Leonhard Euler and the mechanics of rigid bodies^*

In 2015 we commemorated the 250th anniversary of the publication of Theoria motus corporum solidorum seu rigidorum (Theory of the motion of solid or rigid bodies) (Euler 1765b)³ , published by Leonhard Euler in 1765.

When the reader hears this title, he/she tends to think that this text features all the very important contributions made by Euler to rigid body dynamics, but a careful study of this work reveals, as pointed out by Truesdell (1968b, p 603), that it is disperse and full of examples that deviate the reader's attention away from the theory corpus. In fact the relevant results for the treatment of rigid bodies are found in previous manuscripts. Euler somehow tried, in this book, to make a compendium of all his previous results, but he clearly could not accomplish this goal. In this work we present the lengthy path travelled by Euler to build his rigid body theory, from his Scientia Navalis (Naval science) (Euler 1749)⁴ probably finished in 1738, up to his Nova methodus motum corporum rigidorum determinandi (A new method for generating the motion of a rigid body) (Euler 1776)⁵ , written in 1775. We ought to state that we have tried to maintain, as much as is possible, the original notation that Euler used at each step.

It could be said that Euler began to develop the theory of rigid bodies due to his great interest in the motion of ships. Therefore in 1735 or 1736 he began to write his second treatise on mechanics⁶ , Scientia Navalis (Euler 1736). It is in this work that Euler proposed that the movement of a ship could be described by a translation plus a rotation about an axis that passes through the ship's centre of gravity. This was achieved by thinking in terms of a force opposite to the resultant of all the forces acting on the ship's centre of gravity, so this point would be at rest. It is possible, then, to think of the separation of the progressive movement (translational) from the rotational one about an axis passing through the gravity centre, because the force he introduced did not affect the rotation (i.e. it did not generate a torque). This way, two independent principles were required to describe the motion of the ship (and of a rigid body in general). The translation was described by Newton in his second law, so an equation to describe the rotation movement was now necessary.

One could intuitively think that the equation for rotation should be similar to the translational one, i.e. if the second law (of motion) is written as:

$$\begin{eqnarray*}&&F={ma}\end{eqnarray*}$$

then an equivalent to the mass and the angular acceleration should appear in the rotation equation. In the work called Dissertation sur la meilleure construction du cabestan (Dissertation on the best construction of a winch) (Euler 1745)—in the same period as Scientia Navalis—Euler gave the explicit relation between the equation describing the translational and the relative one to the rotation when he pointed out that: 'Thus [the moment] of the acting forces divided by the moment of the matter [the moment of inertia], gives the force of rotation [the angular acceleration], exactly in the same way as in the rectilinar motion the [...] force divided by the [...] mass of the body gives the acceleration. This remarkable analogy well deserves to be underlined' (Maltese 2000, p 327). This way and for the first time, Euler set, for the movement of rotation of a ship about a rotation axis passing through its gravity centre and perpendicular to the movement plane, that:

$$\begin{eqnarray}&&\left[\int {r}^{2}{\rm{d}}{m}\right]{\rm{d}}\omega ={M}{\rm{d}}{t}\end{eqnarray} \tag{ 1 }$$

M being the moment of all the forces producing rotation (i.e. the torque), $\left[\int {r}^{2}{\rm{d}}{m}\right]=I$ the inertia moment and ω the angular velocity, so that:

$$\begin{eqnarray}&&M={Fr}=I\displaystyle \frac{{\rm{d}}\omega }{{\rm{d}}{t}}\end{eqnarray} \tag{ 2 }$$

where F is the magnitude of the resultant of the forces producing the torque, and r is the lever arm.

The problem concerning ships was important for Euler, so he spent much time researching it. However, there was another more fundamental problem at that time: the precession of the equinoxes. Newton in his Principia (Newton 1999, pp 751–7)⁷ was the first one to give an explanation of this phenomenon. Afterwards, and parallel to Euler, D'Alembert approached the problem, improving the Newtonian approach, but it was Euler who approached it in a general way, giving birth to rigid body mechanics.

