Mechanics of Planar Particle Motion - Fictitious Forces in Curvilinear Coordinates

Fictitious Forces in Curvilinear Coordinates

See also: Curvilinear coordinate system and Covariant derivative

To quote Bullo and Lewis: "Only in exceptional circumstances can the configuration of Lagrangian system be described by a vector in a vector space. In the natural mathematical setting, the system's configuration space is described loosely as a curved space, or more accurately as a differentiable manifold."

Instead of Cartesian coordinates, when equations of motion are expressed in a curvilinear coordinate system, Christoffel symbols appear in the acceleration of a particle expressed in this coordinate system, as described below in more detail. Consider description of a particle motion from the viewpoint of an inertial frame of reference in curvilinear coordinates. Suppose the position of a point P in Cartesian coordinates is (x, y, z) and in curvilinear coordinates is (q₁, q₂. q₃). Then functions exist that relate these descriptions:

 

and so forth. (The number of dimensions may be larger than three.) An important aspect of such coordinate systems is the element of arc length that allows distances to be determined. If the curvilinear coordinates form an orthogonal coordinate system, the element of arc length ds is expressed as:

where the quantities h_k are called scale factors. A change dq_k in q_k causes a displacement h_k dq_k along the coordinate line for q_k. At a point P, we place unit vectors e_k each tangent to a coordinate line of a variable q_k. Then any vector can be expressed in terms of these basis vectors, for example, from an inertial frame of reference, the position vector of a moving particle r located at time t at position P becomes:

where q_k is the vector dot product of r and e_k. The velocity v of a particle at P, can be expressed at P as:

where v_k is the vector dot product of v and e_k, and over dots indicate time differentiation. The time derivatives of the basis vectors can be expressed in terms of the scale factors introduced above. for example:

 or, in general,  

in which the coefficients of the unit vectors are the Christoffel symbols for the coordinate system. The general notation and formulas for the Christoffel symbols are:

  

and the symbol is zero when all the indices are different. Despite appearances to the contrary, the Christoffel symbols do not form the components of a tensor. For example, they are zero in Cartesian coordinates, but not in polar coordinates.

Using relations like this one,

which allows all the time derivatives to be evaluated. For example, for the velocity:

with the Γ-notation for the Christoffel symbols replacing the curly bracket notation. Using the same approach, the acceleration is then

Looking at the relation for acceleration, the first summation contains the time derivatives of velocity, which would be associated with acceleration if these were Cartesian coordinates, and the second summation (the one with Christoffel symbols) contains terms related to the way the unit vectors change with time.