mSSM
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The following actually comes from Landau's 3rd Edit. Statistical Physics Part 1, Paragraph on Adiabatic Processes, Page 39.
I have the following two equations, where [itex]\lambda=\lambda(t)[/itex]. I am not so sure about [itex]S[/itex] (which is somewhat my problem):
[tex]\frac{\mathrm{d}S}{\mathrm{d}t} = \left( \frac{\mathrm{d}\lambda}{\mathrm{d}t} \right)^2[/tex]
Which is supposed to mean:
[tex]\frac{\mathrm{d}S}{\mathrm{d}\lambda} = \frac{\mathrm{d}\lambda}{\mathrm{d}t}[/tex]
Now, I thought that what is essentially being done there is multiplying the first equation such that we get:
[tex]\frac{\mathrm{d}S}{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}\lambda} = \frac{\mathrm{d}\lambda}{\mathrm{d}t}[/tex]
But if I now assume that:
[tex]\frac{\mathrm{d}S}{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}\lambda} = \frac{\mathrm{d}S}{\mathrm{d}\lambda}[/tex]
doesn't that in turn mean that [itex]t[/itex] is a function of [itex]\lambda[/itex]? Mathematically this seams sounds (to me), but physically this does not make so much sense, if [itex]t[/itex] is the time.
I have the following two equations, where [itex]\lambda=\lambda(t)[/itex]. I am not so sure about [itex]S[/itex] (which is somewhat my problem):
[tex]\frac{\mathrm{d}S}{\mathrm{d}t} = \left( \frac{\mathrm{d}\lambda}{\mathrm{d}t} \right)^2[/tex]
Which is supposed to mean:
[tex]\frac{\mathrm{d}S}{\mathrm{d}\lambda} = \frac{\mathrm{d}\lambda}{\mathrm{d}t}[/tex]
Now, I thought that what is essentially being done there is multiplying the first equation such that we get:
[tex]\frac{\mathrm{d}S}{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}\lambda} = \frac{\mathrm{d}\lambda}{\mathrm{d}t}[/tex]
But if I now assume that:
[tex]\frac{\mathrm{d}S}{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}\lambda} = \frac{\mathrm{d}S}{\mathrm{d}\lambda}[/tex]
doesn't that in turn mean that [itex]t[/itex] is a function of [itex]\lambda[/itex]? Mathematically this seams sounds (to me), but physically this does not make so much sense, if [itex]t[/itex] is the time.
