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Hamiltonian Dynamics

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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.

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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.

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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

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

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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

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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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