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i`d like to consider system with Coulomb potential: $U = -\frac{\alpha}{r}$ and constant magnetic field. It is easy to write Lagrangian function and get trajectory equation in cylindrical coordinate system. $$\phi = \sqrt{\frac{m}{2}}\int \frac{\frac{M}{mr^2} - \Omega}{\sqrt{E-U_{eff}(r)}}dr,$$ where $\Omega = \frac{eH}{2mc}$, $$U_{eff} = -\frac{\alpha}{r} + \frac{mr^2}{2}(\Omega - \frac{M}{mr^2})^2.$$

If we expand brackets, we can get such integral, that i want to analyze: $$\phi = \frac{M}{\sqrt{2m}}\int\frac{dr}{r^2\sqrt{E+M\Omega + \frac{\alpha}{r} - \frac{M^2}{2mr^2} - \frac{m\Omega^2r^2}{2}}} - \Omega\cdot t $$

So we can see, that effect of the magnetic field is reduced to the replacement of energy by the expression $E' = E + M\Omega$.

We have field: $$U = U_1 + U_2 = -\frac{\alpha}{r} + \frac{M^2}{2mr^2} + \frac{m\Omega^2r^2}{2},$$

Where $U_1 = -\frac{\alpha}{r} + \frac{M^2}{2mr^2}$ and $U_2 = \frac{m\Omega^2r^2}{2}$

i want to consider the case when $|U_2|<<|U_1|$ and $|U_1|<<|U_2|$.

First case:

we can Expand in taylor series denominator, but I expect divergences at the turning points. Also i know, that i can say "$|U_2|$ gives our system precession", and i can calculate it. But for the second case i can`t do the same thing, i think that there will be a drift of trajectory,but i'm not sure.

So i want to know, how we can expand denominator and get approximation of trajectory directly(including coordinates near the turning points).The problem of decomposition into a perturbation series interests me more than the solution of the problem itself.

i`d like to consider system with Coulomb potential: $U = -\frac{\alpha}{r}$ and constant magnetic field.It is easy to write Lagrangian function: $$ L = \frac{m}{2}(\dot{\rho}^2 + \rho^2\dot{\phi}^2) + \frac{eH}{2c}\rho^2 \dot{\phi} + \frac{\alpha}{r} $$ and get trajectory equation in cylindrical coordinate system. $$\phi = \sqrt{\frac{m}{2}}\int \frac{\frac{M}{mr^2} - \Omega}{\sqrt{E-U_{eff}(r)}}dr,$$ where $\Omega = \frac{eH}{2mc}$, $$U_{eff} = -\frac{\alpha}{r} + \frac{mr^2}{2}(\Omega - \frac{M}{mr^2})^2.$$

If we expand brackets, we can get such integral, that i want to analyze: $$\phi = \frac{M}{\sqrt{2m}}\int\frac{dr}{r^2\sqrt{E+M\Omega + \frac{\alpha}{r} - \frac{M^2}{2mr^2} - \frac{m\Omega^2r^2}{2}}} - \Omega\cdot t $$

So we can see, that effect of the magnetic field is reduced to the replacement of energy by the expression $E' = E + M\Omega$.

We have field: $$U = U_1 + U_2 = -\frac{\alpha}{r} + \frac{M^2}{2mr^2} + \frac{m\Omega^2r^2}{2},$$

Where $U_1 = -\frac{\alpha}{r} + \frac{M^2}{2mr^2}$ and $U_2 = \frac{m\Omega^2r^2}{2}$

i want to consider the case when $|U_2|<<|U_1|$ and $|U_1|<<|U_2|$.

First case:

we can Expand in taylor series denominator, but I expect divergences at the turning points. Also i know, that i can say "$|U_2|$ gives our system precession", and i can calculate it. But for the second case i can`t do the same thing, i think that there will be a drift of trajectory,but i'm not sure.

