Dynamics: Cylindrical and Polar Coordinates

In engineering dynamics, choosing the right coordinate system is the difference between a simple solution and an algebraic nightmare. While Cartesian (x-y-z) coordinates are intuitive for straight-line motion, they become painfully complex when analyzing rotation, orbits, or any path that naturally curves around a point. Polar coordinates and their 3D extension, cylindrical coordinates, provide a powerful language to describe these motions directly, aligning the mathematics with the physics of rotation and radial change.

Polar Coordinates: The Foundation

A point in a plane is located using two polar coordinates: the radial distance $r$ (from a fixed origin O) and the angular position $\theta$ (measured from a fixed reference line, usually the positive x-axis). The critical, non-Cartesian concept here is that the unit vectors defining the directions are not fixed; they rotate with the particle's angular position.

The two fundamental unit vectors are:

• ${\mathbf{e}}_r$: The radial unit vector, always pointing directly away from the origin along the radial line.
• ${\mathbf{e}}_{\theta }$: The transverse (or tangential) unit vector, perpendicular to ${\mathbf{e}}_r$ and pointing in the direction of increasing $\theta$.

Their time derivatives are the key to everything that follows. Since these vectors rotate with angular velocity $\dot{\theta }$, their rates of change are: $${\dot{\mathbf{e}}}_r=\dot{\theta }{\mathbf{e}}_{\theta }\text{and}{\dot{\mathbf{e}}}_{\theta }=-\dot{\theta }{\mathbf{e}}_r$$ This tells us the radial vector's direction changes in the transverse direction, and the transverse vector's direction changes opposite the radial direction, both proportional to the angular speed.

Velocity and Acceleration in Polar Coordinates

The position vector is simply $\mathbf{r}=r{\mathbf{e}}_r$. Velocity is the time derivative of position: $$\mathbf{v}=\dot{\mathbf{r}}=\frac{d}{dt}(r{\mathbf{e}}_r)=\dot{r}{\mathbf{e}}_r+r{\dot{\mathbf{e}}}_r$$ Substituting our expression for ${\dot{\mathbf{e}}}_r$ gives the velocity in polar components: $$\mathbf{v}=v_r{\mathbf{e}}_r+v_{\theta }{\mathbf{e}}_{\theta }=\dot{r}{\mathbf{e}}_r+r\dot{\theta }{\mathbf{e}}_{\theta }$$

Here, $v_r=\dot{r}$ is the radial velocity component (rate of elongation or shortening of the radial line). $v_{\theta }=r\dot{\theta }$ is the transverse velocity component (speed perpendicular to the radial line, due to rotation).

Acceleration is the derivative of velocity: $$\mathbf{a}=\dot{\mathbf{v}}=\frac{d}{dt}(\dot{r}{\mathbf{e}}_r+r\dot{\theta }{\mathbf{e}}_{\theta })$$ Carefully applying the product rule and the derivatives of the unit vectors yields the polar acceleration expression: $$\mathbf{a}=(\ddot{r}-r{\dot{\theta }}^2){\mathbf{e}}_r+(r\ddot{\theta }+2\dot{r}\dot{\theta }){\mathbf{e}}_{\theta }$$

This result contains the four classic terms:

1. $\ddot{r}$: Radial acceleration due to a change in radial speed.
2. $-r{\dot{\theta }}^2$: Centripetal acceleration, always directed inward toward the origin (hence the negative sign).
3. $r\ddot{\theta }$: Transverse acceleration due to angular acceleration.
4. $2\dot{r}\dot{\theta }$: Coriolis acceleration, a transverse component that appears whenever there is both radial motion ($\dot{r}$) and rotation ($\dot{\theta }$).

Extension to Cylindrical Coordinates

Cylindrical coordinates are the three-dimensional generalization of polar coordinates. We simply add the familiar Cartesian z-coordinate and its fixed unit vector $\mathbf{k}$ perpendicular to the $r$-$\theta$ plane. The coordinates are $(r,\theta ,z)$.

The position, velocity, and acceleration vectors gain a third, independent component: $$\mathbf{r}=r{\mathbf{e}}_r+z\mathbf{k}$$ $$\mathbf{v}=\dot{r}{\mathbf{e}}_r+r\dot{\theta }{\mathbf{e}}_{\theta }+\dot{z}\mathbf{k}$$ $$\mathbf{a}=(\ddot{r}-r{\dot{\theta }}^2){\mathbf{e}}_r+(r\ddot{\theta }+2\dot{r}\dot{\theta }){\mathbf{e}}_{\theta }+\ddot{z}\mathbf{k}$$

This system is ideal for analyzing motion along a helical path, the dynamics of a particle in a rotating pipe, or any scenario with cylindrical symmetry.

