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Physics · General Physics and Classical Mechanics

Kinematics in Cartesian, Polar and Spherical Coordinates

In coordinate systems with position-dependent basis vectors, differentiating the position vector twice generates named acceleration terms: centripetal, Coriolis and Euler; this chapter derives the velocity and acceleration in Cartesian, plane polar, cylindrical and spherical coordinates.

Kinematics in Cartesian, Polar and Spherical Coordinates

High-school kinematics lives almost entirely in Cartesian coordinates, where the unit vectors x^,y^,z^\hat{x},\hat{y},\hat{z} are the same at every point and differentiating a position vector means differentiating three independent components. University mechanics cannot afford that luxury. Orbits, rotating machinery, central-force problems and anything with spherical or cylindrical symmetry are far more naturally described in coordinates whose basis vectors change from point to point — and the moment a basis vector depends on position, the product rule adds terms to every velocity and acceleration that have real physical names: the centripetal term, the Coriolis term, the Euler term.

This chapter derives those terms from scratch. The method is always the same: write the position vector in the chosen coordinates, then differentiate with respect to time, being careful to differentiate the basis vectors too. Once you have done this honestly for plane polar coordinates, the cylindrical and spherical results follow with more algebra but no new ideas.

The one habit that matters. In any non-Cartesian system, never write "a⃗=r¨ r^\vec{a} = \ddot{r}\,\hat{r}" by analogy with the Cartesian case. Always start from r⃗=r r^\vec{r} = r\,\hat{r} (or its spherical analogue) and differentiate twice, tracking r^˙\dot{\hat{r}}, θ^˙\dot{\hat{\theta}} and so on. The extra terms are the physics.

1. Position, Velocity and Acceleration as Vectors

For a particle tracing a path r⃗(t)\vec{r}(t), velocity and acceleration are defined coordinate-free as

v⃗=dr⃗dt,a⃗=dv⃗dt=d2r⃗dt2\vec{v} = \frac{d\vec{r}}{dt}, \qquad \vec{a} = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}

These definitions are independent of the coordinate system; only their components change from one system to another. The speed is ∣v⃗∣|\vec{v}| and the velocity is always tangent to the path. What differs between coordinate systems is how much of the work is done by changing components and how much by changing basis vectors.

2. Cartesian Coordinates: The Easy Case

With r⃗=x x^+y y^+z z^\vec{r} = x\,\hat{x} + y\,\hat{y} + z\,\hat{z} and constant basis vectors (x^˙=y^˙=z^˙=0⃗\dot{\hat{x}} = \dot{\hat{y}} = \dot{\hat{z}} = \vec{0}),

v⃗=x˙ x^+y˙ y^+z˙ z^,a⃗=x¨ x^+y¨ y^+z¨ z^\vec{v} = \dot{x}\,\hat{x} + \dot{y}\,\hat{y} + \dot{z}\,\hat{z}, \qquad \vec{a} = \ddot{x}\,\hat{x} + \ddot{y}\,\hat{y} + \ddot{z}\,\hat{z}

Every component evolves independently — the statement that motion along perpendicular Cartesian axes is decoupled. This is why projectile motion splits into two one-dimensional problems, and it is exactly the feature the other coordinate systems lack.

3. Plane Polar Coordinates: A Rotating Basis

In the plane, a point is located by (r,θ)(r,\theta) with basis vectors r^\hat{r} (radially outward) and θ^\hat{\theta} (in the direction of increasing θ\theta, i.e. 90∘90^\circ counter-clockwise from r^\hat{r}). In terms of the fixed Cartesian basis,

r^=cos⁡θ x^+sin⁡θ y^,θ^=−sin⁡θ x^+cos⁡θ y^\hat{r} = \cos\theta\,\hat{x} + \sin\theta\,\hat{y}, \qquad \hat{\theta} = -\sin\theta\,\hat{x} + \cos\theta\,\hat{y}

Both depend on θ\theta, and θ\theta depends on time, so by the chain rule

dr^dt=θ˙ (−sin⁡θ x^+cos⁡θ y^)=θ˙ θ^,dθ^dt=θ˙ (−cos⁡θ x^−sin⁡θ y^)=−θ˙ r^\frac{d\hat{r}}{dt} = \dot\theta\,(-\sin\theta\,\hat{x} + \cos\theta\,\hat{y}) = \dot\theta\,\hat{\theta}, \qquad \frac{d\hat{\theta}}{dt} = \dot\theta\,(-\cos\theta\,\hat{x} - \sin\theta\,\hat{y}) = -\dot\theta\,\hat{r}

These two little results — r^˙=θ˙ θ^\dot{\hat{r}} = \dot\theta\,\hat{\theta} and θ^˙=−θ˙ r^\dot{\hat{\theta}} = -\dot\theta\,\hat{r} — are the entire content of polar kinematics. Everything below is the product rule applied with them.