So i want to know, how we can expand denominator and get approximation of trajectory directly(including coordinates near the turning points).The problem of decomposition into a perturbation series interests me more than the solution of the problem itself.

i`d like to consider system with Coulomb potential: $U = -\frac{\alpha}{r}$ and constant magnetic field. It is easy to write Lagrangian function and get trajectory equation in cylindrical coordinate system. $$\phi = \sqrt{\frac{m}{2}}\int \frac{\frac{M}{mr^2} - \Omega}{\sqrt{E-U_{eff}(r)}}dr,$$ where $\Omega = \frac{eH}{2mc}$, $U_{eff} = -\frac{\alpha}{r} + \frac{mr^2}{2}(\Omega - \frac{M}{mr^2})^2$

If we expand brackets, we can get such integral, that i want to analyze: $$\phi = \frac{M}{\sqrt{2m}}\int\frac{dr}{r^2\sqrt{E+M\Omega + \frac{\alpha}{r} - \frac{M^2}{2mr^2} - \frac{m\Omega^2r^2}{2}}} - \Omega\cdot t $$

So we can see, that effect of the magnetic field is reduced to the replacement of energy by the expression $E' = E + M\Omega$.

We have field: $U = U_1 + U_2 = -\frac{\alpha}{r} + \frac{M^2}{2mr^2} + \frac{m\Omega^2r^2}{2}$,

Where $U_1 = -\frac{\alpha}{r} + \frac{M^2}{2mr^2}$ and $U_2 = \frac{m\Omega^2r^2}{2}$

i want to consider the case when $|U_2|<<|U_1|$ and $|U_1|<<|U_2|$.

First case:

we can Expand in taylor series denominator, but I expect divergences at the turning points. Also i know, that i can say "$|U_2|$ gives our system precession", and i can calculate it. But for the second case i can`t do the same thing, i think that there will be a drift of trajectory,but i'm not sure.

So i want to know, how we can expand denominator and get approximation of trajectory directly(including coordinates near the turning points).The problem of decomposition into a perturbation series interests me more than the solution of the problem itself.

i`d like to consider system with Coulomb potential: $U = -\frac{\alpha}{r}$ and constant magnetic field. It is easy to write Lagrangian function and get trajectory equation in cylindrical coordinate system. $$\phi = \sqrt{\frac{m}{2}}\int \frac{\frac{M}{mr^2} - \Omega}{\sqrt{E-U_{eff}(r)}}dr,$$ where $\Omega = \frac{eH}{2mc}$, $$U_{eff} = -\frac{\alpha}{r} + \frac{mr^2}{2}(\Omega - \frac{M}{mr^2})^2.$$

If we expand brackets, we can get such integral, that i want to analyze: $$\phi = \frac{M}{\sqrt{2m}}\int\frac{dr}{r^2\sqrt{E+M\Omega + \frac{\alpha}{r} - \frac{M^2}{2mr^2} - \frac{m\Omega^2r^2}{2}}} - \Omega\cdot t $$

So we can see, that effect of the magnetic field is reduced to the replacement of energy by the expression $E' = E + M\Omega$.

We have field: $$U = U_1 + U_2 = -\frac{\alpha}{r} + \frac{M^2}{2mr^2} + \frac{m\Omega^2r^2}{2},$$

Where $U_1 = -\frac{\alpha}{r} + \frac{M^2}{2mr^2}$ and $U_2 = \frac{m\Omega^2r^2}{2}$

i want to consider the case when $|U_2|<<|U_1|$ and $|U_1|<<|U_2|$.

First case:

we can Expand in taylor series denominator, but I expect divergences at the turning points. Also i know, that i can say "$|U_2|$ gives our system precession", and i can calculate it. But for the second case i can`t do the same thing, i think that there will be a drift of trajectory,but i'm not sure.

