Hi,
On Tue, Jul 7, 2009 at 10:29 PM, William Stein<wst...@gmail.com> wrote:
>> In new symbolics, if I do the same I get
>> --------
>> sage: g
>> D[0](f)(x)
>> sage: g.subs_expr(f(x)==f(x)+df(x))
>> D[0](f)(x)
>> ---------
>>
>> Is there a way to do this in new symbolics? It seems
>> to be a road block in implementing calculus of variations
>> in new symbolics.
>
> I don't see any way to do this yet, which is frustrating. This was as
> close as I got, which is really awful:
>
> sage: g.operator().change_function(f(x)+df(x))(x)
> D[0](f)(x) + D[0](df)(x)
I got another workaround
-----
for n in range(max):
S = S.subs_expr(f(x).diff(x,n)==(f(x)+df(x)).diff(x,n))
-----
It gets my work done as I know 'max' for my problems.
However, I think we need a proper solution not a workaround.
> I really hope once we get these issues sorted out and you implement
> calculus of variations, that you include it in Sage.
Definitely! Right now, I see four issues that need to be sorted
out before calculus of variations can be implemented in Sage.
(1) Improved .subs_expr() method so that one can vary a
functional of the form
S0 = L( q(t), q(t).diff(t) ).integral(t, -oo, oo)
to
S = L( q(t) + dq(t), (q(t)+dq(t)).diff(t) ).integral(t, -oo, oo)
where dq(t) = epsilon*f(t), f(t) a test function.
(2) New D derivative operator does not know how to act
on symbolic integration ( #6465 ). So right now it is not
possible to compute "S.diff(epsilon)"
I have spent last two days in trying to fix this. I believe
this is much more serious issue than the previous one
and this requires changes in both pynac and sage.
Pynac needs to check for special differentiation rules for
function like "integrate( f(x,y), x)" before applying chain rules.
Current "_derivative_" method of SFunction class is inadequate
for this as it is called after chain rule has been applied.
(3) In physics, we always use test function "f(t)" to be Dirac delta.
This is DONE and will be part of sage-4.1
(4) We also need integration algorithm where integrand involves
Dirac delta. This is mostly ready and I will post a ticket once
new symbolic integration class is cleaned up so that it can
be hooked up properly.
So, I guess (1) and (2) need real work mostly in pynac. I was
glancing through ginac code. Ginac seems to apply diff() method
properly to its own integral() method.
Cheers,
Golam
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