Gaspard-Gustave Coriolis

How to use this course

Every student of meteorology and oceanography learns the Coriolis force, and most assume that Gaspard-Gustave Coriolis studied winds or the rotation of the Earth. He did not. He was an engineer who wanted to know how much useful effect a machine delivers. That question led him to fix the modern meanings of work and kinetic energy, and then to ask how the energy balance changes when parts of a machine rotate. The force that bears his name came out of waterwheels and turbines, not weather.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The lessons follow his work: the theory of work (1829), machines and friction, energy in rotating frames (1831–1832), the compound centrifugal force (1835), its later life on the rotating Earth, billiards (1835), and turbines.

Timeline

Paris and Nancy, 1792–1808

Gaspard-Gustave de Coriolis was born in Paris on 21 May 1792, the eldest of six children. His father, Jean-Baptiste-Elzéar de Coriolis, came from a noble family of Provence. He had served as an officer in the Rochambeau expedition during the American War of Independence and in 1792 was in the King’s guard. His mother, Marie-Sophie de Maillet, came from a noble family of Lorraine.

The year of his birth was the year the monarchy fell. The King’s guard was dissolved, and the family, exposed by its rank, left Paris for Nancy in Lorraine. There his father turned to industry, running a tobacco manufactory and later a wallpaper business. Coriolis grew up and went to school in Nancy, where his talent for mathematics showed early.

Polytechnique and the bridges corps, 1808–1816

In 1808 Coriolis passed the entrance examination of the École polytechnique, near the top of his year. The school, founded in 1794, gave a common mathematical and scientific training to the future engineers of the state. Its best graduates chose the corps of Ponts et Chaussées, which built France’s roads, bridges and canals. Coriolis left the school in 1810 and entered this corps.

His years as a working engineer were spent in the provinces, and they ended in the Vosges, where he served near his family in Lorraine. His health was already fragile: a serious fever in 1812 left him weak, and illness recurred throughout his life. He missed the theoretical work of the school. The chance to return came through misfortune: his father died, the family needed support, and in 1816 a teaching post opened in Paris.

Répétiteur at the École polytechnique, 1816–1829

In 1816 Augustin-Louis Cauchy recommended Coriolis for a post as répétiteur in analysis at the École polytechnique. A répétiteur held small-group sessions that complemented the professor’s lectures, so Coriolis spent the next decade teaching Cauchy’s rigorous analysis to future engineers. He used the time to experiment on friction and hydraulics, and to think about what mechanics should offer to people who built machines.

The tradition he joined had roots in Lazare Carnot’s essay on machines of 1783, which analysed machines in terms of the products of forces and displacements. In the 1820s three engineers trained at Polytechnique took up the project of a mechanics for industry: Claude-Louis Navier, Jean-Victor Poncelet and Coriolis. Poncelet showed Coriolis his work on the subject in 1824. In 1829 both published books in which the product of force and displacement was the central quantity. Coriolis gave it its name, travail, and Poncelet acknowledged his priority in the word.

Work and vis viva (1829)

For the general reader. How much has a horse, a waterfall or a steam engine accomplished? Coriolis answered with a single number: the force exerted multiplied by the distance moved. He called it work. He then adjusted the old notion of “living force” by a factor of one half, so that the work done on a body equals exactly the change in its energy of motion. Physics has used both choices ever since.

Definitions

Gottfried Wilhelm Leibniz had called \(mv^2\) the vis viva of a body. With this convention the energy balance of a falling body reads \(mv^2=2mgh\), and a factor of two appears in every balance. In Du calcul de l’effet des machines (1829), Coriolis proposed to call \(\tfrac12mv^2\) the vis viva. With his convention, the work of the forces equals the change in vis viva with no factor at all. He also proposed a unit, the dynamode, equal to 1000 kilogram-force metres. The word work survived; the dynamode did not.

Proof. \(\mathbf F\cdot d\mathbf r=m\dot{\mathbf v}\cdot\mathbf v\,dt=\frac{d}{dt}\big(\tfrac12m\,\mathbf v\cdot\mathbf v\big)dt\). Integrate, and sum over particles.

Transmission of work

A machine receives work from a motor (water, animals, steam) and delivers it to a resistance (a load to lift, a millstone to turn). Over one cycle of steady operation, the vis viva returns to its starting value, so the theorem gives an exact budget.

This is the precise content of the old engineering maxim that what is gained in force is lost in speed. It tells the engineer where to look for improvement: at friction and shocks, never at mechanisms promising more work out than in.

Machines, friction and the chain of equal strength

For the general reader. Engineers of Coriolis’s time lived with friction. Every axle, sliding block and rope loses work to it. Charles-Augustin de Coulomb had found that dry friction is roughly proportional to the load. Using that rule, the work budget of Lesson 04 becomes a tool for designing inclines, screws and hoists. A related design question, which Coriolis studied in 1836, asks how to shape a hanging cable so that it is equally stressed everywhere.

