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Hamilton's Equations and Force

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1) A particle of mass m moves in a plane, under the influence of a central force that depends only on it's distance from the origin. Write the Hamiltonian and Hamilton's equations.

2) A particle of mass m moves in a force filed whose potential in spherical coordinates is V = -(K cos theta)/ r^2. Obtain the canonical equations of motion.

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This solution shows detailed steps in writing the Hamiltonian and Hamilton's equations are also obtaining the canoncial equations of motion.

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(1) The Lagrangian is

L = m(dr/dt)^2/2 + mr^2(dθ/dt)^2/2 - V(r)

The momenta are the radial momentum

P = m(dr/dt)

and the angular momentum

M = mr^2(dθ/dt).

The Hamiltonian is

H = P^2/(2m) + ...

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