Curvilinear motion in various coordinate systems
I examine the different coordinate systems that enable the analysis of curvilinear motion, focusing on Cartesian, cylindrical, spherical, and intrinsic coordinate systems. Beginning with Cartesian coordinates, where orthonormal vectors remain fixed, I progress to cylindrical and spherical systems, which offer insights into curvilinear motion through position-dependent unit vectors. For more complex motions, I use the tangential and normal coordinate systems, also known as the Frenet-Serret framework, to describe path-based motion. My aim is to provide a clear and practical reference for graduate students in mathematics, engineering, and physics seeking an advanced understanding of these systems.
Cartesian Coordinates
The Cartesian coordinate system, with fixed orthonormal vectors \mathbf{i}, \mathbf{j}, \mathbf{k}, provides a straightforward basis for motion analysis. Here, the position \mathbf{r}_P, velocity \mathbf{v}_P, and acceleration \mathbf{a}_P of a point P are given by:
\begin{aligned} \mathbf{r}_P & = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \\ \mathbf{v}_P & = \dot{x} \mathbf{i} + \dot{y} \mathbf{j} + \dot{z} \mathbf{k} \\ \mathbf{a}_P & = \ddot{x} \mathbf{i} + \ddot{y} \mathbf{j} + \ddot{z} \mathbf{k} \end{aligned}
Cylindrical Coordinates
In cylindrical coordinates, a point is described by (\rho, \phi, z), where \rho is the radial distance, \phi is the azimuthal angle, and z is the height. The unit vectors \mathbf{e}_\rho, \mathbf{e}_\phi, and \mathbf{e}_z vary with position. The position, velocity, and acceleration are:
\begin{aligned} \mathbf{r}_P =& \rho \mathbf{e}_\rho + z \mathbf{e}_z \\ \mathbf{v}_P =& \dot{\rho} \mathbf{e}_\rho + \rho \dot{\phi} \mathbf{e}_\phi + \dot{z} \mathbf{e}_z \\ \mathbf{a}_P =& \left( \ddot{\rho} - \rho \dot{\phi}^2 \right) \mathbf{e}_\rho \\ & + \left( 2 \dot{\rho} \dot{\phi} + \rho \ddot{\phi} \right) \mathbf{e}_\phi + \ddot{z} \mathbf{e}_z \end{aligned}
Spherical Coordinates
The spherical coordinate system uses (r, \theta, \phi) with the radial distance r, polar angle \theta, and azimuthal angle \phi. The unit vectors \mathbf{e}_r, \mathbf{e}_\theta, and \mathbf{e}_\phi depend on position. Expressions for position, velocity, and acceleration are:
\begin{aligned} \mathbf{r}_P =& r \mathbf{e}_r \\ \mathbf{v}_P =& \dot{r} \mathbf{e}_r + r \dot{\theta} \mathbf{e}_\theta + r \sin \theta \, \dot{\phi} \mathbf{e}_\phi \\ \mathbf{a}_P =& \left( \ddot{r} - r \left( \dot{\theta}^2 + \sin^2 \theta \, \dot{\phi}^2 \right) \right) \mathbf{e}_r \\ & + \left( 2 \dot{r} \dot{\theta} + r \ddot{\theta} - r \sin \theta \cos \theta \, \dot{\phi}^2 \right) \mathbf{e}_\theta \\ & + \left( 2 \dot{r} \sin \theta \, \dot{\phi} + 2 r \dot{\theta} \cos \theta \, \dot{\phi} + r \sin \theta \, \ddot{\phi} \right) \mathbf{e}_\phi \end{aligned}
Intrinsic Coordinates
For motion along a path, I use the tangential and normal (or Frenet-Serret) coordinate system, where \mathbf{e}_t, \mathbf{e}_n, and \mathbf{e}_b are the unit tangent, normal, and binormal vectors, respectively. These vectors adapt to the particle’s motion. The velocity and acceleration in intrinsic coordinates are:
\begin{aligned} \mathbf{v}_P &= v \, \mathbf{e}_t \\\\ \mathbf{a}_P &= \frac{d v}{d t} \, \mathbf{e}_t + \frac{v^2}{\rho} \, \mathbf{e}_n \end{aligned}
where v is the particle’s speed, \frac{d v}{d t} is the tangential acceleration, and \frac{v^2}{\rho} is the normal acceleration, with \rho representing the path’s radius of curvature. This system is ideal for describing curvilinear motion along a trajectory.
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