Separability
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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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Prinicipal function
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.


Hamiltonian separation

Simpler separability occurs when H is a sum of independent parts.

The Hamilton-Jacobi equation separates into N equations.

Hamiltonian Separation


Staeckel conditions
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)


Combined potentials
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


Parabolic coordinates

Select coordinates

Constant value xh describe paraboloids of revolution

Other coordinate is f

Equate to cartesian system

Find differentials to get velocity.

Parabolic Coordinates


Energy and momentum
Energy and Momentum

Substituting for the new variables:


Separation of variables
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


Generator separation

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

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