So i want to know, how we can expand denominator and get approximation of trajectory directly(including coordinates near the turning points).The problem of decomposition into a perturbation series interests me more than the solution of the problem itself.

i`d like to consider system with Coulomb potential: $U = -\frac{\alpha}{r}$ and constant magnetic field. It is easy to write Lagrangian function and get trajectory equation in cylindrical coordinate system. $$\phi = \sqrt{\frac{m}{2}}\int \frac{\frac{M}{mr^2} - \Omega}{\sqrt{E-U_{eff}(r)}}dr,$$ where $\Omega = \frac{eH}{2mc}$, $U_{eff} = -\frac{\alpha}{r} + \frac{mr^2}{2}(\Omega - \frac{M}{mr^2})^2$

If we expand brackets, we can get such integral, that i want to analyze: $$\phi = \frac{p_{\phi}}{\sqrt{2m}}\int\frac{dr}{r^2\sqrt{E+M\Omega + \frac{\alpha}{r} - \frac{M^2}{2mr^2} - \frac{m\Omega^2r^2}{2}}} - \Omega\cdot t $$

So we can see, that effect of the magnetic field is reduced to the replacement of energy by the expression $E' = E + M\Omega$.

We have field: $U = U_1 + U_2 = -\frac{\alpha}{r} + \frac{M^2}{2mr^2} + \frac{m\Omega^2r^2}{2}$,

Where $U_1 = -\frac{\alpha}{r} + \frac{M^2}{2mr^2}$ and $U_2 = \frac{m\Omega^2r^2}{2}$

i want to consider the case when $|U_2|<<|U_1|$ and $|U_1|<<|U_2|$.

First case:

we can Expand in taylor series denominator, but I expect divergences at the turning points. Also i know, that i can say "$|U_2|$ gives our system precession", and i can calculate it. But for the second case i can`t do the same thing, i think that there will be a drift of trajectory,but i'm not sure.

So i want to know, how we can expand denominator and get approximation of trajectory directly(including coordinates near the turning points).The problem of decomposition into a perturbation series interests me more than the solution of the problem itself.

i`d like to consider system with Coulomb potential: $U = -\frac{\alpha}{r}$ and constant magnetic field. It is easy to write Lagrangian function and get trajectory equation in cylindrical coordinate system. $$\phi = \sqrt{\frac{m}{2}}\int \frac{\frac{M}{mr^2} - \Omega}{\sqrt{E-U_{eff}(r)}}dr,$$ where $\Omega = \frac{eH}{2mc}$, $U_{eff} = -\frac{\alpha}{r} + \frac{mr^2}{2}(\Omega - \frac{M}{mr^2})^2$

If we expand brackets, we can get such integral, that i want to analyze: $$\phi = \frac{M}{\sqrt{2m}}\int\frac{dr}{r^2\sqrt{E+M\Omega + \frac{\alpha}{r} - \frac{M^2}{2mr^2} - \frac{m\Omega^2r^2}{2}}} - \Omega\cdot t $$

So we can see, that effect of the magnetic field is reduced to the replacement of energy by the expression $E' = E + M\Omega$.

We have field: $U = U_1 + U_2 = -\frac{\alpha}{r} + \frac{M^2}{2mr^2} + \frac{m\Omega^2r^2}{2}$,

Where $U_1 = -\frac{\alpha}{r} + \frac{M^2}{2mr^2}$ and $U_2 = \frac{m\Omega^2r^2}{2}$

i want to consider the case when $|U_2|<<|U_1|$ and $|U_1|<<|U_2|$.

First case:

we can Expand in taylor series denominator, but I expect divergences at the turning points. Also i know, that i can say "$|U_2|$ gives our system precession", and i can calculate it. But for the second case i can`t do the same thing, i think that there will be a drift of trajectory,but i'm not sure.

So i want to know, how we can expand denominator and get approximation of trajectory directly(including coordinates near the turning points).The problem of decomposition into a perturbation series interests me more than the solution of the problem itself.