Transport Theorem
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The transport theorem (or transport equation, rate of change transport theorem or basic kinematic equation) is a vector equation that relates the
time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...
of a
Euclidean vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
as evaluated in a non-rotating coordinate system to its time derivative in a
rotating reference frame A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. (This article considers onl ...
. It has important applications in
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
and
analytical dynamics In classical mechanics, analytical dynamics, also known as classical dynamics or simply dynamics, is concerned with the relationship between Motion (physics), motion of bodies and its causes, namely the force (physics), forces acting on the bodies ...
and diverse fields of engineering. A
Euclidean vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
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 (a rotating reference system) 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 A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. (This article considers onl ...
and can be written as: \frac\boldsymbol = \left \left(\frac\right)_ + \boldsymbol \times \right\boldsymbol \ . 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. Reference states: "Despite of its importance in classical mechanics and its ubiquitous application in engineering, there is no universally-accepted name for the Euler derivative transformation formula ..Several terminology are used: kinematic theorem, transport theorem, and transport equation. These terms, although terminologically correct, are more prevalent in the subject of fluid mechanics to refer to entirely different physics concepts." An example of such a different physics concept is
Reynolds transport theorem In differential calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or simply the Reynolds theorem, named after Osborne Reynolds (1842–1912), is a three-dimensional generalization of the Leibniz in ...
.


Derivation

Let _i:=T^E_B_i be the
basis vectors In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as componen ...
of B, as seen from the reference frame E, and denote the components of a vector in B by just f_i. Let :G:=T' \cdot T^ so that this
coordinate transformation In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
is generated, in time, according to in T'=G\cdot T. Such a generator
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
is important for trajectories in Lie group theory. Applying the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
with implict
summation convention In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
, :' = (f_i _i)' = (f_iT)'_i = (f_i'T + f_i G\cdot T)\,_i = (f_i' + f_i G)\,_i = \left( \left(\tfrac\right)_B + G \right) \boldsymbol For the rotation groups (n), one has T^B_E:=(T^E_B)^=(T^E_B)^T. In three dimensions, the generator G then equals the cross product operation from the left, a skew-symmetric
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
E\times := _E\times for any vector . As a matrix, it is also related to the vector as seen from B via : E\times = ^E_B_B\times = T^E_B\cdot B\times\cdot T^B_E


References

{{Reflist Mathematical theorems