Equation Solving - Bug?

Questions
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1

Hi all,

I’m trying to solve a set of three equations in three variables: ax, ay, and az. However, Mathematica’s ability to solve the equations seems to depend on the way in which I write them, in a trivial way.

This one works:

works.jpg
works.jpg491×255 26 KB

but this one doesn’t:

doesntwork.JPG
doesntwork.JPG1132×112 14.3 KB

The only difference between the two is that I moved things around (solved for the v dot a expression), and used a1 instead of ax, v1 to mean vx, etc.

Why does this happen? How can I structure my equations in a way that Mathematica will be able to solve them?

Thanks a ton!

2

On the menu bar try clicking on Evaluation, Quit Kernel, Local, Quit
and then try evaluating your two problems again.

In[1]:= Solve[{
  c^2 (F1 - m*g*a1)/(g^3*v1*m) == v1*a1 + v2*a2 + v3*a3,
  c^2 (F2 - m*g*a2)/(g^3*v2*m) == v1*a1 + v2*a2 + v3*a3,
  c^2 (F3 - m*g*a3)/(g^3*v3*m) == v1*a1 + v2*a2 + v3*a3},
 {a1, a2, a3}]

Out[1]= {{
  a1 -> -((-c^2 F1 + F2 g^2 v1 v2 - F1 g^2 v2^2 + F3 g^2 v1 v3 - F1 g^2 v3^2)/
   (g m (c^2 + g^2 v1^2 + g^2 v2^2 + g^2 v3^2))), 
  a2 -> -((-c^2 F2 - F2 g^2 v1^2 + F1 g^2 v1 v2 + F3 g^2 v2 v3 - F2 g^2 v3^2)/
   (g m (c^2 + g^2 v1^2 + g^2 v2^2 + g^2 v3^2))), 
  a3 -> -((-c^2 F3 - F3 g^2 v1^2 - F3 g^2 v2^2 + F1 g^2 v1 v3 + F2 g^2 v2 v3)/
   (g m (c^2 + g^2 v1^2 + g^2 v2^2 + g^2 v3^2)))}}

In[2]:= Solve[{
  fx == m*vx*g^3 (vx*ax + vy*ay + vz*az)/c^2 + g*m*ax,
  fy == m*vy*g^3 (vx*ax + vy*ay + vz*az)/c^2 + g*m*ay,
  fz == m*vz*g^3 (vx*ax + vy*ay + vz*az)/c^2 + g*m*az},
 {ax, ay, az}]

Out[2]= {{
   ax -> -((-c^2 fx + fy g^2 vx vy - fx g^2 vy^2 + fz g^2 vx vz - fx g^2 vz^2)/
   (g m (c^2 + g^2 vx^2 + g^2 vy^2 + g^2 vz^2))), 
  ay -> -((-c^2 fy - fy g^2 vx^2 + fx g^2 vx vy + fz g^2 vy vz - fy g^2 vz^2)/
   (g m (c^2 + g^2 vx^2 + g^2 vy^2 + g^2 vz^2))), 
  az -> -((-c^2 fz - fz g^2 vx^2 - fz g^2 vy^2 + fx g^2 vx vz + fy g^2 vy vz)/
   (g m (c^2 + g^2 vx^2 + g^2 vy^2 + g^2 vz^2)))}}

It is likely, but impossible to verify now, that you had some value assigned to something which gave no solution to the second problem.
This is a somewhat common problem and often extremely difficult to recognize when it happens unless you know the result is incorrect.

In the future, a powerful method for checking this is to scrape-n-paste your three equations into a new cell and tapping <shift><enter>.
If the result appears to be exactly what your system was then it is unlikely that something had accidentally been assigned to some variable.
But if the result looks somewhat or very different then you can try to track down which variable might have have some unexpected value.

Or perhaps it is a difference in behavior between one version of Mathematica and another.

3

Is there a way of displaying all variables with assigned values?

4

Hi Frank,

this might help:

DeleteCases[
Table[{Names["Global`*"][[k4313]], (Names["Global`*"] // ToExpression)[[k4313]]}, {k4313, 1, Length[Names["Global`*"]]}], 
{"k4313", ___}] // TableForm

I am quite sure that here is an easier way though. This particular function works only for the global context.

M.