In 1750 Euler presented, in the Academy of Berlin, a manuscript entitled Decouverte d'un noveau principe de mecanique (Discovery of a new principle in mechanics) (Euler 1752)⁸ , in which he poses the first version of the equations that feature his name, the 'Euler equations'. For their deduction, he begins with a 'determination of the movement in general of which a solid body is capable, while its centre of gravity remains at rest' (Wilson 1987, p 259). To achieve this, Euler introduces the idea of an instantaneous rotation axis. In fact, the concept had already be proposed by D'Alembert in his Recherches sur la precession des equinoxes (Researches on the precession of equinoxes) (D'Alambert 1749)⁹ for the case of Earth, but it is Euler who demonstrates the existence of such an axis in a general way. In Euler's words: 'Assuming, then, the gravity centre of whichever solid body at rest ... I shall demonstrate in what follows that independently of the movement of such body, it shall occur that not only the gravity centre shall remain at rest, but that there would also exist an infinity of points situated along a straight line passing through the gravity centre which would likewise find themselves without movement. This is, that independently of the movement of the body, it shall exist in each instant, a rotation movement about an axis passing thought the gravity centre' (Euler 1752, p 188).

For the demonstration of the existence of the instantaneous rotation axis (Euler 1752, pp 197–204), Euler introduced a cartesian axes system fixed in absolute space, in which the origin is the gravity centre of a rigid body. Afterwards he considered a point Z of the body that moves with a velocity (P, Q, R) with coordinates (x, y, z), and a point Z₀¹⁰ with coordinates $(x+{\rm{d}}{x},y+{\rm{d}}{y},z+{\rm{d}}{z})$, that moves with a velocity $(P+{\rm{d}}P,Q+{\rm{d}}Q,R+{\rm{d}}R)$ ¹¹ separated by a distance d, given by

$$\begin{eqnarray}&&{d}^{2}(Z,{Z}_{0})={{\rm{d}}{x}}^{2}+{{\rm{d}}{y}}^{2}+{{\rm{d}}{z}}^{2}.\end{eqnarray} \tag{ 3 }$$

After a time interval dt, the coordinates of the two points shall be

$$\begin{eqnarray*}&&Z=(x+{P}{\rm{d}}{t},y+{Q}{\rm{d}}{t},z+{R}{\rm{d}}{t})\end{eqnarray*}$$

and

$$\begin{eqnarray*}&&{Z}_{0}=(x+{\rm{d}}x+(P+{\rm{d}}P){\rm{d}}t,y+{\rm{d}}y+(Q+{\rm{d}}Q){\rm{d}}t,z+{\rm{d}}z+(R+{\rm{d}}R){\rm{d}}t).\end{eqnarray*}$$

Due to the fact that it is a rigid body, the distance between the two points ought to be the same, so after the time interval dt elapses, we have:

$$\begin{eqnarray}\begin{array}{rcl}{d}^{2}(Z,{Z}_{0}) & = & {[x+P{\rm{d}}t-(x+{\rm{d}}x+(P+{\rm{d}}P){\rm{d}}t)]}^{2}\\ & & +\,[y+Q{\rm{d}}t-(y+{\rm{d}}y+(Q+{\rm{d}}Q)){\rm{d}}t{)}^{2}]\\ & & +\,[z+R{\rm{d}}t-(z+{\rm{d}}z+(R+{\rm{d}}R)){\rm{d}}t{)}^{2}]\\ & = & [{(-{\rm{d}}x-\mathrm{dPd}t)}^{2}+{(-{\rm{d}}y-{\rm{d}}Q{\rm{d}}t)}^{2}+{(-{\rm{d}}z-{\rm{d}}R{\rm{d}}t)}^{2}]\\ & = & {{\rm{d}}x}^{2}+{{\rm{d}}y}^{2}+{{\rm{d}}z}^{2}+2({\rm{d}}x{\rm{d}}P+{\rm{d}}y{\rm{d}}Q+{\rm{d}}z{\rm{d}}R){\rm{d}}t\\ & & +\,({{\rm{d}}P}^{2}+{{\rm{d}}Q}^{2}+{{\rm{d}}R}^{2}){{\rm{d}}t}^{2}\mathrm{.}\end{array}\end{eqnarray} \tag{ 4 }$$

