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Re: Differentiating with respect to an expression

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rouben@shadow.(none) (Rouben Rostamian) writes:

> In article <87sihpl61w.fsf@san.rr.com>, Joe Riel  <joer@san.rr.com> wrote:
>>rouben@shadow.(none) (Rouben Rostamian) writes:
>>
>>> The following issue comes up quite often in the context of
>>> analytical mechanics.  I have a clunky solution for it.
>>> I wonder if there is a clever way.
>>>
>>> Let's say we have L = x^2 + x'^2 * x''^2, where x is a function of t,
>>> and in the usual mathematical notation, x' and x'' are the first and
>>> second derivatives of x.
>>>
>>> We want to compute the derivative of L with respect to x'.  The
>>> answer should be 2*x'*x''^2.  Here is the way I do it in Maple:
>>>
>>> restart;
>>> L := x(t)^2 + diff(x(t),t)^2 * diff(x(t),t,t)^2;
>>> subs(diff(x(t),t,t)=Z2, diff(x(t),t)=Z1, %);
>>> diff(%, Z1);
>>> subs(Z1=diff(x(t),t), Z2=diff(x(t),t,t), %);
>>>
>>> The L shown above is simple enough so that we don't need a
>>> CAS to compute the derivative.  The L in a real example will
>>> be the result of a long chain of calculations, will depend on
>>> several functions and their derivatives, and will take up a
>>> couple of screenfuls.
>>>
>>> If there is a clever way to compute that derivative, I would
>>> like to know.
>>
>>A low-level way to do this is with frontend:
>>
>>frontend(diff, [L,diff(x(t),t)]);
>>       2*diff(x(t),t)*diff(diff(x(t),t),t)^2
>>
>>An alternative method is to use the Physics package:
>>
>>with(Physics):
>>
>>diff(L, diff(x(t),t));
>>       2*diff(x(t),t)*diff(diff(x(t),t),t)^2
>>
>># alternatively, with Physics
>>
>>diff(L, D(x)(t));
>>     2*D(x)(t)*`@@`(D,2)(x)(t)^2
>>
>
> Thanks much, Joe, this is exactly what I was hoping for.
> I had no idea about frontend() or the Physics package.
>
> This brings up a somewhat related question.  Doing
>    frontend(int, [x(t)^2, x(t)]);
> we get x(t)^3/3, which is fine.  The following, however,
> issues an Error message:
>     frontend(int, [x(t)^2, x(t)=a..b]);
>
> Is there a way to get that to work too?
>
> Rouben

The optional third argument of frontend can be used for that.
See the help page.  It consists of a list of two sets.
The first set is significant here, it contains the types 
that frontend does *not* freeze.  By default the `+`, `*`, and `^` 
types are not frozen.  In this case, you also do *not* want
to freeze `=` and `..`.  So the following works:

  frontend(int, [x(t)^2, x(t)=a..b], [{`+`,`^`,`*`,`..`,`=`},{}]);
       1/3*b^3-1/3*a^3

Frequently one can use the Not type to specify what you do
want frozen.  For example,

  frontend(int, [x(t)^2, x(t)=a..b], [{Not(identical(x(t)))},{}]);
       1/3*b^3-1/3*a^3



-- 
Joe Riel

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Thread

Differentiating with respect to an expression rouben@shadow.(none) (Rouben Rostamian) - 2014-11-11 22:10 +0000
  Re: Differentiating with respect to an expression Joe Riel <joer@san.rr.com> - 2014-11-11 14:35 -0800
    Re: Differentiating with respect to an expression rouben@shadow.(none) (Rouben Rostamian) - 2014-11-11 23:39 +0000
      Re: Differentiating with respect to an expression "Nasser M. Abbasi" <nma@12000.org> - 2014-11-11 18:50 -0600
        Re: Differentiating with respect to an expression "Nasser M. Abbasi" <nma@12000.org> - 2014-11-11 19:44 -0600
        Re: Differentiating with respect to an expression rouben@shadow.(none) (Rouben Rostamian) - 2014-11-12 01:55 +0000
          Re: Differentiating with respect to an expression Joe Riel <joer@san.rr.com> - 2014-11-11 18:00 -0800
            Re: Differentiating with respect to an expression Joe Riel <joer@san.rr.com> - 2014-11-11 18:03 -0800
              Re: Differentiating with respect to an expression rouben@shadow.(none) (Rouben Rostamian) - 2014-11-12 02:44 +0000
                Re: Differentiating with respect to an expression Joe Riel <joer@san.rr.com> - 2014-11-11 19:16 -0800
                Re: Differentiating with respect to an expression rouben@shadow.(none) (Rouben Rostamian) - 2014-11-12 04:53 +0000
      Re: Differentiating with respect to an expression Joe Riel <joer@san.rr.com> - 2014-11-11 17:45 -0800
        Re: Differentiating with respect to an expression Joe Riel <joer@san.rr.com> - 2014-11-11 17:52 -0800
        Re: Differentiating with respect to an expression rouben@shadow.(none) (Rouben Rostamian) - 2014-11-12 02:03 +0000

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