Classical Mechanics

Motion on the Rotating Earth

Applications of rotating-frame dynamics to motion near Earth's surface.

This essay applies the dynamics of rotating reference frames to motion near Earth's surface. Treating the Earth as a rotating frame, we derive the Coriolis and centrifugal contributions in terrestrial coordinates and examine how the general equations of rotating-frame dynamics acquire concrete physical meaning.

Prerequisites

Dynamics in Rotating Reference Frames
Vector calculus
Introductory classical mechanics

Earth-fixed reference frames
Earth's angular velocity
Local terrestrial coordinates
Coriolis effects near Earth's surface
Centrifugal effects near Earth's surface
Latitude dependence
Physical interpretation

I. Rotating-Frame Preliminaries
II. Centrifugal and Coriolis Forces on Earth
1. Forces in Spherical Coordinates
2. Lagrangian in Spherical Coordinates
3. Linearized Equations of Motion

I. Rotating-Frame Preliminaries

For a reference frame rotating with constant angular velocity

ω

, the acceleration measured in the rotating frame is

Earth:Acceleration Transformation

a^{'} = A a - ω \times (ω \times r^{'}) - 2 ω \times \frac{d r ^{'}}{d t} .

The corresponding inertial forces are therefore the centrifugal force,

Earth:Centrifugal Force General

F_{cf} = - m ω \times (ω \times r^{'}),

and the Coriolis force,

Earth:Coriolis Force General

F_{cor} = - 2 m ω \times \frac{d r ^{'}}{d t} .

These expressions, derived in the previous essay, provide the rotating-frame dynamics used throughout the derivation of the Foucault pendulum.

II. Centrifugal and Coriolis Forces on Earth

In this section we study the centrifugal and Coriolis forces near the Earth's surface. To keep the analysis as transparent as possible, we model the Earth as a perfect sphere. Although this approximation neglects effects such as the Earth's oblateness, it captures the essential physics and provides a convenient setting for deriving the mathematical expressions used throughout this essay.

We consider a reference frame fixed to the Earth with origin at its center. The

z

-axis points from the South Pole toward the North Pol to the North Pole, with positive direction from South to North. Earth's angular velocity is along the

z

axis:

Earth:Angular Velocity

ω = ω k

where

ω

is the angular velocity of the Earth equal to

7.29 \times 1 0^{- 5} s^{- 1}

We will be interested in the motion of a body in the vicinity of the Earth surface, and the ideal coordinates to describe such a motion are the spherical coordinates

(ρ, θ, ϕ)

, such that,

x y z = ρ sin ϕ cos θ = ρ sin ϕ sin θ = ρ cos ϕ

Earth-fixed spherical coordinates.

where

ρ \geq 0

is the radial distance from the center of the reference system,

0 \leq θ \leq 2 π

is the azimuthal angle, and

0 \leq ϕ \leq π

is the polar angle. Associated with the spherical coordinates are the unit vectors

e_{r}

e_{θ}

, and

e_{ϕ}

given by,

e_{ρ} e_{θ} e_{ϕ} = sin ϕ cos θ i + sin ϕ sin θ j + cos ϕ k = - sin θ i + cos θ j = cos ϕ cos θ i + cos ϕ sin θ j - sin ϕ k

where

i

j

, and

k

are the unit vectors of the rotating reference system along the

x

y

, and

z

axes, respectively. The unit vectors define a local orthonormal basis and maintain the orientation of the reference system:

e_{θ} \times e_{ρ} e_{ϕ} \times e_{θ} e_{ρ} \times e_{ϕ} = e_{ϕ} = e_{ρ} = e_{θ}

and they are related to the latitude and longitude of the Earth:

•
$e_{ρ}$ : Radial unit vector pointing outward from the center of the Earth, through the point under consideration.
•
$e_{θ}$ : Points tangentially along lines of latitude towards the east.
•
$e_{ϕ}$ : Points tangentially along lines of longitude towards the South Pole.

The position vector in spherical coordinates is,

r = ρ e_{ρ}

and the angular velocity is represented by,

ω = ω (cos (ϕ) e_{ρ} - sin (ϕ) e_{ϕ}) .

and finally the velocity of the body in spherical coordinates is given by,

\dot{r} = \overset{ρ}{˙} e_{ρ} + ρ \dot{θ} sin (ϕ) e_{θ} + ρ \dot{ϕ} e_{ϕ}

Since the Earth's surface is naturally described by spherical coordinates, expressing both the angular velocity and the particle velocity in the local spherical basis considerably simplifies the derivation of the centrifugal and Coriolis forces.