Equating (3) and (4), and neglecting the terms $({{\rm{d}}{P}}^{2}+{{\rm{d}}{Q}}^{2}+{{\rm{d}}{R}}^{2}){{\rm{d}}{t}}^{2}$, we have that:

$$\begin{eqnarray*}&&({\rm{d}}{x}{\rm{d}}{P}+{\rm{d}}{y}{\rm{d}}{Q}+{\rm{d}}{z}{\rm{d}}{R}){\rm{d}}{t}=0\end{eqnarray*}$$

so

$$\begin{eqnarray}&&{\rm{d}}{x}{\rm{d}}{P}+{\rm{d}}{y}{\rm{d}}{Q}+{\rm{d}}{z}{\rm{d}}{R}=0.\end{eqnarray} \tag{ 5 }$$

Afterwards, Euler assumed the case in which dx = dy = 0, which implies dR = 0, so we have that R does not depend on z. Repeating the same argument, Euler finds that P does not depend on x, and that Q does not depend on y, so that:

$$\begin{eqnarray}P & = & {Ay}+{Bz}\\ Q & = & {Cz}-{Ax}\\ R & = & -\,{Bx}-{Cy}\end{eqnarray} \tag{ 6 }$$

with A, B, C being constants.

Inasmuch as the instantaneous rotation axis ought to be instantaneously at rest, Euler determined, by a variable change, that the points for which the speed (P, Q, R) is equal to zero in the dt interval are: x = Cu, y = −Bu, z = Au, where u is the new variable. These points determine a direct line passing throught the origin, the instantaneous rotation axis. Afterwards, Euler identified the angular velocity of the body as¹² :

$$\begin{eqnarray}&&{[{A}^{2}+{B}^{2}+{C}^{2}]}^{1/2}.\end{eqnarray} \tag{ 7 }$$

Once he had demonstrated the existence of the instantaneous rotation axis, Euler was interested in calculating the acceleration of a mass element dM, so as to later be able to apply the second law of Newton. With this goal (Euler 1752, pp 205–10) he assumed a system of three fixed axes in absolute space, mutually perpendicular. From equation (6), the displacement of a point with coordinates x, y, z during the interval dt, could be expressed as:

$$\begin{eqnarray}{\rm{d}}{x} & = & (\lambda y-\mu z){\rm{d}}{t}\\ {\rm{d}}{y} & = & (\nu z-\lambda x){\rm{d}}{t}\\ {\rm{d}}{z} & = & (\mu x-\nu y){\rm{d}}{t}\end{eqnarray} \tag{ 8 }$$

where λ = A, μ = −B and ν = C, so now the angular velocity is

$$\begin{eqnarray}&&{[{\nu }^{2}+{\mu }^{2}+{\lambda }^{2}]}^{1/2}\end{eqnarray} \tag{ 9 }$$

and differentiating equation (8) it is obtained that:

$$\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{\rm{d}}x & = & (y{\rm{d}}\lambda -z{\rm{d}}\mu ){\rm{d}}t+(\lambda \nu z+\mu \nu y-(\lambda \lambda +\mu \mu )x){{\rm{d}}t}^{2}\\ {\rm{d}}{\rm{d}}y & = & (z{\rm{d}}\nu -x{\rm{d}}\lambda ){\rm{d}}t+(\mu \nu x+\lambda \mu z-(\nu \nu +\lambda \lambda )y){{\rm{d}}t}^{2}\\ {\rm{d}}{\rm{d}}z & = & (x{\rm{d}}\mu -y{\rm{d}}\nu ){\rm{d}}t+(\lambda \mu y+\lambda \nu x-(\mu \mu +\nu \nu )z){{\rm{d}}t}^{2}\end{array}.\end{eqnarray} \tag{ 10 }$$

