The speed of a curve in different coordinate systems

[kkz23691]

Hello,

If for a curve in Cartesian coordinates $||\dot{{\mathbf r}}||=\mbox{const}$ (i.e. the curve is constant speed) will the speed of the curve change in cylindrical and spherical coordinates?
Could someone experienced share how the transition from flat Euclidian space to curved space alters the speed of the curve?

Thanks!

 

[HallsofIvy]

I am puzzled by the expression "speed of a curve". A curve just sits there! It doesn't have any "speed". But I assume you mean the length of the tangent vector to the curve at any point which could also be thought of as the speed of an object moving along the curve. That is an "intrinsic" property of the curve, it does not depend on the coordinate system.

 

[kkz23691]

HallsofIvy, I hear you :) "Unit speed" seems to be a term used in math.
It turns out, $||\ddot{{\mathbf r}}||$ depends on the coordinate system; it wasn't obvious to me that $||\dot{{\mathbf r}}||$ doesn't.

 

[kkz23691]

To add this - even in the same coordinate system a curve can be reparametrized from a non-unit-speed to unit-speed. If a curve is already unit-speed in some coordinate system, I am uncertain why it would still be unit-speed in any other coordinate system.

 

[HallsofIvy]

HallsofIvy, I hear you :) "Unit speed" seems to be a term used in math.
It turns out, $||\ddot{{\mathbf r}}||$ depends on the coordinate system; it wasn't obvious to me that $||\dot{{\mathbf r}}||$ doesn't.

I would be very surprised if "speed", a physics term, were used in math. What you are calling a "unit speed curve" seems to be, mathematically, a "curve parameterized by arc length". Such an expression for a curve always has the property that the derivative (x, y, z), the "direction vector" of the curve, with respect to the parameter, arclength, is a unit-length tangent vector. If you think of the parameter as "time", t, the that derivative is the velocity vector but that is again a "physics", not "math" concept.