1. Forces in Spherical Coordinates

In the rotating frame the body is subject to three forces: the gravitational force, the centrifugal force, and the Coriolis force.

•
The gravitational force is directed radially inward and is given by,
$F_{g} = - m g e_{ρ}$
where $m$ is the mass of the body and $g$ is the acceleration due to gravity.
•
The centrifugal force is directed radially outward from the axis of rotation and in spherical coordinates it is represented by the formula:
Earth:Centrifugal Force Spherical
$F_{cf} = m ω^{2} ρ (sin^{2} (ϕ) e_{ρ} + sin (ϕ) cos (ϕ) e_{ϕ})$
•
The Coriolis force is perpendicular to the velocity of the body and is quantified by the formula:
$F_{cor} = - 2 m ω \times \dot{r} = - 2 mω (cos (ϕ) e_{ρ} - sin (ϕ) e_{ϕ}) \times (\overset{ρ}{˙} e_{ρ} + ρ \dot{θ} sin (ϕ) e_{θ} + ρ \dot{ϕ} e_{ϕ}) = - 2 mω (ρ \dot{θ} sin (ϕ) cos (ϕ) e_{ρ} \times e_{θ} + (cos (ϕ) ρ \dot{ϕ} + sin (ϕ) \overset{ρ}{˙}) e_{ρ} \times e_{ϕ} - sin^{2} (ϕ) ρ \dot{θ} e_{ϕ} \times e_{θ}) = - 2 mω (- ρ \dot{θ} sin (ϕ) cos (ϕ) e_{ϕ} + (cos (ϕ) ρ \dot{ϕ} + sin (ϕ) \overset{ρ}{˙}) e_{θ} - sin^{2} (ϕ) ρ \dot{θ} e_{ρ})$
which simplifies to,
Earth:Coriolis Force Spherical
$F_{cor} = mω (2 ρ \dot{θ} sin^{2} (ϕ) e_{ρ} - 2 (\overset{ρ}{˙} sin (ϕ) + ρ \dot{ϕ} cos (ϕ)) e_{θ} + ρ \dot{θ} sin (2 ϕ) e_{ϕ})$

Before proceeding, it is instructive to examine the combined inertial force obtained by adding the centrifugal and Coriolis contributions,

F = mω ρ (t) sin^{2} (ϕ (t)) (2 \dot{θ} (t) + ω) e_{ρ} - 2 mω (\overset{ρ}{˙} (t) sin (ϕ (t)) + ρ (t) \dot{ϕ} (t) cos (ϕ (t))) e_{θ} + mω ρ (t) sin (ϕ (t)) cos (ϕ (t)) (2 \dot{θ} (t) + ω) e_{ϕ}

One particularly interesting situation emerge from these expressions, which corresponds to a body at rest in the inertial frame. In such a situation the body at rest in the inertial frame appears in the rotating frame to move with angular velocity

\dot{θ} = - ω

, while

\overset{ρ}{˙} = \dot{ϕ} = 0

. Under these conditions, the combined inertial force is

F = - m ω^{2} ρ (sin^{2} (ϕ) e_{ρ} + sin (ϕ) cos (ϕ) e_{ϕ})

The Coriolis force is therefore equal to twice the centrifugal force in magnitude and points in the opposite direction. Their resultant is an inward force equal to the negative of the centrifugal force. This resultant produces the centripetal acceleration observed in the rotating frame as the body appears to revolve around the Earth's axis with velocity

- ω

2. Lagrangian in Spherical Coordinates

We now formulate the dynamics of the system in spherical coordinates. Under the assumption that the gravitational acceleration

g

is constant and directed radially inward, the gravitational potential is

V_{g} = m g ρ .

The centrifugal force derives from the position-dependent potential

V_{cf} = - \frac{1}{2} m ω^{2} ρ^{2} sin^{2} (ϕ) .

The Coriolis force cannot be derived from an ordinary position-dependent potential. It can, however, be represented by the velocity-dependent generalized potential

V_{cor} = - mω ρ^{2} \dot{θ} sin^{2} (ϕ) .

