Trajectories with Conserved Quantities

1. The problem statement, all variables and given/known data
A particle moves along a trajectory with constant magnitude of the velocity |[itex]\stackrel{→}{v}[/itex]|=[itex]\stackrel{→}{v0}[/itex] and constant angular momentum L⃗ = L⃗0. Determine the possible trajectories.

2. Relevant equations

d(L⃗)/(dt)=[itex]\stackrel{→}{N}[/itex] where [itex]\stackrel{→}{N}[/itex]=torque



3. The attempt at a solution

I thought that there would be 3 paths that it could take

(i) a strait line if its in a strait line it can have constant velocity and no change in angular momentum, because torque is 0, because there is no Force.

(ii) a circle because force is parallel to the radius, there is no torque, means no change in angular momentum, and uniform circular motion has a constant |v|.

(iii) helix just like a combination of a strait line and a circle. both there is no change in angular momentum, so together there should be no change as well. (no force in any direction besides parallel to radius). also since there is constant |v| in the uniform circle, and now you are applying a constant v in a direction perpendicular to the v of the circle, the new |v| magnitude should be constant as well.

So my question is, are these are correct? if so are they the only three? and also if so, how do I know there all no others? (is there anyway to prove these are the only 3)

if it helps at all the title to this problem as my teacher gave is (Trajectory with conserved quantities)

Thank you for your help!

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