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Understanding Hamilton’s Principle Function and Its Separability Conditions

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This text explores Hamilton's principle function, focusing on cases of separability, where each dependent variable relies solely on one coordinate. It discusses the conditions necessary for Hamiltonian separability, including conservation of H and specific matrix forms of potential. The principles are illustrated with an example involving a charged particle influenced by two forces. It also covers the energy and momentum relationships in separable systems and provides methods to transform variables for effective analysis within the Hamilton-Jacobi framework.

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Understanding Hamilton’s Principle Function and Its Separability Conditions

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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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