Indeed, for each generalized coordinate

q_{i} \in {ρ, ϕ, θ}

, it satisfies

\frac{d}{d t} \frac{\partial V _{cor}}{\partial q ˙ _{i}} - \frac{\partial V _{cor}}{\partial q _{i}} = ⟨ F_{cor} \frac{\partial r}{\partial q _{i}} ⟩ .

Explicitly,

\frac{d}{d t} \frac{\partial V _{cor}}{\partial θ ˙} - \frac{\partial V _{cor}}{\partial θ} \frac{d}{d t} \frac{\partial V _{cor}}{\partial ϕ ˙} - \frac{\partial V _{cor}}{\partial ϕ} \frac{d}{d t} \frac{\partial V _{cor}}{\partial ρ ˙} - \frac{\partial V _{cor}}{\partial ρ} = ⟨ F_{cor} \frac{\partial r}{\partial θ} ⟩, = ⟨ F_{cor} \frac{\partial r}{\partial ϕ} ⟩, = ⟨ F_{cor} \frac{\partial r}{\partial ρ} ⟩ .

Similarly, the centrifugal potential satisfies

- \frac{\partial V _{cf}}{\partial θ} - \frac{\partial V _{cf}}{\partial ϕ} - \frac{\partial V _{cf}}{\partial ρ} = ⟨ F_{cf} \frac{\partial r}{\partial θ} ⟩, = ⟨ F_{cf} \frac{\partial r}{\partial ϕ} ⟩, = ⟨ F_{cf} \frac{\partial r}{\partial ρ} ⟩ .

The kinetic energy is

T = \frac{1}{2} m (\overset{ρ}{˙}^{2} + ρ^{2} \dot{ϕ}^{2} + ρ^{2} \dot{θ}^{2} sin^{2} (ϕ)) .

The Lagrangian is therefore

L = T - V_{g} - V_{cf} - V_{cor},

or, after combining the terms,

L = \frac{1}{2} m (- 2 g ρ + ρ^{2} (sin^{2} (ϕ) (\dot{θ} + ω)^{2} + \dot{ϕ}^{2}) + \overset{ρ}{˙}^{2}) .

The appearance of the combination

(\dot{θ} + ω)

has a direct physical interpretation: it is the angular velocity of the body with respect to the inertial frame. The equations of motion follow from the Euler--Lagrange equations,

\frac{d}{d t} \frac{\partial L}{\partial ρ ˙} - \frac{\partial L}{\partial ρ} \frac{d}{d t} \frac{\partial L}{\partial ϕ ˙} - \frac{\partial L}{\partial ϕ} \frac{d}{d t} \frac{\partial L}{\partial θ ˙} - \frac{\partial L}{\partial θ} = 0, = 0, = 0.

Since the Lagrangian does not depend explicitly on

θ

, this coordinate is cyclic and its conjugate momentum is conserved,

m ρ^{2} sin^{2} (ϕ) (\dot{θ} + ω) = constant .

Expanding the Euler--Lagrange equations yields the nonlinear system

Earth:Spherical Equations Of Motion

g - ρ (sin^{2} (ϕ) (\dot{θ} + ω)^{2} + \dot{ϕ}^{2}) + \overset{ρ}{¨} = 0, ρ (2 \ddot{ϕ} - sin (2 ϕ) (\dot{θ} + ω)^{2}) + 4 \overset{ρ}{˙} \dot{ϕ} = 0, ρ \ddot{θ} sin (ϕ) + 2 (\dot{θ} + ω) (\overset{ρ}{˙} sin (ϕ) + ρ \dot{ϕ} cos (ϕ)) = 0.

3. Linearized Equations of Motion

We consider now the motion of an object covering distances over the Earth surface much smaller than Earth radius

R

in any direction. We will assume the dynamics is in a neighbourhood of the position identified by spherical coordinates