4. Velocity and Acceleration in Polar Form

Start from r⃗=r r^\vec{r} = r\,\hat{r}. Differentiating once, with the product rule:

v⃗=r˙ r^+r r^˙=r˙ r^+rθ˙ θ^\vec{v} = \dot{r}\,\hat{r} + r\,\dot{\hat{r}} = \dot{r}\,\hat{r} + r\dot\theta\,\hat{\theta}

The first term is the radial velocity (moving in or out); the second is the transverse velocity rθ˙r\dot\theta (swinging around). Differentiating again:

a⃗=r¨ r^+r˙ r^˙+r˙θ˙ θ^+rθ¨ θ^+rθ˙ θ^˙\vec{a} = \ddot{r}\,\hat{r} + \dot{r}\,\dot{\hat{r}} + \dot{r}\dot\theta\,\hat{\theta} + r\ddot\theta\,\hat{\theta} + r\dot\theta\,\dot{\hat{\theta}}

Substituting r^˙=θ˙ θ^\dot{\hat{r}} = \dot\theta\,\hat{\theta} and θ^˙=−θ˙ r^\dot{\hat{\theta}} = -\dot\theta\,\hat{r} and collecting components:

a⃗=(r¨−rθ˙ 2)r^+(rθ¨+2r˙θ˙)θ^\vec{a} = \left(\ddot{r} - r\dot\theta^{\,2}\right)\hat{r} + \left(r\ddot\theta + 2\dot{r}\dot\theta\right)\hat{\theta}

Each term has a name:

  • r¨\ddot{r} — genuine radial acceleration from changing radial speed.
  • −rθ˙ 2-r\dot\theta^{\,2} — the centripetal term: present whenever the particle is going around, directed inward, equal to −vθ2/r-v_\theta^2/r.
  • rθ¨r\ddot\theta — the angular or Euler term: from angular acceleration.
  • 2r˙θ˙2\dot{r}\dot\theta — the Coriolis term: present only when the particle is simultaneously moving radially and rotating. It is the reason a figure skater spins up when pulling their arms in.

Worked example — uniform circular motion. If r=Rr = R is constant and θ˙=ω\dot\theta = \omega is constant, then r˙=r¨=0\dot{r} = \ddot{r} = 0 and θ¨=0\ddot\theta = 0, leaving

v⃗=Rω θ^,a⃗=−Rω2 r^=−v2R r^\vec{v} = R\omega\,\hat{\theta}, \qquad \vec{a} = -R\omega^2\,\hat{r} = -\frac{v^2}{R}\,\hat{r}

exactly the familiar centripetal result — but now derived, not asserted, and as a special case of a formula that also handles spiralling and non-uniform rotation.

Worked example — a bead on a rotating rod. A rod pivoted at the origin rotates at constant θ˙=ω\dot\theta = \omega while a frictionless bead slides outward with r(t)r(t). The transverse component of a⃗\vec{a} is rθ¨+2r˙θ˙=2r˙ωr\ddot\theta + 2\dot{r}\dot\theta = 2\dot{r}\omega: even with zero angular acceleration, a transverse (Coriolis) acceleration 2r˙ω2\dot{r}\omega is required, supplied by the normal force from the rod. If no transverse force acted, angular momentum L=mr2θ˙L = mr^2\dot\theta would be conserved and θ˙\dot\theta would have to fall as the bead moved out; here the rod does push transversely, so LL grows and ω\omega stays fixed.

5. Cylindrical Coordinates

Cylindrical coordinates (ρ,ϕ,z)(\rho,\phi,z) are plane polars in the xyxy-plane plus an independent Cartesian zz. Since z^\hat{z} is constant, the results simply append a zz-term to the polar expressions:

v⃗=ρ˙ ρ^+ρϕ˙ ϕ^+z˙ z^\vec{v} = \dot\rho\,\hat{\rho} + \rho\dot\phi\,\hat{\phi} + \dot{z}\,\hat{z} a⃗=(ρ¨−ρϕ˙ 2)ρ^+(ρϕ¨+2ρ˙ϕ˙)ϕ^+z¨ z^\vec{a} = \left(\ddot\rho - \rho\dot\phi^{\,2}\right)\hat{\rho} + \left(\rho\ddot\phi + 2\dot\rho\dot\phi\right)\hat{\phi} + \ddot{z}\,\hat{z}

These are the natural coordinates for helical motion, for a charged particle in a uniform magnetic field, and for anything with axial symmetry.