At this point, Euler was able to apply the second law for a mass element dM. In this way, the force in the direction of the three axes is (Euler 1716)¹³ :

$$\begin{eqnarray}&&\displaystyle \frac{2{\rm{d}}{M}{\rm{d}}{\rm{d}}{x}}{{{\rm{d}}{t}}^{2}},\displaystyle \frac{2{\rm{d}}{M}{\rm{d}}{\rm{d}}{y}}{{{\rm{d}}{t}}^{2}},\displaystyle \frac{2{\rm{d}}{M}{\rm{d}}{\rm{d}}{z}}{{{\rm{d}}{t}}^{2}}\end{eqnarray} \tag{ 11 }$$

i.e.:

$$\begin{eqnarray}&&2\displaystyle \frac{{\rm{d}}{M}}{{\rm{d}}{t}}({y}{\rm{d}}\lambda -{z}{\rm{d}}\mu )+2{\rm{d}}{M}(\lambda \nu z+\mu \nu y-(\lambda \lambda +\mu \mu )x)\\ &&2\displaystyle \frac{{\rm{d}}{M}}{{\rm{d}}{t}}({z}{\rm{d}}\nu -{x}{\rm{d}}\lambda )+2{\rm{d}}{M}(\mu \nu x+\lambda \mu z-(\nu \nu +\lambda \lambda )y)\\ &&2\displaystyle \frac{{\rm{d}}{M}}{{\rm{d}}{t}}({x}{\rm{d}}\mu -{y}{\rm{d}}\nu ){\rm{d}}{t}+2{\rm{d}}{M}(\lambda \mu y+\lambda \nu x-(\mu \mu +\nu \nu )z).\end{eqnarray} \tag{ 12 }$$

From this point, Euler stated that '... in order to know exactly the status of these forces, it is necessary to take into account only their moments with respect to our three axes ...' (Euler 1752, p 208). Each one of the three components of the force that acts on the mass dM, shall have a moment (the torque components, in modern language), about two of the three axes.

Thereby, Euler determined the moments on the three axes (Euler 1752, pp 208–9)¹⁴ in such a way that the moment on z is:

$$\begin{eqnarray}&&2\displaystyle \frac{{\rm{d}}{M}}{{\rm{d}}{t}}({y}{y}{\rm{d}}\lambda +{x}{x}{\rm{d}}\lambda -{y}{z}{\rm{d}}\mu -{x}{z}{\rm{d}}\nu )\\ &&\quad +2{\rm{d}}{M}(\lambda \nu {yz}-\lambda \mu {xz}+\mu \nu {yy}-\mu \nu {xx}-(\mu \mu -\nu \nu ){xy})\end{eqnarray} \tag{ 13 }$$

the y component is

$$\begin{eqnarray}&&2\displaystyle \frac{{\rm{d}}{M}}{{\rm{d}}{t}}({x}{x}{\rm{d}}\mu +{z}{z}{\rm{d}}\mu -{x}{y}{\rm{d}}\nu -{y}{z}{\rm{d}}\lambda )\\ &&\quad +2{\rm{d}}{M}(\lambda \mu {xy}-\mu \nu {yz}+\lambda \nu {xx}-\lambda \nu {zz}-(\nu \nu -\lambda \lambda ){xz})\end{eqnarray} \tag{ 14 }$$

and lastly, on x

$$\begin{eqnarray}&&\begin{array}{l}2\displaystyle \frac{{\rm{d}}M}{{\rm{d}}t}(zz{\rm{d}}\nu +yy{\rm{d}}\nu -xz{\rm{d}}\lambda -xy{\rm{d}}\mu )\\ \quad +2{\rm{d}}M(\mu \nu \mathrm{xz}-\lambda \nu \mathrm{xy}+\lambda \mu \mathrm{zz}-\lambda \mu \mathrm{yy}-(\lambda \lambda -\mu \mu )\mathrm{yz}).\end{array}\end{eqnarray} \tag{ 15 }$$

