# Thread: Converting from Cartesian to Polar Coordinates

1. ## Converting from Cartesian to Polar Coordinates

Hey there. I've had a problem that I've been stumbling over for the past 5 hours or so. Any help would be really, really appreciated.

Make the following change of variables from rectangular coordinates to polar coordinates:

x = r*cos(@), y = r*sin(@), r^2 = x^2 + y^2, @ = arctan(y/x)

Then Show:

r * dr/dt = x * dx/dt + y * dy/dt and (r^2) * d@/dt = x * dy/dt - y * dx/dt

Then express the following in terms of polar coordinates, where dr/dt = r^3 and d@/dt = -1

dx/dt = y + x * (x^2 + y^2)
dy/dt = -x + y * (x^2 + y^2)

Here's the work I have so far. I fear a good percentage of it is wrong.

r^2 = x^2 + y^2
d/dt(r^2 = x^2 + y^2) = [2r(dr/dt) = 2x(dx/dt) + 2y(dy/dt)] -> r(dr/dt) = x(dx/dt) + y(dy/dt)

As for (r^2) * d@/dt = x * dy/dt - y * dx/dt, I have no idea.

dx/dt = y + x * (x^2 + y^2)
rsin(t) + (r^3)cos(t) = rsin(t) + rcos(t)[(rcos(t))^2 + (rsin(t))^2]
rsin(t) + (r^3)cos(t) = rsin(t) + (r^3)cos(t)
0 = 0

dy/dt = -x + y * (x^2 + y^2)
-rcos(t) + (r^3)sin(t) = -rcos(t) + (r^3)sin(t)
0 = 0

Obviously those last two parts are wrong, as from there I am supposed to graph the phase line diagram for r, then find a solution in terms of r and @ when x(0)=1 and y(0)=0. Any help would be greatly appreciated, as I've been working on this problem since 3:30 today. Seeing as it is now almost 1am, I'm getting some rest. I have another problem I'm stumped on as well, but it's rather difficult, and I already feel like a bother for asking help on this at such a late hour. Thank you to anyone who does help. I'll be awake again in about 7 hours to answer any questions regarding this.

2. ## Re: Converting from Cartesian to Polar Coordinates

Well, I've worked on it a bit more, though it doesn't really seem to help.

r*dr/dt = x*dx/dt + y*dy/dt -> r^4 = r*cos(t) * [r*sin(t) + (r^3)*cos(t)] + r*sin(t)*[-r*cos(t) + (r^3)*sin(t)]
r^4 = (r^4) * [cos(t)^2 + sin(t)^2]
r^4 = r^4
0 = 0

3. ## Re: Converting from Cartesian to Polar Coordinates

Originally Posted by Rukar
Hey there. I've had a problem that I've been stumbling over for the past 5 hours or so. Any help would be really, really appreciated.

Make the following change of variables from rectangular coordinates to polar coordinates:

x = r*cos(@), y = r*sin(@), r^2 = x^2 + y^2, @ = arctan(y/x)

Then Show:

r * dr/dt = x * dx/dt + y * dy/dt and (r^2) * d@/dt = x * dy/dt - y * dx/dt

Then express the following in terms of polar coordinates, where dr/dt = r^3 and d@/dt = -1

dx/dt = y + x * (x^2 + y^2)
dy/dt = -x + y * (x^2 + y^2)
_____________________________________

? = tan[sup:bvegt3uf]-1[/sup:bvegt3uf](y/x)

d?/dt = 1/[1+(y/x)[sup:bvegt3uf]2[/sup:bvegt3uf]] * [(-y*(dx/dt) + x*(dy/dt)]/x[sup:bvegt3uf]2[/sup:bvegt3uf]

Now finish the algebra to get

r[sup:bvegt3uf]2[/sup:bvegt3uf] * d?/dt = -y*(dy/dt) + x*(dx/dt)

_________________________________________

Here's the work I have so far. I fear a good percentage of it is wrong.

r^2 = x^2 + y^2
d/dt(r^2 = x^2 + y^2) = [2r(dr/dt) = 2x(dx/dt) + 2y(dy/dt)] -> r(dr/dt) = x(dx/dt) + y(dy/dt)

As for (r^2) * d@/dt = x * dy/dt - y * dx/dt, I have no idea.

dx/dt = y + x * (x^2 + y^2)
rsin(t) + (r^3)cos(t) = rsin(t) + rcos(t)[(rcos(t))^2 + (rsin(t))^2]
rsin(t) + (r^3)cos(t) = rsin(t) + (r^3)cos(t)
0 = 0

dy/dt = -x + y * (x^2 + y^2)
-rcos(t) + (r^3)sin(t) = -rcos(t) + (r^3)sin(t)
0 = 0

Obviously those last two parts are wrong, as from there I am supposed to graph the phase line diagram for r, then find a solution in terms of r and @ when x(0)=1 and y(0)=0. Any help would be greatly appreciated, as I've been working on this problem since 3:30 today. Seeing as it is now almost 1am, I'm getting some rest. I have another problem I'm stumped on as well, but it's rather difficult, and I already feel like a bother for asking help on this at such a late hour. Thank you to anyone who does help. I'll be awake again in about 7 hours to answer any questions regarding this.

4. ## Re: Converting from Cartesian to Polar Coordinates

Originally Posted by Rukar
Hey there. I've had a problem that I've been stumbling over for the past 5 hours or so. Any help would be really, really appreciated.

Make the following change of variables from rectangular coordinates to polar coordinates:

x = r*cos(@), y = r*sin(@), r^2 = x^2 + y^2, @ = arctan(y/x)

Then Show:

r * dr/dt = x * dx/dt + y * dy/dt and

(r^2) * d@/dt = x * dy/dt - y * dx/dt

Then express the following in terms of polar coordinates, where dr/dt = r^3 and d@/dt = -1

dx/dt = y + x * (x^2 + y^2)
dy/dt = -x + y * (x^2 + y^2)

r * dr/dt = x * dx/dt + y * dy/dt ..........................(1)

and

(r^2) * d@/dt = x * dy/dt - y * dx/dt .......................(2)

Treat equations (1) & (2) as two equations with two unknowns (unknowns being dy/dt and dx/dt)

Then solve for dx/dt and dy/dt

.

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