Coordinate Systems for Mechanics

A coordinate system is a mathematical scheme for assigning numbers to positions and directions. The physical motion does not depend on which coordinates are chosen, but a good coordinate system can make the geometry, constraints, and equations of a mechanics problem much simpler.

The central principle is therefore not that one coordinate system is more “correct” than another. Instead,

(1)

Cartesian coordinates are natural for straight-line motion, projectiles, rectangular geometry, and many Free-body diagrams. Polar and cylindrical coordinates are natural when distance from an axis and rotation about that axis are important. Spherical coordinates are natural when distance from a point is the dominant geometric variable, as in central-force problems.

This article develops the geometry needed to use these systems in mechanics without yet deriving the full velocity and acceleration formulas of curvilinear kinematics. Those derivatives belong to later kinematics articles.

1 Coordinate system versus reference frame

A coordinate system and a reference frame are related but distinct ideas.

A reference frame identifies the physical observer or laboratory whose origin and axes have a specified motion. A coordinate system is the mathematical labeling used inside that frame.

For example, one inertial laboratory frame may describe a particle using Cartesian coordinates (x,y,z) or cylindrical coordinates (ρ,ϕ,z). Changing from one of those coordinate systems to the other does not by itself create a new physical observer.

This distinction becomes important later when mechanics is written in accelerating or rotating reference frames.

2 Cartesian coordinates

In a three-dimensional Cartesian coordinate system, mutually perpendicular axes define fixed unit vectors

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A position vector is

(3)

The coordinates x, y, and z are signed distances measured along the three axes.

Because the Cartesian basis vectors are fixed in orientation, they are especially convenient when differentiating vector components with respect to time.

The infinitesimal displacement is

(4)

and the differential distance satisfies

(5)

2.1 Rotated Cartesian axes

The axes need not be horizontal and vertical. In an inclined-plane problem, for example, it is often convenient to choose one Cartesian axis parallel to the incline and another normal to it.

If a new x′ axis is rotated counterclockwise through an angle α from the original x axis, then

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and

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For a vector

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the components in the rotated basis are

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(10)

The geometric vector is unchanged; only its numerical components have changed.

3 Plane polar coordinates

For two-dimensional problems with circular or radial geometry, a point may be described by

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where r is the nonnegative distance from the origin and 𝜃 is the angular coordinate measured from the positive x axis.

The Cartesian and polar coordinates are related by

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(13)

Conversely,

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and the angle should be determined with the signs of both x and y taken into account. Computationally, a quadrant-aware two-argument arctangent is preferred.

3.1 The polar basis

Polar coordinates use position-dependent unit vectors. The radial unit vector points away from the origin,

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and the transverse unit vector points in the direction of increasing 𝜃,

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The position vector is particularly simple:

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Unlike e_x and e_y, the polar basis vectors change direction when 𝜃 changes. This fact is responsible for important extra terms when velocity and acceleration are differentiated in polar coordinates. Those terms are geometric consequences of the moving basis, not fictitious forces created merely by choosing polar coordinates.

4 Deriving the polar differential displacement

The geometry of polar coordinates becomes especially clear by differentiating the coordinate transformation. Starting with

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we obtain

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Similarly,

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Insert these into

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Collecting terms gives

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The two perpendicular differential distances are therefore

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and

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Hence

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For motion along a circle of fixed radius,

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so

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The angle must be expressed in radians for this differential arc-length relation.

5 Cylindrical coordinates

Cylindrical coordinates extend plane polar coordinates by retaining an independent vertical coordinate:

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Here ρ is distance from the z axis, ϕ is the azimuth about the z axis, and z is the usual Cartesian height.

The transformation is

(29)

(30)

(31)

Conversely,

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The local basis is

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and the differential displacement is

(34)

Therefore

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The differential volume is

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The factor ρ is geometric: an angular increment dϕ corresponds to a longer physical arc when the point lies farther from the axis.

Cylindrical coordinates are useful for shafts, disks, cylinders, circular or helical motion, and systems with symmetry about an axis.

