Separability

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

Staeckel Conditions  Specific conditions exist for separability. H is conserved.H is conserved. L is no more than quadratic in dq j /dt, so that in matrix form: H=1/2 (p  a)T -1 (p  a)+ V(q j )L is no more than quadratic in dq j /dt, so that in matrix form: H=1/2 (p  a)T -1 (p  a)+ V(q j ) The coordinates are orthogonal, so T is diagonal.The coordinates are orthogonal, so T is diagonal. The vector a has a j = a j ( q j )The vector a has a j = a j ( q j ) The potential is separable.The potential is separable. There exists a matrix  with  ij =  ij ( q i )There exists a matrix  with  ij =  ij ( q i )

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. X Y Z

Parabolic Coordinates  Select coordinates Constant value   describe paraboloids of revolutionConstant value   describe paraboloids of revolution Other coordinate is Other coordinate is  Equate to cartesian systemEquate to cartesian system  Find differentials to get velocity.

Energy and Momentum Substituting for the new variables:

Separation of Variables  Hamiltonian is not directly separable. Set E = T + VSet E = T + V Multiply by Multiply by   There are parts depending just on .  There is a cyclic coordinate . Constant of motion p Constant of motion p  Reduce to two degrees of freedomReduce to two degrees of freedom

Generator Separation  Set Hamilton’s function. Use momentum definitionUse momentum definition Expect two constants Expect two constants   Find one variable  Do the same for the other variable.  And get the last constant. next