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Separability

Separability. Prinicipal Function. In some cases Hamilton’s principal function can be separated. Each W depends on only one coordinate. This is totally separable. Function can be partially separable. Simpler separability occurs when H is a sum of independent parts.

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Separability

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  1. Separability

  2. Prinicipal Function • In some cases Hamilton’s principal function can be separated. • Each W depends on only one coordinate. • This is totally separable. Function can be partially separable.

  3. Simpler separability occurs when H is a sum of independent parts. The Hamilton-Jacobi equation separates into N equations. Hamiltonian Separation

  4. Staeckel Conditions • Specific conditions exist for separability. • H is conserved. • L is no more than quadratic in dqj/dt, so that in matrix form: H=1/2(p - a)T-1(p -a)+V(qj) • The coordinates are orthogonal, so T is diagonal. • The vector a has aj = aj (qj) • The potential is separable. • There exists a matrix F with Fij = Fij(qi)

  5. Combined Potentials • Particle under two forces • Attractive central force • Uniform field along z • Eg: charged particle with another fixed point charge in a uniform electric field. Z Y X

  6. Select coordinates Constant value xh describe paraboloids of revolution Other coordinate is f Equate to cartesian system Find differentials to get velocity. Parabolic Coordinates

  7. Energy and Momentum Substituting for the new variables:

  8. Separation of Variables • Hamiltonian is not directly separable. • Set E = T + V • Multiply by (x + h)/2 • There are parts depending just on x, h. • There is a cyclic coordinate f. • Constant of motion pf • Reduce to two degrees of freedom

  9. Set Hamilton’s function. Use momentum definition Expect two constants a, b Find one variable Do the same for the other variable. And get the last constant. Generator Separation next

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