Remarks On Alfvens Perturbation Method
Remarks On Alfvens Perturbation Method
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3), the magnetic moment of the 'equivalent dipole' is a function of R. We would therefore expect p, (R) to give a contribution to (3. U) when R is varied. In actuality, however, this is not the case: p. May be treated as a constant in (3. U) if the variations of R are siifficiently slow. We shall now carry out the reasoning that leads to this conclusion.
- 7 - We first consider a homogeneous field along the z-axLs, ajid we intro- duce the cylindrical coordinates (p, ©, z). The z-coraponent of (...canonical) angular moraenttan, P_, is constant. That is, (U. L) Pq -= mp2© + i Agp ■ mp 6 + s— Hp = const. But, from (2, 9), mc Therefore, (U. 2) p^ » - 2|!. JI . - I^ ^, - const.
The action variable associated with the coordinate 9 is 2n >o (U. 3) Jq - Pe '^^ " 2" ^« = -^2n, .
For a homogeneous field, therefore, the magnetic moment of the equivalent dipole is a constant of the motion and is an action variable. It follows from the adiabatic theorem that it is also an adiabatic invariant, that is, the magnetic moment is unaffected by all changes that are 'adiabatically slow* compared with the period of the motion corresponding to an homogeneous field.
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To quickly assess the difficulty of the text, read a short excerpt:
3), the magnetic moment of the 'equivalent dipole' is a function of R. We would therefore expect p, (R) to give a contribution to (3. U) when R is varied. In actuality, however, this is not the case: p. May be treated as a constant in (3. U) if the variations of R are siifficiently slow. We shall now carry out the reasoning that leads to this conclusion.
- 7 - We first consider a homogeneous field along the z-axLs, ajid we intro- duce the cylindrical coordinates (p, ©, z). The z-coraponent of (...canonical) angular moraenttan, P_, is constant. That is, (U. L) Pq -= mp2© + i Agp ■ mp 6 + s— Hp = const. But, from (2, 9), mc Therefore, (U. 2) p^ » - 2|!. JI . - I^ ^, - const.
The action variable associated with the coordinate 9 is 2n >o (U. 3) Jq - Pe '^^ " 2" ^« = -^2n, .
For a homogeneous field, therefore, the magnetic moment of the equivalent dipole is a constant of the motion and is an action variable. It follows from the adiabatic theorem that it is also an adiabatic invariant, that is, the magnetic moment is unaffected by all changes that are 'adiabatically slow* compared with the period of the motion corresponding to an homogeneous field.
What to read after Remarks On Alfvens Perturbation Method?
You can find similar books in the "Read Also" column, or choose other free books by Owen Fleischman to read onlineMoreLess
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