Coulomb friction

For a load of weight \(P\) pulled up a slope of angle \(\alpha\) at constant speed, the useful work over a distance \(s\) is \(Ps\sin\alpha\) and the friction loss is \(\mu Ps\cos\alpha\). The efficiency is

The catenary of equal strength

A uniform cable hangs in an ordinary catenary, and its tension is largest at the supports. If the cross-section \(A\) is allowed to vary, it can be chosen so that the stress \(\sigma=T/A\) is the same everywhere. Coriolis treated this problem in 1836.

Derivation. Let \(\theta\) be the angle of the cable with the horizontal. The horizontal tension \(H\) is constant, so \(T=H\sec\theta\) and \(A=T/\sigma=(H/\sigma)\sec\theta\). Vertical balance of an element of length \(ds\) gives \(d(H\tan\theta)=\rho gA\,ds\). With \(ds=\sec\theta\,dx\), this becomes \(H\sec^2\theta\,d\theta=(\rho gH/\sigma)\sec^2\theta\,dx\), so \(d\theta/dx=1/c\) and \(\theta=x/c\). Then \(dy/dx=\tan(x/c)\), which integrates to the stated curve. As \(x\to\pm\pi c/2\) the slope and section become infinite.

Energy in rotating machines (1831–1832)

For the general reader. In a waterwheel or a turbine, water moves along vanes that are themselves turning. An engineer would like to write the energy balance as seen by someone riding on the wheel. Coriolis showed in 1831 that this is possible: the usual rule that work equals change of kinetic energy still holds in the rotating frame, provided one adds the familiar centrifugal force to the real forces. Nothing else is needed for the energy balance.

Let a frame rotate with constant angular velocity \(\boldsymbol\omega\) about a fixed axis. Write \(\mathbf r\) for the position relative to a point on the axis, and \(\mathbf v_r\) for the velocity measured in the rotating frame.

Proof. The relative equation of motion (Lesson 07) is \(m\dot{\mathbf v}_r=\mathbf F-m\,\boldsymbol\omega\times(\boldsymbol\omega\times\mathbf r)-2m\,\boldsymbol\omega\times\mathbf v_r\). Dot with \(\mathbf v_r\). The last term vanishes because \((\boldsymbol\omega\times\mathbf v_r)\cdot\mathbf v_r=0\). For the centrifugal term, \(\frac{d}{dt}\tfrac12|\boldsymbol\omega\times\mathbf r|^2=(\boldsymbol\omega\times\mathbf r)\cdot(\boldsymbol\omega\times\mathbf v_r)=-\big(\boldsymbol\omega\times(\boldsymbol\omega\times\mathbf r)\big)\cdot\mathbf v_r\), so its work is the change in \(\tfrac12m|\boldsymbol\omega\times\mathbf r|^2\). Coriolis’s 1832 memoir proved this before he had isolated the extra term; the 1835 memoir explained why it drops out.

Bernoulli in a rotating channel

For steady flow of an ideal incompressible fluid relative to the rotating frame, the same argument applied to a fluid particle gives, along a relative streamline,

where \(s\) is the distance to the axis. This is the starting point of the theory of turbines and centrifugal pumps.

The compound centrifugal force (1835)

For the general reader. On a merry-go-round, a ball rolled straight outward seems to swerve sideways. Newton’s laws fail in a rotating frame unless one adds fictitious forces. Coriolis found exactly what must be added. One part is the familiar centrifugal force. The other, which he called the compound centrifugal force, pushes sideways on anything that moves relative to the rotating frame. It is the force now named after him.

Kinematics

Let a frame with orthonormal vectors \(\mathbf e_i(t)\) rotate with angular velocity \(\boldsymbol\omega(t)\), so that \(\dot{\mathbf e}_i=\boldsymbol\omega\times\mathbf e_i\). For any vector \(\mathbf A=\sum A_i\mathbf e_i\),

Apply this twice to the position \(\mathbf r\) of a particle, measured from a point on the axis.

Proof. \(\mathbf v_{\text{in}}=\mathbf v_r+\boldsymbol\omega\times\mathbf r\). Differentiate in the inertial frame and use the rule again: \(\mathbf a_{\text{in}}=(\dot{\mathbf v}_r)_{\text{rot}}+\boldsymbol\omega\times\mathbf v_r+\dot{\boldsymbol\omega}\times\mathbf r+\boldsymbol\omega\times(\mathbf v_r+\boldsymbol\omega\times\mathbf r)\). Collect the two \(\boldsymbol\omega\times\mathbf v_r\) terms.

