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# Separability - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about ' Separability' - ciara-crosby

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Presentation Transcript

### Separability

• 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.

The Hamilton-Jacobi equation separates into N equations.

Hamiltonian Separation

• 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)

• 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

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:

• 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

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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