Integrating these equations on a mass element, and defining:

$$\begin{eqnarray*}\displaystyle \int {\rm{d}}{M}({xx}+{yy}) & = & {Mff}\qquad \qquad \displaystyle \int {x}{y}{\rm{d}}{M}={Mll}\\ \displaystyle \int {\rm{d}}{M}({xx}+{zz}) & = & {Mgg}\qquad \qquad \displaystyle \int {x}{z}{\rm{d}}{M}={Mmm}\\ \displaystyle \int {\rm{d}}{M}({yy}+{zz}) & = & {Mhh}\qquad \qquad \displaystyle \int {y}{z}{\rm{d}}{M}={Mnn}\end{eqnarray*}$$

where Euler emphasised that Mff, Mgg and Mhh are the inertia moments of the body along the three fixed axes, it was obtained that (Euler 1752, pp 209–10):

$$\begin{eqnarray}&&2M\left({ff}\displaystyle \frac{{\rm{d}}\lambda }{{\rm{d}}{t}}-{nn}\displaystyle \frac{{\rm{d}}\mu }{{\rm{d}}{t}}-{mm}\displaystyle \frac{{\rm{d}}\nu }{{\rm{d}}{t}}+\lambda \nu {nn}-\lambda \mu {mm}-(\mu \mu -\nu \nu ){ll}+\mu \nu ({hh}-{gg})\right)\\ &&2M\left({gg}\displaystyle \frac{{\rm{d}}\mu }{{\rm{d}}{t}}-{ll}\displaystyle \frac{{\rm{d}}\nu }{{\rm{d}}{t}}-{nn}\displaystyle \frac{{\rm{d}}\lambda }{{\rm{d}}{t}}+\lambda \mu {ll}-\mu \nu {nn}-(\nu \nu -\lambda \lambda ){mm}+\lambda \nu ({ff}-{hh})\right)\\ &&2M\left({hh}\displaystyle \frac{{\rm{d}}\nu }{{\rm{d}}{t}}-{mm}\displaystyle \frac{{\rm{d}}\lambda }{{\rm{d}}{t}}-{ll}\displaystyle \frac{{\rm{d}}\mu }{{\rm{d}}{t}}+\mu \nu {mm}-\lambda \nu {ll}-(\lambda \lambda -\mu \mu ){nn}+\lambda \mu ({gg}-{ff})\right).\end{eqnarray} \tag{ 16 }$$

It could be said that these expressions are the first version of the Euler equations, which would be expressed in a modern fashion as:

$$\begin{eqnarray}\begin{array}{rcl}{N}_{{z}} & = & {I}_{{zz}}{\dot{\omega }}_{{z}}-{I}_{{yz}}{\dot{\omega }}_{{y}}-{I}_{{xz}}{\dot{\omega }}_{{x}}+{I}_{{yz}}{\omega }_{{z}}{\omega }_{{x}}-{I}_{{xz}}{\omega }_{{z}}{\omega }_{{y}}-{I}_{{xy}}({\omega }_{{y}}^{2}-{\omega }_{{x}}^{2})+({I}_{{xx}}-{I}_{{yy}}){\omega }_{{y}}{\omega }_{{x}}\\ {N}_{{y}} & = & {I}_{{yy}}{\dot{\omega }}_{{y}}-{I}_{{xy}}{\dot{\omega }}_{{x}}-{I}_{{yz}}{\dot{\omega }}_{{z}}+{I}_{{xy}}{\omega }_{{y}}{\omega }_{{z}}-{I}_{{yz}}{\omega }_{{x}}{\omega }_{{y}}-{I}_{{xz}}({\omega }_{{x}}^{2}-{\omega }_{{z}}^{2})+({I}_{{zz}}-{I}_{{xx}}){\omega }_{{z}}{\omega }_{{x}}\\ {N}_{{x}} & = & {I}_{{xx}}{\dot{\omega }}_{{x}}-{I}_{{xz}}{\dot{\omega }}_{{z}}-{I}_{{xy}}{\dot{\omega }}_{{y}}+{I}_{{xz}}{\omega }_{{y}}{\omega }_{{x}}-{I}_{{xy}}{\omega }_{{z}}{\omega }_{{x}}-{I}_{{yz}}({\omega }_{{z}}^{2}-{\omega }_{{y}}^{2})+({I}_{{yy}}-{I}_{{zz}}){\omega }_{{z}}{\omega }_{{y}}\end{array}\end{eqnarray} \tag{ 17 }$$

