of an area, expression manipulation, that show serious weaknesses of Maple.  It cannot  be expected  that  normal users will come out with so fined tuned tricks to perform such simple mathematical steps.

normal(x + y/z);
                               x z + y
                               -------
                                  z

VectorCalculus:-ArcLength(<x,(x/2)^(2/3)>, x=0..u) assuming u>0;
eval(%,u=2);


               (2/3)    (2/3) 1/2  2/3       (2/3)    (2/3) 1/2  2/3
           (9 u      + 2     )    u      (9 u      + 2     )    2
  - 2/27 + --------------------------- + ---------------------------
                        3                            27


                                        1/2
                                   20 10
                          - 2/27 + --------
                                      27

As an alternative:

with(RealDomain):
Re(:-exp(I*k*x));
                               cos(k x)
:-sqrt(-3);
                                 1/2
                                3    I

evalc(Re(exp(I*k*x)));
                              cos(k x)

is useful here:

with(RootFinding):
with(RootFinding[Parametric]):
m:=CellDecomposition([a*x^3 + x^2  - a*x + 1=0],[x],[a]);

m := Record(Equations = [a*x^3+x^2-a*x+1],Inequalities = [],
Variables = [x],Parameters = [a],
DiscriminantVariety = [[a], [a^4-1-11*a^2]],
ProjectionPolynomials = [[a, a^4-1-11*a^2]],
SamplePoints = [[a = -4], [a = -2], [a = 2], [a = 4]])

The information on the number of solutions in each region of the parameter line can be condensed in this way:

f1:=proc(x)
if has(x[1,1],infinity) then x[1,1]
else 
Isolate(x[1,1])[x[1,2]];
end if:
end proc:

f4:=proc(x)
if has(x[1,4],infinity) then x[1,4]
else 
Isolate(x[1,4])[x[1,5]];
end if:
end proc:

for i to 4 do
NumberOfSolutions(m)[i],CellDescription(m,i),
[f1,f4](CellDescription(m,i));
end do;

[1, 3], [[-infinity, 0, a, a^4-1-11*a^2, 1]], [-infinity, a = -3.330190677]

[2, 1], [[a^4-1-11*a^2, 1, a, a, 1]], [a = -3.330190677, a = 0.]

[3, 1], [[a, 1, a, a^4-1-11*a^2, 2]], [a = 0., a = 3.330190677]

[4, 3], [[a^4-1-11*a^2, 2, a, infinity, 0]], [a = 3.330190677, infinity]

Which reads as number of cell (1 to 4), number of real solutions in the cell (1 or 3), root of the polynomial in the parameter determining the boundary of the cell (eg 1st root of a^4-1-11*a^2), and numerical evaluation of these boundaries.

In fact, assuming N>=1 and c>=1, the expected result is 2*u[s,t]*d[s,t]^2 provided 1<=s<=c and 1<=t<=N, integers, otherwise 0.

I think that there is no such command for sums. It would be desirable.

seems to depend on the history of evaluations. Eg in Maple 10.06:

for i to 30 do
F(i/2*10^(-10));
end do;

produces this other error message:

Error, (in F) cannot evaluate the solution further right of .12961725e-8, probably a singularity

constants, the general answer is yes. Eg, you can calculate the eigenvalues as roots of a polynomial, expressed as a determinant. The determinant has sums of terms and all these terms, products with the same number of factors, ought to have the same dimension (eg the same power of length).

(-2*H1*H2-H2^2-H1^2)*`ϕt`+(H1^2+2*H1*H2+H2^2)*`ηt`*Ut
+4*H2*H1*Uc*`ηc` = 0;
collect(%,[`ϕt`,`ηt`,`ηc`],(x->x/(H1+H2)^2)@factor);

You can use <maple> tags, but sometimes this mechanism does not work properly.

If yes, as it is massless, it holds  T=2*T1.

Eg. in 3D space with x a cartesian coordinate, x=3 is a plane. In 1D it is a point, etc.

that I have met time ago.

with(VariationalCalculus):
F:= diff(x(t),t)^2/t^3:
EL:=EulerLagrange(F, t, x(t));

EL := {-2*diff(x(t),`$`(t,2))/t^3+6*diff(x(t),t)/t^4, 2*diff(x(t),t)/t^3 = K[1]};

s:=dsolve(EL[1]);
	s := x(t) = _C1+_C2*t^4

eq1:=subs(x(t)=2,t=1,s);
eq2:=subs(x(t)=17,t=2,s);
sC:=solve({eq1,eq2});
subs(sC,s);

	x(t) = 1+t^4

in the file name string