6 Spherical coordinates

Spherical coordinates are natural for problems centered on a point. Several notation conventions exist, so every article or problem should state its convention explicitly.

This mechanics article uses

(37)

where

• r is distance from the origin;
• 𝜃 is the polar angle measured downward from the positive z axis;
• ϕ is the azimuth measured in the xy plane from the positive x axis.

The Cartesian transformation is

(38)

(39)

(40)

Conversely,

(41)

(42)

The azimuth ϕ is determined from x and y with quadrant information retained.

The differential displacement is

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Thus

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and the differential volume is

(45)

Spherical coordinates are especially useful for gravitational and other central-force fields, spherical bodies, and three-dimensional systems with symmetry about a point.

7 Scale factors and orthogonal curvilinear coordinates

Cartesian, polar, cylindrical, and spherical coordinates are all examples of orthogonal coordinate systems: their local basis directions are mutually perpendicular.

A useful general form for the differential distance is

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where the quantities h_i are called scale factors.

For Cartesian coordinates,

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For cylindrical coordinates,

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For the spherical convention used here,

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These factors encode how a coordinate increment translates into an actual physical distance. They later enter velocity, acceleration, gradients, divergence, curl, and volume integration in curvilinear coordinates.

8 Worked example 1: Cartesian to polar

A point in the plane has Cartesian coordinates

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Find its polar coordinates.

The radial coordinate is

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so

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The point lies in the first quadrant, and

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Therefore

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Thus the same point may be written

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9 Worked example 2: Cartesian to cylindrical

A point has Cartesian coordinates

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Find its cylindrical coordinates.

The distance from the z axis is

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Because x < 0 and y > 0, the point is in the second quadrant of the xy plane. The azimuth is therefore

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The vertical coordinate is unchanged:

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Hence

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10 Worked example 3: Cartesian to spherical

A point has coordinates

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Using the spherical convention of this article, find (r,𝜃,ϕ).

First,

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Next,

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so

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Since x = y > 0,

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Thus

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11 Worked example 4: choose axes that follow an incline

A 2.00 kg block rests on a frictionless plane inclined at

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above the horizontal. Choose a rotated Cartesian basis with +x′ up the incline and +y′ outward normal to the surface. Resolve the weight into this basis.

The weight magnitude is

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The component along the incline is opposite +x′:

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Thus

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The normal component points into the plane, opposite +y′:

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Therefore

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The physics has not changed. The coordinate choice simply aligns the components with the natural directions of the constraint.

12 How to choose a coordinate system

A useful coordinate system usually reflects either a geometric constraint or a symmetry.

Typical choices include:

• Cartesian coordinates for projectiles, blocks, straight tracks, and rectangular geometries;
• rotated Cartesian coordinates for inclined planes or motion constrained along a straight line;
• polar coordinates for planar rotation or central motion;
• cylindrical coordinates for rotation about an axis, disks, cylinders, and helical paths;
• spherical coordinates for gravitational or other central fields and spherical bodies.

A good choice often reduces the number of nonzero components or turns a constraint into the statement that one coordinate is constant.

13 Coordinate singularities

Curvilinear coordinate systems contain locations where one or more coordinates become ambiguous.

At r = 0 in polar coordinates, the angle 𝜃 is undefined. In cylindrical coordinates, ϕ is undefined on the z axis. In spherical coordinates, the azimuth ϕ is undefined at the poles, and both angles are undefined at the origin.

These are often coordinate singularities, not singularities in the underlying physical space. A particle can pass smoothly through the origin even though polar coordinates temporarily become a poor description there.

14 Coordinates, components, and basis vectors

Three related ideas should not be confused:

1. coordinates label the location of a point;
2. components are numerical coefficients of a vector relative to a basis;
3. basis vectors specify the directions used to construct the vector.

In Cartesian coordinates, the position coordinates and the coefficients in

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happen to look very similar. In spherical coordinates the position vector is instead simply

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even though three coordinates (r,𝜃,ϕ) are needed to locate the point. The angles determine the direction of e_r rather than appearing as separate additive components of r.