where N = (N_x, N_y, N_z) is the torque, ${\boldsymbol{\omega }}=({\omega }_{x},{\omega }_{y},{\omega }_{z})$ the angular velocity, I_xx, I_yy, I_zz the inertia moments and I_xy, I_yz, I_zx the inertia products. The components are represented by a two order tensor, expressed by a 3 × 3 symmetric matrix, where the diagonal elements are the inertia moments and the remainder are the inertia products. The I tensor receives the inertia tensor name.

These equations were a big step in the understanding of a rigid body, but it is clear that their integration is a complex problem and aditionally, as it is pointed out by Euler, '... from these formulae it shall be known for each instant, the change in the position of the rotation axis and of the angular velocity. It is then necessary to change each instant the position of the three axes ... This coerces to calculate for each instant the values ll, mm, nn, ff, gg, hh, insomuch as the change in the position of the body with respect to the three axes shall cause continuous variations' (Euler 1752, p 214).

In 1751, Euler in Du mouvement de rotation des corps solides autour d'un axe variable (On the movement of rotation of solid bodies around a variable axis) (Euler 1765a)¹⁵ , considering the problem that represents the calculation with time of the inertia tensor components, changed the scope, introducing axes fixed to the body, choosing them as the principal of inertia, i.e. the axes in which the inertia tensor is diagonal (Wilson 1987, pp 264–70)¹⁶ .

Euler chose a system of axes fixed in absolute space IA, IB, IC and determined the position of the rotation axis IO through the angles AIO = α, AIB = β and AIC = γ. Afterwards, he considered a point Z of the rigid body, with coordinates (x, y, z) with respect to axes IA, IB, IC, and he chooses three mutually perpendicular axes Za, Zb and Zc, parallel to the axes in absolute space, in such a way that they passed through Z and were the principal of inertia. If a body is rotating along the IO axis with an angular velocity ω, the components on Za, Zb and Zc could be expressed by means of the direction cosines

$$\begin{eqnarray*}&&{\boldsymbol{\omega }}=(\omega \cos \alpha ,\omega \cos \beta ,\omega \cos \gamma )\end{eqnarray*}$$

with ${\cos }^{2}\alpha +{\cos }^{2}\beta +{\cos }^{2}\gamma =1$.

The velocity (u, v, w) of the point Z, with respect to the principal axes, is:

$$\begin{eqnarray}\begin{array}{rcl}u & = & \displaystyle \frac{{\rm{d}}x}{{\rm{d}}t}=\omega (z\cos \beta -y\cos \gamma )\\ v & = & \displaystyle \frac{{\rm{d}}y}{{\rm{d}}t}=\omega (x\cos \gamma -z\cos \alpha )\\ w & = & \displaystyle \frac{{\rm{d}}z}{{\rm{d}}t}=\omega (y\cos \alpha -x\cos \beta )\end{array}.\end{eqnarray} \tag{ 18 }$$

