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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Function can be partially separable.
Simpler separability occurs when H is a sum of independent parts.
The Hamilton-Jacobi equation separates into N equations.Hamiltonian Separation
Constant value xh describe paraboloids of revolution
Other coordinate is f
Equate to cartesian system
Find differentials to get velocity.Parabolic Coordinates
Substituting for the new variables:
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