(ρ_{0}, θ_{0}, ϕ_{0})

r_{0} = ρ_{0} e_{ρ_{0}}

where

ρ_{0} \approx R

. Around this position a local system of reference can be constructed by the unit vectors

e_{ρ_{0}}

e_{θ_{0}}

, and

e_{ϕ_{0}}

as,

e_{ρ_{0}} e_{θ_{0}} e_{ϕ_{0}} = sin ϕ_{0} cos θ_{0} i + sin ϕ_{0} sin θ_{0} j + cos ϕ_{0} k = - sin θ_{0} i + cos θ_{0} j = cos ϕ_{0} cos θ_{0} i + cos ϕ_{0} sin θ_{0} j - sin ϕ_{0} k

and a point in the vicinity of the position

r_{0}

has spherical coordinates

(ρ_{0} + δ ρ, θ_{0} + δ θ, ϕ_{0} + δ ϕ)

and position,

r = (ρ_{0} + δ ρ) e_{ρ} (θ_{0} + δ θ, ϕ_{0} + δ ϕ) .

which up to first order reads,

r = r_{0} + δ ρ e_{ρ_{0}} + ρ_{0} sin (ϕ_{0}) δ θ e_{θ_{0}} + ρ_{0} δ ϕ e_{ϕ_{0}} .

Under the assumption that

∥ r - r_{0} ∥ ≪ R

, we have for the displacements in each direction,

δ ρ ρ_{0} δ ϕ ρ_{0} sin (ϕ_{0}) δ θ ≪ ρ_{0} ≪ ρ_{0} or δ ϕ ≪ 1 ≪ ρ_{0} or sin (ϕ_{0}) δ θ ≪ 1

We will therefore assume that the dynamics can be described by the perturbations,

(ρ_{0} + ρ (t), θ_{0} + θ (t), ϕ_{0} + ϕ (t))

where at all times

ρ (t) ≪ ρ_{0}

ϕ (t) ≪ 1

, and

sin (ϕ_{0}) θ (t) ≪ 1

. The equations of motion spherical equations of motion can be expressed in terms of the displacements

(ρ (t), θ (t), ϕ (t))

and simplified to first order as,

Earth:Linearized Spherical Equations

\overset{ρ}{¨} (t) 2 ρ_{0} \ddot{ϕ} (t) ρ_{0} sin (ϕ_{0}) \ddot{θ} (t) = = = - g ω^{2} ρ_{0} sin (2 ϕ_{0}) + ω^{2} ρ_{0} sin^{2} (ϕ_{0}) + ω^{2} sin (2 ϕ_{0}) ρ (t) - 2 ω sin (ϕ_{0}) \overset{ρ}{˙} (t) + ω^{2} sin^{2} (ϕ_{0}) ρ (t) + 2 ω^{2} ρ_{0} cos (2 ϕ_{0}) ϕ (t) - 2 ω ρ_{0} cos (ϕ_{0}) \dot{ϕ} (t) + ω^{2} ρ_{0} sin (2 ϕ_{0}) ϕ (t) + 2 ω ρ_{0} sin (2 ϕ_{0}) \dot{θ} (t) + 2 ω ρ_{0} sin^{2} (ϕ_{0}) \dot{θ} (t)

Defining dimensional variables of length as,

x = ρ_{0} ϕ y = ρ_{0} sin (ϕ_{0}) θ z = ρ

and defining a local coordinate system as,

e_{z} = e_{ρ_{0}} e_{y} = e_{θ_{0}} e_{x} = e_{ϕ_{0}}

the linearized equations of motion can be expressed as,

Earth:Local Cartesian Equations

\overset{z}{¨} \overset{x}{¨} \overset{y}{¨} = - g + ω^{2} ρ_{0} sin^{2} ϕ_{0} + ω^{2} sin^{2} ϕ_{0} z + ω^{2} sin (2 ϕ_{0}) x + 2 ω sin ϕ_{0} \overset{y}{˙}, = \frac{1}{2} ω^{2} ρ_{0} sin (2 ϕ_{0}) + \frac{1}{2} ω^{2} sin (2 ϕ_{0}) z + ω^{2} cos (2 ϕ_{0}) x + 2 ω cos ϕ_{0} \overset{y}{˙}, = - 2 ω sin ϕ_{0} \overset{z}{˙} - 2 ω cos ϕ_{0} \overset{x}{˙} .

Motion on the Rotating Earth

Applications of rotating-frame dynamics to motion near Earth's surface.

Prerequisites

Contents

Contents

I. Rotating-Frame Preliminaries

II. Centrifugal and Coriolis Forces on Earth

1. Forces in Spherical Coordinates

2. Lagrangian in Spherical Coordinates

3. Linearized Equations of Motion