Differentiating the component u with respect to time, it is obtained that:

$$\begin{eqnarray*}&&\displaystyle \frac{{\rm{d}}u}{{\rm{d}}t}=z\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}(\omega \cos \beta )-y\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}(\omega \cos \gamma )+\omega \cos \beta \displaystyle \frac{{\rm{d}}z}{{\rm{d}}t}-\omega \cos \gamma \displaystyle \frac{{\rm{d}}y}{{\rm{d}}t}.\end{eqnarray*}$$

Substituting dy = vdt and dz = wdt, and using the fact that the sum of the square of the direction cosines is 1, arrives at:

$$\begin{eqnarray}&&{\rm{d}}{u}={z}{\rm{d}}(\omega \cos \beta )-{y}{\rm{d}}(\omega \cos \gamma )+{\omega }^{2}{\rm{d}}{t}(y\cos \alpha \cos \beta +z\cos \alpha \cos \gamma -x{\sin }^{2}\alpha )\end{eqnarray} \tag{ 19 }$$

and, analogously, differentiating the other two components and substituting dx = udt, dy = vdt and dz = wdt, obtains, respectively:

$$\begin{eqnarray}&&{\rm{d}}{v}={x}{\rm{d}}(\omega \cos \gamma )-{z}{\rm{d}}(\omega \cos \alpha )+{\omega }^{2}{\rm{d}}{t}(x\cos \alpha \cos \beta +z\cos \beta \cos \gamma -y{\sin }^{2}\beta )\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{\rm{d}}w=y{\rm{d}}(\omega \cos \alpha )-x{\rm{d}}(\omega \cos \beta )+{\omega }^{2}{\rm{d}}t(x\cos \alpha \cos \beta +y\cos \beta \cos \gamma -z{\sin }^{2}\gamma ).\end{eqnarray} \tag{ 21 }$$

At this point, Euler was able to apply the second law of Newton, so the components of the force on a mass element dM (let us remember that, accordingly to Euler, M is the weight) is: $\tfrac{{\rm{d}}{M}}{2g}\tfrac{{\rm{d}}{u}}{{\rm{d}}{t}},\tfrac{{\rm{d}}{M}}{2g}\tfrac{{\rm{d}}{v}}{{\rm{d}}{t}},\tfrac{{\rm{d}}{M}}{2g}\tfrac{{\rm{d}}{w}}{{\rm{d}}{t}}$ ¹⁷ .

Substituting equations (19), (20) and (21), we have:

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{M}}{2g}\displaystyle \frac{{\rm{d}}{u}}{{\rm{d}}{t}} & = & \displaystyle \frac{{\rm{d}}{M}}{2g}\left[z\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}(\omega \cos \beta )-y\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}(\omega \cos \gamma )\right.\\ & & +\,\left.{\omega }^{2}(y\cos \alpha \cos \beta +z\cos \alpha \cos \gamma -x{\sin }^{2}\alpha )\phantom{\rule[-0000]{}{3ex}}\right]\\ \displaystyle \frac{{\rm{d}}{M}}{2g}\displaystyle \frac{{\rm{d}}{v}}{{\rm{d}}{t}} & = & \displaystyle \frac{{\rm{d}}{M}}{2g}\left[x\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}(\omega \cos \gamma )-z\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}(\omega \cos \alpha )\right.\\ & & +\left.\,{\omega }^{2}(x\cos \alpha \cos \beta +z\cos \beta \cos \gamma -y{\sin }^{2}\beta )\phantom{\rule[-0000]{}{3ex}}\right]\\ \displaystyle \frac{{\rm{d}}{M}}{2g}\displaystyle \frac{{\rm{d}}{w}}{{\rm{d}}{t}} & = & \displaystyle \frac{{\rm{d}}{M}}{2g}\left[y\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}(\omega \cos \alpha )-x\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}(\omega \cos \beta )\right.\\ & & +\,{\omega }^{2}(x\cos \alpha \cos \beta +y\cos \beta \cos \gamma -z{\sin }^{2}\gamma )\left.\phantom{\rule[-0000]{}{3ex}}\right].\end{eqnarray} \tag{ 22 }$$

