What is Transport theorem?

The transport theorem is a vector equation that relates the time derivative of a Euclidean vector as evaluated in a non-rotating coordinate system to its time derivative in a rotating reference frame. It has important applications in classical mechanics and analytical dynamics and diverse fields of engineering. A Euclidean vector represents a certain magnitude and direction in space that is independent of the coordinate system in which it is measured. However, when taking a time derivative of such a vector one actually takes the difference between two vectors measured at two different times t and t+dt. In a rotating coordinate system, the coordinate axes can have different directions at these two times, such that even a constant vector can have a non-zero time derivative. As a consequence, the time derivative of a vector measured in a rotating coordinate system can be different from the time derivative of the same vector in a non-rotating reference system. For example, the velocity vector of an airplane as evaluated using a coordinate system that is fixed to the earth is different from its velocity as evaluated using a coordinate system that is fixed in space. The transport theorem provides a way to relate time derivatives of vectors between a rotating and non-rotating coordinate system, it is derived and explained in more detail in rotating reference frame and can be written as:

Here f is the vector of which the time derivative is evaluated in both the non-rotating, and rotating coordinate system. The subscript r designates its time derivative in the rotating coordinate system and the vector Ω is the angular velocity of the rotating coordinate system.

The Transport Theorem is particularly useful for relating velocities and acceleration vectors between rotating and non-rotating coordinate systems.