What causes a solution to be dependent on a parameter in differential equations?

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Discussion Overview

The discussion revolves around the dependence of solutions to a system of differential equations on a parameter, specifically focusing on the equations governing the dynamics of two vector quantities, \(\mathbf{m_1}\) and \(\mathbf{m_2}\). Participants explore the implications of parameter \(\gamma_{cr}\) and its role in the solutions, particularly when expressed in different forms or coordinate systems.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that the derivative \(\frac{d (\mathbf{m_1} + \mathbf{m_2})}{dt}\) does not explicitly depend on \(\gamma_{cr}\), but the dynamics of \(\mathbf{m_1}\) and \(\mathbf{m_2}\) are intertwined through the equations.
  • Another participant introduces a change of variables to show that the dependence on \(\gamma_{cr}\) arises implicitly through the dynamics of \(\mathbf{m_-}\), which is defined as the difference between \(\mathbf{m_1}\) and \(\mathbf{m_2}\).
  • A follow-up question is raised about the conditions under which \(\frac{d(\mathbf{m_1}+\mathbf{m_2})}{dt} \neq 0\) would affect the dependence on \(\gamma_{cr}\).
  • It is clarified that if the equation for \(\frac{d\mathbf{m_+}}{dt}\) is independent of \(\mathbf{m_-}\), then \(\mathbf{m_+}\) will not depend on \(\gamma_{cr}\), specifically occurring when \(\gamma_1 = \gamma_2\).
  • A participant inquires about general methods to determine when solutions are independent of parameters and whether there are established names for such systems.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which solutions depend on parameters, with some proposing specific cases while others challenge or refine these ideas. The discussion remains unresolved regarding a general method for identifying parameter independence in various systems.

Contextual Notes

The discussion highlights the complexity of parameter dependence in differential equations, with specific attention to the implications of variable changes and the relationships between different components of the system. Limitations include the need for further exploration of general principles and definitions related to parameter dependence.

Who May Find This Useful

Readers interested in differential equations, particularly those studying systems with multiple interacting components and their parameter dependencies, may find this discussion relevant.

BeauGeste
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I'm not sure where to put this but I though DEQ would be a good start.
I have been dealing with the following differential equations:

<br /> <br /> \frac{d \mathbf{m_1}}{dt} = \gamma_1 \mathbf{m_1} \times \mathbf{B} - \frac{n_{2}}{\gamma_{cr}} \mathbf{m_1} +<br /> \frac{n_1}{\gamma_{cr}} \mathbf{m_2}<br /> <br /> <br />


<br /> <br /> \frac{d \mathbf{m_2}}{dt} = \gamma_2 \mathbf{m_2} \times \mathbf{B} - \frac{n_{1}}{\gamma_{cr}} \mathbf{m_2} +<br /> \frac{n_2}{\gamma_{cr}} \mathbf{m_1}<br /> <br />

The weird things is that \frac{d (\mathbf{m_1} + \mathbf{m_2})}{dt} is not independent of \gamma_{cr} even though when you add bottom and top deq's the \gamma_{cr} terms cancel out. All I know is that it has something to do with \gamma_{1} \neq \gamma_{2} .

Also when you convert to polar coordinates (i assume the m-vectors are only in the x-y direction and B is in the z-direction), the 2 diff. eq.'s take on a form where it's obvious that the \gamma_{cr}'s stick around.

I'm not too fluent in differential equations so I'm not sure if there's an obvious answer here.
Thanks.
 
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\frac{d(\boldsymbol{m_1}+\boldsymbol{m_2})}{dt} certainly has no explicit dependence on \gamma_{cr}. However, the differential equation you get for it is not sufficient to uniquely specify \boldsymbol{m_1}+\boldsymbol{m_2}, even beyond the usual integration constant. We can see this by a change a variable.

So, let \boldsymbol{m_+} = \boldsymbol{m_1}+\boldsymbol{m_2} and \boldsymbol{m_-} = \boldsymbol{m_1}-\boldsymbol{m_2}.

Then, it's not too hard to see that

\frac{d\boldsymbol{m_+}}{dt} = (\gamma_+ \boldsymbol{m_+} + \gamma_- \boldsymbol{m_-})\times \boldsymbol{B}

where \gamma_+ = \frac{\gamma_1 + \gamma_2}{2} and \gamma_- = \frac{\gamma_1 - \gamma_2}{2}. This immediately means that it is necessary to know the dynamics of \boldsymbol{m_-}, as well. But, since that will be found by subtracting the two equations, it will automatically depend, explicitly, on \gamma_{cr}. Thus, \boldsymbol{m_+} will depend on \gamma_{cr} implicitly through its dependence on \boldsymbol{m_-}.
 
Wow. Thanks a lot for that. I've been banging my head for a while about that

A follow up question, if I may:

Will your reasoning be true whenever

\frac{d(\boldsymbol{m_1}+\boldsymbol{m_2})}{dt} \neq 0?
 
No, it won't. If the equation for \frac{d\boldsymbol{m_+}}{dt} is totally independent of \boldsymbol{m_-}, then \boldsymbol{m_+} will not depend on \gamma_{cr}. As your system is defined, this will happen whenever \gamma_- = 0. And, this will happen in exactly those cases where \gamma_1 = \gamma_2.
 
OK, I see. I guess the better question to ask is
When in general will you know that a solution will be independent of some parameter? For this case you made a change in variable and realized that the parameter in question will be an important part of the solutions. Is that something you would do for any system of equations? Or is there a more general way to tell?

Is there a general name for these type of systems? Would I find this info in a Diff. Eq. textbook?

Thanks again.
 

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