## physicsdiff.mpl Version 0.1

This procedure allows you to perform a first variation of some expression (presumed to be inside an integral, e.g. the Lagrangian density in an action). It will differentiate the expression treating the specified function as the differentiation parameter, with the additional feature that it automatically attempts to integrate first order derivatives of that function by parts.
To call the procedure, run:
```
physicsdiff(expression,function,diffparam);
```

### Example

```
expr := h(r)*g(r)*2 + h(r)*diff(h(r),r)^2*3 + diff(h(r),r):
physicsdiff(expr,h(r),r);
2*g(r) + 3*(diff(h(r),r))^2 - 2*diff(3*h(r)*diff(h(r),r),r)
```

### Source Code

```
#This maple procedure does a functional variation, much like diff does when used
#with(Physics):, but with the added feature that it can vary simple derivatives
#with respect to that function, ie. vary(c*diff(f(r),r)^2,f(r)) using
#integration by parts.
_physicsdiff2_term := proc(function,term,innerterm,dparam)
local counter, thecoeff,tmp;
if is(op(0,innerterm) = `^`) then
thecoeff := coeff(term,diff(function,dparam)^(op(2,innerterm)));
RETURN(op(2,innerterm)*diff(thecoeff*diff(function,dparam)^(op(2,innerterm)-1),dparam));
elif is(op(0,innerterm)=`diff`) then
thecoeff := coeff(term,diff(function,dparam));
RETURN(diff(thecoeff,dparam));
else
print("Error - could not determine power of term.");
RETURN(0);
fi:
end proc:
_physicsdiff_term := proc(expr,function,term,dparam)
local counter,tmp,retterm;
retterm := 0;
if (has(term,diff(function,dparam)) and not(has(term,diff(function,dparam $ 2)))) then
if is(op(0,term)=`*`) then
for counter from 1 to nops(term) do
tmp := op(counter,term);
if (has(tmp,diff(function,dparam))) then
retterm := retterm - _physicsdiff2_term(function,term,tmp,dparam);
fi:
end do:
else
#There is nothing to be done - the term is just a lone derivative of "function" - it will be killed.
print("Found a term that's just a lone derivative. Integration by parts failed.");
fi:
RETURN(retterm);
elif has(term,diff(function,dparam $ 2)) then
print("Error, found a derivative of order greater than 2. This term is killed.");
RETURN(0);
else
RETURN(0);
fi:
end proc:
physicsdiff := proc(expr,function,dparam)
local expr2,newexpr,exprlength,term,tmp,counter;
description "This Maple procedure will perform a physicists' functional differentiation, neglecting boundary terms and integrating by parts to functionally vary an expression with respect to a function, even if it contains powers of first derivatives of that function. Written by W. Brenna, Sept 3 2012. Version 0.1.";
expr2 := expand(expr);
exprlength := nops(expr2);
#newexpr := expr2;
newexpr := 0;
if is(op(0,expr2)=`+`) then
for counter from 1 to exprlength do
term := op(counter,expr2);
#This just does the first order integration by parts.
newexpr := newexpr + _physicsdiff_term(newexpr,function,term,dparam):
end do:
else
term := expr2;
newexpr := newexpr + _physicsdiff_term(newexpr,function,term,dparam):
fi:
#Finally, we do the regular differentiation by parts to build up the final
#terms.
#RETURN(newexpr + frontend(diff,[expr2,function]));
#frontend is hopeless for this - non-integer powers make it cry.
RETURN(newexpr + Physics:-diff(expr2,function));
end proc:
```