As I understand your problem, you are looking to solve for motion with a constraint like ##|\dot{\vec{r}}|=bt+c##.
You could take a polar parametrization like ##r(t)=R_0(1+d\sin{\theta(t)})##, substitute it into the polar form of the constraint...
What a question, haha! I will suggest another question to you that you can look into. How, given the time-symmetric equations of motion in both classical (Newton's physics) and quantum physics, can the arrow of time (thermodynamics) arise?
The last integral here looks like a multiple of the complete elliptic integral of the second kind. If a special function counts as an analytical solution, then you might be in luck. As for the logarithmic integral, I'm unclear if you can even find special functions.
Do you happen to know the source for the quote? I did a quick Google search and couldn't find it. People rarely proclaim a sentence or two and leave it at that---context would help. The quote is sufficiently woowoo that you could read all kinds of meaning into it.
It also strikes me as a little strange. The physics Nobel appears to be shared between work specifically done in climate science (two of the winners share half the prize for this) and work for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to...
Two people share half the prize for climate models, the other half goes to Parisi. Climate is probably just a better publicity generator than "theory of disordered materials and random processes."
I think what makes thermodynamics seem metaphysical is the willy-nilly use of limit taking combined with the fact that thermodynamics is the subject whose sole purpose is to relate physical variables; i.e. if you take a limit here you're probably taking it for any number of other variables...
The hard part is deciding what is "relevant" as used in the rule. The tricky part of conflicts is that it is hard to determine from the perspective of the person who may have a conflict. That means to me that you always have to lean in the direction of full disclosure. Outside of ethics, I think...
I appreciate the explanation but I think it still ignores the issue which I believe complicates the analysis. That is, the dependence of the force/potential on the relative orientation of various vector quantities in the problem.
I share the intuition of many of those who have posted about the behavior for central potentials. However, as a nitpick, I'm not entirely convinced that just because something isn't 1/r or r^2 for potential there can't be closed orbits. Bertrand's theorem is the mathematical statement of what...
How do you distinguish the example problems solved in a standard physics textbook from what you are looking for? In general, "interesting" is not a good proxy for what will teach you the skills to solve problems yourself.
Maybe the Feynman lectures would suit your purpose. They are kind of...
Just because you can find planck's constant in some expression and isolate it doesn't mean the variables on the other side of the equals sign relate to spin. If you study quantum mechanics, you will find that planck's constant appears... well... everywhere.