15 Preview: from coordinates to kinematics

When coordinates depend on time, the position becomes

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In Cartesian coordinates the basis is fixed, while in polar, cylindrical, and spherical coordinates the local basis changes direction as the particle moves.

This distinction produces the characteristic radial and transverse terms in curvilinear velocity and acceleration. Later articles derive, for example, the polar-coordinate velocity

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and the corresponding acceleration. Those are kinematic consequences of differentiating both the coordinates and the moving basis vectors.

At a still more general level, analytical mechanics uses generalized coordinates chosen to match the degrees of freedom and constraints of a system. Cartesian, cylindrical, and spherical coordinates are important examples, but they are not the end of the idea.

16 Common mistakes

1. Confusing a coordinate system with a reference frame. A coordinate transformation need not represent a different observer.
2. Using a one-argument arctangent without checking the quadrant. The ratio y∕x alone does not identify the correct angle.
3. Mixing spherical conventions. Some texts interchange the symbols 𝜃 and ϕ. State the convention before calculating.
4. Treating curvilinear basis vectors as fixed. e_r, e_𝜃, e_ρ, and e_ϕ change direction with position.
5. Using degrees in differential arc formulas. Relations such as ds = r d𝜃 assume radians.
6. Mistaking coordinate singularities for physical singularities. An undefined angular coordinate may reflect only a poor labeling at that point.

17 Practice problems

MECH-COORD-001 — Choose the natural coordinates

For each system, identify a natural coordinate system and explain why: (a) a projectile near Earth’s surface, (b) a bead constrained to a circular hoop in a plane, and (c) a particle moving in a central gravitational field.

MECH-COORD-002 — Cartesian to polar

Convert

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to polar coordinates.

MECH-COORD-003 — Polar to Cartesian

Convert

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to Cartesian coordinates.

MECH-COORD-004 — Cylindrical to Cartesian

Convert

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to Cartesian coordinates.

MECH-COORD-005 — Cartesian to spherical

Using the convention of this article, convert

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to spherical coordinates.

MECH-COORD-006 — Arc length in polar coordinates

A particle moves along a circle of radius 2.00 m through an angular increment of 0.300 rad. Find the path length.

MECH-COORD-007 — Differential distance in cylindrical coordinates

At ρ = 3.00 m, a small coordinate change is

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Estimate ds.

MECH-COORD-008 — Differential distance in spherical coordinates

At r = 4.00 m and 𝜃 = 60^∘, let

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Estimate ds.

MECH-COORD-009 — Cylindrical volume element

At ρ = 2.00 m, estimate the volume of a small cylindrical-coordinate cell with

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MECH-COORD-010 — Gravity in incline coordinates

A 5.00 kg block lies on a plane inclined at 20^∘. Choose +x′ up the plane and +y′ outward normal to it. Find the two components of the weight using g = 9.80 m/s².

18 Short answer key

1. (a) Cartesian; (b) polar; (c) spherical.
2. r = 4.243 m, 𝜃 = 135^∘.
3. x = −3.464 m, y = −2.000 m.
4. x ≈ 3.00 m, y ≈ 4.00 m, z = −2.00 m.
5. r = 2 = 2.828 m, 𝜃 = 45^∘, ϕ = 120^∘.
6. s = 0.600 m.
7. ds ≈ 0.122 m.
8. ds ≈ 0.132 m.
9. dV = 0.00200 m³.
10. W_x′ = −16.8 N, W_y′ = −46.0 N.

19 Connections to later mechanics

Coordinate systems are used throughout the mechanics learning path. The next foundations article distinguishes coordinate descriptions from physical reference frames. Later kinematics articles derive velocity and acceleration in polar, cylindrical, and spherical coordinates. Central-force mechanics, rotating frames, rigid-body dynamics, and Lagrangian mechanics all rely on the coordinate ideas introduced here.

References