How to continually simplify derivatives of function defined in a system?

Question

For example, if I input something like

b'[t_] = a[t];
a'[t_] = -Sin[b[t]];
a''[t]

it outputs just a''[t].

Is there a way to get Mathematica to always simplify any derivatives of a and b to expressions only involving a and b (with no derivatives), so that whenever I type in something like a''[t] it outputs -Cos[b[t]]a[t]?

Yes I did clear, but it still doesn't work. Does it work for you? — keagan_callis, Feb 08 '21 at 00:54
If clearing does not work, you need to start again in a fresh kernel. If @cvgmt's and your example still doesn't evaluate as expected, then please post screenshots and version information. — J. M.'s missing motivation, Feb 08 '21 at 02:06
To evaluate a''[t] requires a[t] to be defined. Since only a'[t] is defined, you would need to enter a''[t]as D[a'[t], t] — Bob Hanlon, Feb 08 '21 at 02:23
Sorry - I see that cvgmt's example defines a[t] and not a'[t], and so it does work as his example intends but is different from my problem does not solve my problem — keagan_callis, Feb 08 '21 at 14:55

score 2 · Answer 1 · answered Feb 08 '21 at 07:01

2

You could try the following:

b' = a;
Derivative[n_?Positive][a] := Derivative[n-1][Function[-Sin[b[#]]]]

Then:

a''[t]

-a[t] Cos[b[t]]

answered Feb 08 '21 at 07:01

Carl Woll

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Yes thank you this does work if I do Derivative[n_?Positive][b] := Derivative[n - 1][Function[a[#]]]; Derivative[n_?Positive][a] := Derivative[n - 1][Function[-Sin[b[#]]]]; – keagan_callis Feb 08 '21 at 14:57
Would you mind explaining what the ?Positive part and the # is for? I don't know much about mathematica. – keagan_callis Feb 08 '21 at 15:04
@keagan_callis [`#](https://mathematica.stackexchange.com/a/25616/4999) is short forSlot[1], which represents the first argument toFunction.Derivativeoperates only on functions, not expressions (one usually usesD, as inD[expr, t]`, to operate on expressions). – Michael E2 Feb 09 '21 at 16:08

How to continually simplify derivatives of function defined in a system?

1 Answers1