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Higher order total derivatives


I am trying to avoid explicit specifying functional dependencies on
time in a dynamical system.
Total derivatives (Dt) seems like the right thing, but I can't get
Mathematica to make the right inferences for higher order derivatives.

Consider a simple example:

Block[{a, v, x, t},
 t /: Dt[v, t] = a;
 t /: Dt[x, t] = v;
 Dt[x, {t, 2}]
 ]

Output:

Dt[x, {t, 2}]

What would I have to do to make Dt[x,{t,2}] come out as "a"?

Nest[Dt[#, t] &, x, 2] gives the right answer, but I would rather not
have to go this way

/Janus



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