Applications in Rotating Mechanisms

Consider a robotic arm where a link of length $L$ rotates in a vertical plane with a controlled angle $\theta (t)$, while a slider block moves along the link with a prescribed distance $r(t)$ from the pivot. To find the acceleration of the block's tool point:

1. Establish the origin at the pivot.
2. The known kinematic variables are $r(t)$, $\dot{r}$, $\ddot{r}$, $\theta (t)$, $\dot{\theta }$, and $\ddot{\theta }$.
3. Directly apply the polar acceleration formula: $\mathbf{a}=(\ddot{r}-r{\dot{\theta }}^2){\mathbf{e}}_r+(r\ddot{\theta }+2\dot{r}\dot{\theta }){\mathbf{e}}_{\theta }$.

The $-r{\dot{\theta }}^2$ term gives the centripetal force required to keep the block on the rotating link, and the $2\dot{r}\dot{\theta }$ (Coriolis) term represents the additional transverse force the link must exert if the block is sliding radially while the link rotates. Using Cartesian coordinates for this problem would involve far more complex trigonometry.

Applications to Orbital Motion

Polar coordinates are the natural choice for analyzing orbital motion under a central force like gravity. Kepler's laws and Newton's law of gravitation are most elegantly expressed in this frame. For a planet orbiting a sun fixed at the origin, the gravitational force has only a radial component: $\mathbf{F}=-(GMm/r^2){\mathbf{e}}_r$.

Applying Newton's second law, $\mathbf{F}=m\mathbf{a}$, using our polar acceleration components gives two scalar equations of motion:

• Radial: $m(\ddot{r}-r{\dot{\theta }}^2)=-\frac{GMm}{r^2}$
• Transverse: $m(r\ddot{\theta }+2\dot{r}\dot{\theta })=0$

The transverse equation simplifies to $\frac{d}{dt}(m{r}^2\dot{\theta })=0$, which states that the angular momentum $m{r}^2\dot{\theta }$ is constant—Kepler's second law of equal areas swept in equal times. These polar equations form the starting point for deriving elliptical orbits and all other conic-section trajectories.

Common Pitfalls

1. Forgetting the Unit Vectors Rotate: The most fundamental error is treating ${\mathbf{e}}_r$ and ${\mathbf{e}}_{\theta }$ as constants when taking time derivatives. Always remember ${\dot{\mathbf{e}}}_r=\dot{\theta }{\mathbf{e}}_{\theta }$ and ${\dot{\mathbf{e}}}_{\theta }=-\dot{\theta }{\mathbf{e}}_r$.
2. Misidentifying Centripetal Acceleration: The centripetal term is $-r{\dot{\theta }}^2$ and it is always directed inward (the negative radial direction). It is present anytime there is rotation ($\dot{\theta }\ne 0$), even if the radial distance $r$ is constant (like uniform circular motion, where $\ddot{r}=0$).
3. Omitting the Coriolis Term: The $2\dot{r}\dot{\theta }$ component is easy to miss if you don't carefully apply the product rule. Remember, it arises whenever an object moves radially within a rotating frame. In orbital motion, it explains why a satellite speeds up as it falls closer to the planet.
4. Mixing 2D and 3D Formulations: When adding the third dimension in cylindrical coordinates, ensure you are not incorrectly adding a $z$-component to the polar terms for $r$ and $\theta$. The ${\mathbf{e}}_r$ and ${\mathbf{e}}_{\theta }$ components remain exactly as derived in 2D, with the $z$-component operating independently.

Summary

• Polar coordinates $(r,\theta )$ use rotating unit vectors ${\mathbf{e}}_r$ and ${\mathbf{e}}_{\theta }$, whose derivatives are ${\dot{\mathbf{e}}}_r=\dot{\theta }{\mathbf{e}}_{\theta }$ and ${\dot{\mathbf{e}}}_{\theta }=-\dot{\theta }{\mathbf{e}}_r$.
• Velocity has a radial component $v_r=\dot{r}$ and a transverse component $v_{\theta }=r\dot{\theta }$.
• Acceleration consists of four distinct physical terms: radial acceleration ($\ddot{r}$), centripetal acceleration ($-r{\dot{\theta }}^2$), transverse angular acceleration ($r\ddot{\theta }$), and the Coriolis acceleration ($2\dot{r}\dot{\theta }$).
• Cylindrical coordinates $(r,\theta ,z)$ extend this to 3D by simply adding the $z$-component and its fixed unit vector $\mathbf{k}$ to the polar results.
• These coordinates are indispensable for efficiently solving problems involving rotating mechanisms (like linkages and robotics) and orbital motion, where the forces and geometry align naturally with radial and transverse directions.