Coriolis did not separate the terms quite as textbooks now do. He regarded the sum of the ordinary and compound centrifugal forces as one effect of rotation and was interested in the compound part because it disappeared from the energy balance. The form of the acceleration had been found in special cases before him, by Leonhard Euler in 1749 and in Pierre-Simon Laplace’s tidal equations of 1778. Coriolis gave the general theorem for systems of bodies.

The rotating Earth

For the general reader. Coriolis never applied his theorem to the Earth. Others did, mostly without knowing his work. Léon Foucault’s pendulum, the large-scale winds that flow along lines of equal pressure, and the circular drift of ocean currents all follow from the same term. This lesson collects these consequences.

Use local axes at latitude \(\varphi\): \(x\) east, \(y\) north, \(z\) up. The Earth’s angular velocity is \(\boldsymbol\Omega=\Omega(0,\cos\varphi,\sin\varphi)\) with \(\Omega=7.292\times10^{-5}\) s⁻¹. Absorb the centrifugal force into effective gravity \(\mathbf g\). For horizontal motion, the horizontal Coriolis acceleration is \(-f\,\hat{\mathbf z}\times\mathbf u\) with the Coriolis parameter

Sketch for (iii). For small swings, write the horizontal position as a complex number \(\zeta=x+iy\). The equations become \(\ddot\zeta+2i\Omega\sin\varphi\,\dot\zeta+\omega_0^2\zeta=0\), with \(\omega_0^2=g/\ell\). Substituting \(\zeta=e^{-i\Omega\sin\varphi\,t}\eta\) and neglecting \(\Omega^2\) gives \(\ddot\eta+\omega_0^2\eta=0\): an ordinary pendulum in a frame turning at rate \(-\Omega\sin\varphi\).

Foucault showed his pendulum in Paris in 1851. William Ferrel used the deflecting effect of rotation to explain mid-latitude winds in 1856, and Christophorus Buys Ballot stated his empirical law relating wind and pressure in 1857. None of them drew on Coriolis’s paper.

The mathematics of billiards (1835)

For the general reader. Billiards was a passion in Coriolis’s Paris, and he wrote the first full mathematical theory of the game. A ball struck off centre both slides and spins. Coriolis showed that while it slides, friction pushes it in a fixed direction, so it travels along a parabola. Once it starts rolling it goes straight. This explains the curved paths that skilled players produce with spin.

Model a ball of mass \(m\), radius \(R\) and moment of inertia \(I=\tfrac25mR^2\) on a horizontal table with sliding friction coefficient \(\mu\). Let \(\mathbf v\) be the velocity of the centre and \(\boldsymbol\omega\) the angular velocity. The contact point, at \(-R\hat{\mathbf z}\) from the centre, has velocity \(\mathbf u=\mathbf v+\boldsymbol\omega\times(-R\hat{\mathbf z})\). The ball rolls when \(\mathbf u=0\).

Proof. \(m\dot{\mathbf v}=-\mu mg\,\hat{\mathbf u}\) and \(I\dot{\boldsymbol\omega}=(-R\hat{\mathbf z})\times(-\mu mg\,\hat{\mathbf u})=\mu mgR\,\hat{\mathbf z}\times\hat{\mathbf u}\). Then \(\dot{\mathbf u}=\dot{\mathbf v}-R\,\dot{\boldsymbol\omega}\times\hat{\mathbf z}=-\mu g\hat{\mathbf u}-\tfrac52\mu g(\hat{\mathbf z}\times\hat{\mathbf u})\times\hat{\mathbf z}=-\tfrac72\mu g\hat{\mathbf u}\), since \((\hat{\mathbf z}\times\hat{\mathbf u})\times\hat{\mathbf z}=\hat{\mathbf u}\) for horizontal \(\hat{\mathbf u}\). So \(\hat{\mathbf u}\) never changes and \(|\mathbf u|\) decreases linearly to zero at \(t_s\). The acceleration of the centre is constant, which gives a parabola, and \(\mathbf v_f=\mathbf v_0-\mu g\,t_s\hat{\mathbf u}_0=\mathbf v_0-\tfrac27\mathbf u_0\).

Waterwheels and turbines

For the general reader. Coriolis’s theory of relative motion was made for machines like waterwheels and the new reaction turbines of the 1820s and 1830s. Water enters a turning wheel, slides along its vanes and leaves. How much of its energy does the wheel capture? The answer follows from the momentum and energy balances of Lessons 04 and 06, and it explains the design rules engineers still follow: enter without shock, leave without speed.

Angular momentum balance

Consider steady flow through a wheel turning at constant \(\omega\) about a fixed axis. Water enters at radius \(r_1\) and leaves at radius \(r_2\), with mass flow rate \(\dot m\). Let \(c_{\theta1},c_{\theta2}\) be the tangential components of the absolute velocity at entry and exit.