6. Spherical Coordinates: The Basis Vectors

Spherical coordinates (r,θ,ϕ)(r,\theta,\phi) use rr = distance from origin, θ\theta = polar angle from the zz-axis (colatitude), ϕ\phi = azimuthal angle in the xyxy-plane. The orthonormal basis is

r^=sin⁡θcos⁡ϕ x^+sin⁡θsin⁡ϕ y^+cos⁡θ z^\hat{r} = \sin\theta\cos\phi\,\hat{x} + \sin\theta\sin\phi\,\hat{y} + \cos\theta\,\hat{z} θ^=cos⁡θcos⁡ϕ x^+cos⁡θsin⁡ϕ y^−sin⁡θ z^\hat{\theta} = \cos\theta\cos\phi\,\hat{x} + \cos\theta\sin\phi\,\hat{y} - \sin\theta\,\hat{z} ϕ^=−sin⁡ϕ x^+cos⁡ϕ y^\hat{\phi} = -\sin\phi\,\hat{x} + \cos\phi\,\hat{y}

All three depend on the angles, so all three have non-zero time derivatives whenever θ\theta or ϕ\phi changes. Working them out with the chain rule (the same procedure as in Section 3, just longer) gives

r^˙=θ˙ θ^+ϕ˙sin⁡θ ϕ^,θ^˙=−θ˙ r^+ϕ˙cos⁡θ ϕ^,ϕ^˙=−ϕ˙sin⁡θ r^−ϕ˙cos⁡θ θ^\dot{\hat{r}} = \dot\theta\,\hat{\theta} + \dot\phi\sin\theta\,\hat{\phi}, \quad \dot{\hat{\theta}} = -\dot\theta\,\hat{r} + \dot\phi\cos\theta\,\hat{\phi}, \quad \dot{\hat{\phi}} = -\dot\phi\sin\theta\,\hat{r} - \dot\phi\cos\theta\,\hat{\theta}

7. Velocity and Acceleration in Spherical Form

Starting from r⃗=r r^\vec{r} = r\,\hat{r} and differentiating once:

v⃗=r˙ r^+rθ˙ θ^+rϕ˙sin⁡θ ϕ^\vec{v} = \dot{r}\,\hat{r} + r\dot\theta\,\hat{\theta} + r\dot\phi\sin\theta\,\hat{\phi}

The three components are the radial, polar and azimuthal velocities; the factor sin⁡θ\sin\theta in the last reflects that circles of constant θ\theta shrink toward the poles. Differentiating a second time and collecting terms yields the standard result

a⃗=(r¨−rθ˙ 2−rϕ˙ 2sin⁡2θ)r^+(rθ¨+2r˙θ˙−rϕ˙ 2sin⁡θcos⁡θ)θ^\vec{a} = \left(\ddot{r} - r\dot\theta^{\,2} - r\dot\phi^{\,2}\sin^2\theta\right)\hat{r} + \left(r\ddot\theta + 2\dot{r}\dot\theta - r\dot\phi^{\,2}\sin\theta\cos\theta\right)\hat{\theta} +(rϕ¨sin⁡θ+2r˙ϕ˙sin⁡θ+2rθ˙ϕ˙cos⁡θ)ϕ^+ \left(r\ddot\phi\sin\theta + 2\dot{r}\dot\phi\sin\theta + 2r\dot\theta\dot\phi\cos\theta\right)\hat{\phi}

You are rarely asked to reproduce all nine terms from memory, but you are expected to be able to derive them and to recognise the centripetal (−rθ˙2-r\dot\theta^2, −rϕ˙2sin⁡2θ-r\dot\phi^2\sin^2\theta) and Coriolis (2r˙θ˙2\dot{r}\dot\theta, 2r˙ϕ˙sin⁡θ2\dot{r}\dot\phi\sin\theta, 2rθ˙ϕ˙cos⁡θ2r\dot\theta\dot\phi\cos\theta) contributions.

