagnetic Energy  The Jacobean energy integral equals the kinetic plus electrostatic potential. Kinetic energyKinetic energy Kinetic “momentum”Kinetic “momentum”  The kinetic energy and momentum are related the usual way. But they are not conjugateBut they are not conjugate next

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