Andromeda
Note

Motion in Polar Coordinates

Definition

Motion in the polar plane is described using a moving frame of reference defined by radial (ur\mathbf{u}_r) and angular (uθ\mathbf{u}_\theta) unit vectors:

  • ur=(cos⁡θ)i+(sin⁡θ)j\mathbf{u}_r = (\cos \theta)\mathbf{i} + (\sin \theta)\mathbf{j}

    • How to read: “The radial unit vector is equal to the cosine of theta times the unit vector i, plus the sine of theta times the unit vector j.”
    • Meaning: Radial unit vector points outward from the origin at angle θ\theta.

  • uθ=−(sin⁡θ)i+(cos⁡θ)j\mathbf{u}_\theta = -(\sin \theta)\mathbf{i} + (\cos \theta)\mathbf{j}

    • How to read: “The transverse unit vector is equal to negative sine of theta times the unit vector i, plus the cosine of theta times the unit vector j.”
    • Meaning: Angular unit vector is perpendicular to radial, pointing in the direction of increasing θ\theta.

Why It Matters

Analyzing circular or orbital motion in Cartesian coordinates is a mathematical nightmare. Polar coordinates are the natural language of the universe for anything that rotates. Without them, satellite navigation and planetary physics would be virtually impossible to compute accurately.

Core Concepts

  • Position: r=rur\mathbf{r} = r \mathbf{u}_r

    • How to read: “The position vector r is equal to the scalar radius r times the radial unit vector.”
    • Meaning: Position vector has magnitude rr in the radial direction.

  • Velocity: v=r˙ur+rθ˙uθ\mathbf{v} = \dot{r} \mathbf{u}_r + r\dot{\theta} \mathbf{u}_\theta

    • How to read: “The velocity vector is equal to the time derivative of r times the radial unit vector, plus r times the time derivative of theta times the transverse unit vector.”
    • Meaning: Velocity has radial component r˙\dot{r} and tangential component rθ˙r\dot{\theta}.

  • Acceleration: a=(r¨−rθ˙2)ur+(rθ¨+2r˙θ˙)uθ\mathbf{a} = (\ddot{r} - r\dot{\theta}^2)\mathbf{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\mathbf{u}_\theta

    • How to read: “The acceleration vector is equal to the second time derivative of r minus r times the square of the time derivative of theta, all times the radial unit vector, plus r times the second time derivative of theta plus two times the time derivative of r times the time derivative of theta, all times the transverse unit vector.”
    • Meaning: Radial acceleration includes centripetal term −rθ˙2-r\dot{\theta}^2; tangential includes Coriolis term 2r˙θ˙2\dot{r}\dot{\theta}.

  • Time Derivatives: u˙r=θ˙uθ\dot{\mathbf{u}}_r = \dot{\theta}\mathbf{u}_\theta and u˙θ=−θ˙ur\dot{\mathbf{u}}_\theta = -\dot{\theta}\mathbf{u}_r.

    • How to read: “The time derivative of the radial unit vector is equal to the time derivative of theta times the transverse unit vector; and the time derivative of the transverse unit vector is equal to the negative time derivative of theta times the radial unit vector.”
    • Meaning: As the particle moves, the basis vectors rotate—this rotation creates extra acceleration terms.

Connected Concepts

Connected notes