Worked example — the conical pendulum. A bob swings so that r=ℓr = \ell (constant, the string length) and θ=θ0\theta = \theta_0 (constant cone half-angle), with steady azimuthal rate ϕ˙=Ω\dot\phi = \Omega. Then r˙=r¨=0\dot{r} = \ddot{r} = 0, θ˙=θ¨=0\dot\theta = \ddot\theta = 0, ϕ¨=0\ddot\phi = 0, and the acceleration reduces to

a⃗=−ℓΩ2sin⁡2θ0 r^−ℓΩ2sin⁡θ0cos⁡θ0 θ^\vec{a} = -\ell\Omega^2\sin^2\theta_0\,\hat{r} - \ell\Omega^2\sin\theta_0\cos\theta_0\,\hat{\theta}

Its magnitude is ℓΩ2sin⁡θ0\ell\Omega^2\sin\theta_0 and it points horizontally, toward the axis — the centripetal acceleration of uniform circular motion on the circle of radius ℓsin⁡θ0\ell\sin\theta_0. Setting this equal to the horizontal component of tension over mass recovers the standard result Ω2=g/(ℓcos⁡θ0)\Omega^2 = g/(\ell\cos\theta_0).

8. Common Pitfalls

  • Treating r^\hat{r}, θ^\hat{\theta}, ϕ^\hat{\phi} as constant. They are the whole difficulty. Their time derivatives generate every centripetal, Coriolis and Euler term.
  • Writing a⃗=r¨ r^+rθ¨ θ^\vec{a} = \ddot{r}\,\hat{r} + r\ddot\theta\,\hat{\theta} by dropping the "extra" terms. The correct radial component is r¨−rθ˙2\ddot{r} - r\dot\theta^2 and the transverse is rθ¨+2r˙θ˙r\ddot\theta + 2\dot{r}\dot\theta.
  • Confusing the two polar angles in spherical coordinates. In physics convention θ\theta is measured from the zz-axis (colatitude, 00 to π\pi) and ϕ\phi is azimuthal (00 to 2π2\pi); many maths texts swap the symbols.
  • Forgetting the sin⁡θ\sin\theta in the azimuthal velocity. vϕ=rϕ˙sin⁡θv_\phi = r\dot\phi\sin\theta, not rϕ˙r\dot\phi; the discrepancy vanishes only in the equatorial plane.
  • Mixing bases in one equation. Do not add a r^\hat{r} component to an x^\hat{x} component; convert everything to one basis first.
  • Assuming the polar θ^\hat{\theta} and the spherical θ^\hat{\theta} are the same object. They are defined by different geometry; keep track of which coordinate system you are in.

Summary

  • Velocity and acceleration are defined coordinate-free as r⃗˙\dot{\vec{r}} and r⃗¨\ddot{\vec{r}}; only the components and basis vectors change between systems.
  • Cartesian: constant basis, so a⃗=x¨ x^+y¨ y^+z¨ z^\vec{a} = \ddot{x}\,\hat{x} + \ddot{y}\,\hat{y} + \ddot{z}\,\hat{z} and the axes decouple.
  • Plane polar: r^˙=θ˙ θ^\dot{\hat{r}} = \dot\theta\,\hat{\theta}, θ^˙=−θ˙ r^\dot{\hat{\theta}} = -\dot\theta\,\hat{r} give v⃗=r˙ r^+rθ˙ θ^\vec{v} = \dot{r}\,\hat{r} + r\dot\theta\,\hat{\theta} and a⃗=(r¨−rθ˙2) r^+(rθ¨+2r˙θ˙) θ^\vec{a} = (\ddot{r} - r\dot\theta^2)\,\hat{r} + (r\ddot\theta + 2\dot{r}\dot\theta)\,\hat{\theta}.
  • The named terms: −rθ˙2-r\dot\theta^2 centripetal, 2r˙θ˙2\dot{r}\dot\theta Coriolis, rθ¨r\ddot\theta Euler/angular.
  • Cylindrical: the polar results with an independent z¨ z^\ddot{z}\,\hat{z} appended.
  • Spherical: v⃗=r˙ r^+rθ˙ θ^+rϕ˙sin⁡θ ϕ^\vec{v} = \dot{r}\,\hat{r} + r\dot\theta\,\hat{\theta} + r\dot\phi\sin\theta\,\hat{\phi}; the acceleration has centripetal and Coriolis terms in every component, derivable from the three basis-vector derivatives.
  • Method for any system: write r⃗\vec{r} in that system and differentiate twice, always differentiating the basis vectors too.