Difference between revisions of "MIC-Kepler problem"
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''in classical mechanics'' | ''in classical mechanics'' | ||
| − | The [[Hamiltonian system|Hamiltonian system]] | + | The [[Hamiltonian system|Hamiltonian system]] $ ( M, \omega _ \mu ,H _ \mu ) $, |
| + | where $ M = T ^ {*} { {\dot{\mathbf R} } } ^ {3} $, | ||
| + | $ { {\dot{\mathbf R} } } ^ {3} = \mathbf R ^ {3} \setminus \{ 0 \} $, | ||
| − | + | $$ | |
| + | \omega _ \mu = \sum _ {i = 1 } ^ { 3 } dp _ {i} \wedge dq _ {i} - { | ||
| + | \frac \mu {2 \left | q \right | ^ {3} } | ||
| + | } \sum _ {i,j,k = 1 } ^ { 3 } \epsilon _ {ijk } q _ {i} dq _ {j} \wedge dq _ {k} $$ | ||
and | and | ||
| − | + | $$ | |
| + | H _ \mu = { | ||
| + | \frac{\left | p \right | ^ {2} }{2} | ||
| + | } - { | ||
| + | \frac \alpha {\left | q \right | } | ||
| + | } + { | ||
| + | \frac{\mu ^ {2} }{2 \left | q \right | ^ {2} } | ||
| + | } , \alpha, \mu \in \mathbf R, \alpha > 0, | ||
| + | $$ | ||
| − | which can be considered as a one-parameter deformation family of the standard Kepler problem | + | which can be considered as a one-parameter deformation family of the standard Kepler problem $ ( M, \omega _ {0} ,H _ {0} ) $ |
| + | with the remarkable property that it retains its high dynamical symmetries. Physically, the deformation parameter $ \mu $ | ||
| + | is interpreted as the magnetic charge of the particle at rest and measures the pitch of the cone on which trajectories lie. Its genuine mathematical interpretation is as a [[Cohomology|cohomology]] class of the symplectic structure. The global symmetry group of the problem is either $ { \mathop{\rm SO} } ( 4 ) $, | ||
| + | $ E ( 3 ) $ | ||
| + | or $ { \mathop{\rm SO} } ( 3,1 ) $, | ||
| + | depending on whether the energy is negative, zero or positive. The motion satisfies Kepler's three laws (cf. [[Kepler equation|Kepler equation]]). | ||
| − | The Hilbert spaces associated with the quantized problem carry almost all unitary irreducible representations of the respective covering groups, the only exception being the group | + | The Hilbert spaces associated with the quantized problem carry almost all unitary irreducible representations of the respective covering groups, the only exception being the group $ { \mathop{\rm SL} } ( 2, \mathbf C ) $, |
| + | for which only the [[Principal series|principal series]] representations arise. All this allows one to derive the spectrum and multiplicities of the bound states, as well differential cross sections of the scattering process and quantizations of the magnetic charge. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. McIntosh, A. Cisneros, "Degeneracy in the presence of magnetic monopole" ''J. Math. Phys.'' , '''11''' (1970) pp. 896–916</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I. Mladenov, V. Tsanov, "Geometric quantization of the MIC-Kepler problem" ''J. Phys. A: Math. Gen.'' , '''20''' (1987) pp. 5865–5871</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> I. Mladenov, "Scattering of charged particles off dyons" ''J. Physics A Math. and Gen.'' , '''21''' (1988) pp. L1–L4</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. McIntosh, A. Cisneros, "Degeneracy in the presence of magnetic monopole" ''J. Math. Phys.'' , '''11''' (1970) pp. 896–916</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I. Mladenov, V. Tsanov, "Geometric quantization of the MIC-Kepler problem" ''J. Phys. A: Math. Gen.'' , '''20''' (1987) pp. 5865–5871</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> I. Mladenov, "Scattering of charged particles off dyons" ''J. Physics A Math. and Gen.'' , '''21''' (1988) pp. L1–L4</TD></TR></table> | ||
Latest revision as of 04:11, 6 June 2020
in classical mechanics
The Hamiltonian system $ ( M, \omega _ \mu ,H _ \mu ) $, where $ M = T ^ {*} { {\dot{\mathbf R} } } ^ {3} $, $ { {\dot{\mathbf R} } } ^ {3} = \mathbf R ^ {3} \setminus \{ 0 \} $,
$$ \omega _ \mu = \sum _ {i = 1 } ^ { 3 } dp _ {i} \wedge dq _ {i} - { \frac \mu {2 \left | q \right | ^ {3} } } \sum _ {i,j,k = 1 } ^ { 3 } \epsilon _ {ijk } q _ {i} dq _ {j} \wedge dq _ {k} $$
and
$$ H _ \mu = { \frac{\left | p \right | ^ {2} }{2} } - { \frac \alpha {\left | q \right | } } + { \frac{\mu ^ {2} }{2 \left | q \right | ^ {2} } } , \alpha, \mu \in \mathbf R, \alpha > 0, $$
which can be considered as a one-parameter deformation family of the standard Kepler problem $ ( M, \omega _ {0} ,H _ {0} ) $ with the remarkable property that it retains its high dynamical symmetries. Physically, the deformation parameter $ \mu $ is interpreted as the magnetic charge of the particle at rest and measures the pitch of the cone on which trajectories lie. Its genuine mathematical interpretation is as a cohomology class of the symplectic structure. The global symmetry group of the problem is either $ { \mathop{\rm SO} } ( 4 ) $, $ E ( 3 ) $ or $ { \mathop{\rm SO} } ( 3,1 ) $, depending on whether the energy is negative, zero or positive. The motion satisfies Kepler's three laws (cf. Kepler equation).
The Hilbert spaces associated with the quantized problem carry almost all unitary irreducible representations of the respective covering groups, the only exception being the group $ { \mathop{\rm SL} } ( 2, \mathbf C ) $, for which only the principal series representations arise. All this allows one to derive the spectrum and multiplicities of the bound states, as well differential cross sections of the scattering process and quantizations of the magnetic charge.
References
| [a1] | H. McIntosh, A. Cisneros, "Degeneracy in the presence of magnetic monopole" J. Math. Phys. , 11 (1970) pp. 896–916 |
| [a2] | I. Mladenov, V. Tsanov, "Geometric quantization of the MIC-Kepler problem" J. Phys. A: Math. Gen. , 20 (1987) pp. 5865–5871 |
| [a3] | I. Mladenov, "Scattering of charged particles off dyons" J. Physics A Math. and Gen. , 21 (1988) pp. L1–L4 |
MIC-Kepler problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=MIC-Kepler_problem&oldid=47741