Notation

Before proceeding with any mathematical derivations, an introduction to the notation being used
may be helpful. When new key terms are introduced, they will be written in *italics*.

*Parameters* are constants that geometrically describe the track and its elements.
Each element type requires different parameters to be specified. *Variables* change along
the track. Since the car is constrained to move along the track, this is a single degree of freedom
system, with the DOF being the distance along the track. This distance is the independent
variable. *Functions* vary along the track. All functions are dependent on the independent
variable. The term *quantities* will be used to encompass parameters, variables, and functions,
essentially referring to all numeric quantities.

Vectors will be represented in boldface, ; while scalars will be represented in plain font, . A unit vector will be represented as a vector with a hat over it, . In this way, the types of quantities used can easily be distinguished from one another.

Three coordinate systems will be used in this research. All are right-handed, orthogonal systems. See figure 1.3.

**Global**- An inertial coordinate system is established, denoted in capitals as
, and whose axes are in the
directions. The
direction is the
*vertical*direction, and is perpendicular to the surface of the earth. The origin of the global coordinate system is placed at the station, where the track begins. The initial track direction is along the axis. All quantities dealing with the global coordinate system will be expressed in capitals. **Local**- An inertial coordinate system is created for each new element that is added.
The coordinates are represented in lower case by , with the corresponding axes in the
standard
directions. The origin of the local coordinate system is
placed at the beginning of the particular track element being described. The local coordinate
system is aligned such that the axis is parallel to the axis. The initial track direction is in
the - plane.
^{1.3} **Car-Fixed**- A non-inertial coordinate system is created with the origin affixed to the
center of gravity of the car. The unit vectors are represented by
^{1.4}, which represent the forward and radial vectors.^{1.5}The forward vector points in the direction the car is moving, while the radial direction points toward the center of curvature.

The distance a car has traveled along the track will be denoted as , while the distance a car
has traveled into an element will be denoted as . The total length of an element is denoted by
either or . Note that the function is dependent on the independent variable
and the parameters of all previous elements.

The azimuth angle is denoted as , and is defined as the angle between the projection of
the forward vector in the - plane and the axis. Refer to figure 1.4. The element will be created such that the azimuth angle at the beginning of an element is zero. The azimuth angle at the end of an element is denoted as .

The vertical angle is denoted by , and is defined as the angle between the forward direction and the - plane. Refer again to figure 1.4. The vertical angles at the beginning and end of an element are denoted as and , respectively.

In dynamic calculations, the derivative with respect to time, as in , is used frequently and is denoted simply as .