Proof. In steady flow, the rate at which angular momentum about the axis leaves the wheel minus the rate at which it enters equals the torque exerted on the water by the wheel. That torque is \(\dot m(r_2c_{\theta2}-r_1c_{\theta1})\), and the wheel receives the opposite.

Agreement with the relative energy theorem

Write the absolute velocity as \(\mathbf c=\mathbf w+\mathbf U\), where \(\mathbf w\) is the velocity relative to the vanes. From \(w^2=c^2-2Uc_\theta+U^2\), we get \(Uc_\theta=\tfrac12(c^2+U^2-w^2)\), so

The first term is the drop in absolute kinetic energy. The other two are what Coriolis’s theorem of Lesson 06 predicts along a relative streamline: the centrifugal potential and the relative kinetic energy exchange with pressure, and the Coriolis force, doing no relative work, does not appear. It is nevertheless the Coriolis force, balanced by the vanes, that transmits the torque.

Navier’s successor, 1829–1843

The year 1829 changed Coriolis’s position. His book on the effect of machines appeared, and he became professor of mechanics at the new École centrale des arts et manufactures, a private school founded that year to train engineers for industry. When Cauchy left France after the revolution of July 1830, refusing the oath to the new king, Coriolis was offered his post at the École polytechnique. He declined, wanting time for research.

In the early 1830s he taught applied mechanics alongside Claude-Louis Navier at the École des ponts et chaussées. Navier died in August 1836, and Coriolis succeeded him both in the chair and in the mechanics section of the Académie des sciences. In 1838 he succeeded the physicist Pierre Louis Dulong as director of studies at the École polytechnique, responsible for its whole curriculum. He gave up teaching applied mechanics to save his health, and contemporaries judged him an excellent director.

His health had never been good, and in the spring of 1843 it collapsed. He died in Paris on 19 September 1843, aged 51, and was buried in the Montparnasse cemetery. His Traité de la mécanique des corps solides, a revised edition of the 1829 book, appeared in 1844. His sister Cécile married the physicist Eugène Péclet.

Legacy: how a name reached the weather

During his life Coriolis was known for the theory of machines and the definition of work. His 1835 memoir on relative motion was read by students of mechanics but not by meteorologists. George Hadley had explained the trade winds by the Earth’s rotation in 1735, Laplace had written the rotating tidal equations in 1778, and Ferrel and Buys Ballot developed the theory of winds in the 1850s without him. His name began to appear in meteorology near the end of the nineteenth century, and the phrase “Coriolis force” became standard only in the early twentieth century, by around 1920. Today it is one of the most familiar names in geophysics.

Legacy

• Energy: the word work in its scientific sense, the factor \(\tfrac12\) in kinetic energy, and the work–energy budget of machines.
• Relative motion: the energy theorem in rotating frames (1831–1832) and the compound centrifugal force (1835), the basis of turbine theory and of every rotating-frame calculation in geophysics, astrophysics and engineering.
• Billiards: the first full mechanical theory of the game, still cited by physicists of sport.
• Education: a generation of engineers trained at the École centrale, Ponts et Chaussées and Polytechnique.
• Honours: one of the 72 names inscribed on the Eiffel Tower; a lunar crater named in 1970; the asteroid (16564) Coriolis; a French oceanographic research vessel named in 1963.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Gaspard Gustave de Coriolis, MacTutor History of Mathematics.
2. Gaspard-Gustave de Coriolis, Wikipedia; Gaspard-Gustave Coriolis, Wikipédia (family and early career).
3. P. Costabel, Coriolis, Gaspard Gustave de, Dictionary of Scientific Biography, vol. 3, Scribner, 1971.
4. A. Persson, The Coriolis effect: four centuries of conflict between common sense and mathematics, History of Meteorology 2 (2005).

Science

1. G. Coriolis, Du calcul de l’effet des machines, Carilian-Goeury, Paris, 1829.
2. G. Coriolis, Mémoire sur le principe des forces vives dans les mouvements relatifs des machines, Journal de l’École polytechnique 13 (1832).
3. G. Coriolis, Mémoire sur les équations du mouvement relatif des systèmes de corps, Journal de l’École polytechnique 15 (1835), 142–154.
4. G. Coriolis, Théorie mathématique des effets du jeu de billard, Carilian-Goeury, Paris, 1835.
5. A. Persson, How do we understand the Coriolis force?, Bulletin of the American Meteorological Society 79 (1998).
6. Coriolis force, Wikipedia (history and standard formulas).
7. D. G. Alciatore, The Illustrated Principles of Pool and Billiards, Sterling, 2004 (modern treatment of the 30° rule).
8. G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, 2nd ed., Cambridge University Press, 2017 (geostrophy, inertial oscillations).