From here, it is possible to calculate the moments of the forces in each axis, obtaining, for the component along the Za axis

$$\begin{eqnarray*}&&\begin{array}{l}\displaystyle \frac{{\rm{d}}{M}}{2g}\left[{y}^{2}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}(\omega \cos \alpha )-{xy}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}(\omega \cos \beta )+{\omega }^{2}({xy}\cos \alpha \cos \beta +{y}^{2}\cos \beta \cos \gamma -{yz}{\sin }^{2}\gamma )\right]\\ \quad -\,\displaystyle \frac{{\rm{d}}{M}}{2g}\left[{xz}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}(\omega \cos \gamma )-{z}^{2}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}(\omega \cos \alpha )+\,{\omega }^{2}({xz}\cos \alpha \cos \beta +{z}^{2}\cos \beta \cos \gamma -{yz}{\sin }^{2}\beta )\right]\end{array}\end{eqnarray*}$$

from where, integrating on the mass element dM, arrives at

$$\begin{eqnarray*}&&\displaystyle \frac{1}{2g}\left[\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}(\omega \cos \alpha )\displaystyle \int ({y}^{2}+{z}^{2}){\rm{d}}{M}-\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}(\omega \cos \beta )\displaystyle \int ({xy}-{xz}){\rm{d}}{M}\right]\\ &&\quad +\displaystyle \frac{1}{2g}\left[{\omega }^{2}\cos \alpha \cos \beta \displaystyle \int ({xy}+{xz}){\rm{d}}{M}+{\omega }^{2}\cos \beta \cos \gamma \displaystyle \int ({y}^{2}-{z}^{2}){\rm{d}}{M}\right]\end{eqnarray*}$$

which can be rewritten as:

$$\begin{eqnarray*}&&\begin{array}{l}\displaystyle \frac{1}{2g}\left[\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}(\omega \cos \alpha )\displaystyle \int ({y}^{2}+{z}^{2}){\rm{d}}M-\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}(\omega \cos \beta )\displaystyle \int (\mathrm{xy}-\mathrm{xz}){\rm{d}}M\right]\\ \quad +\,\displaystyle \frac{1}{2g}\left[{\omega }^{2}\cos \alpha \cos \beta \displaystyle \int (\mathrm{xy}+\mathrm{xz}){\rm{d}}M\right.\\ \quad +\,\left.{\omega }^{2}\cos \beta \cos \gamma \left[\displaystyle \int ({x}^{2}+{y}^{2}){\rm{d}}M-\displaystyle \int ({x}^{2}+{z}^{2}){\rm{d}}M\right]\right]\mathrm{.}\end{array}\end{eqnarray*}$$

Now, inasmuch as the axes are the principal of inertia, the inertia products are zero, i.e.

$$\begin{eqnarray*}&&\int {x}{y}{\rm{d}}{M}=\int {x}{z}{\rm{d}}{M}=\int {y}{z}{\rm{d}}{M}=0\end{eqnarray*}$$

and defining the moments of inertia as:

$$\begin{eqnarray*}\displaystyle \int {\rm{d}}{M}({x}^{2}+{y}^{2}) & = & {Maa}\\ \displaystyle \int {\rm{d}}{M}({x}^{2}+{z}^{2}) & = & {Mbb}\\ \displaystyle \int {\rm{d}}{M}({y}^{2}+{z}^{2}) & = & {Mcc}\end{eqnarray*}$$

doing the same calculation for the other two components, and defining P, Q and R as the moments of the forces along the principal axes, Euler arrived at:

$$\begin{eqnarray}P & = & {Maa}\displaystyle \frac{{\rm{d}}\omega \cos \alpha }{2{g}{\rm{d}}{t}}+M({cc}-{bb})\displaystyle \frac{{\omega }^{2}\cos \beta \cos \gamma }{2g}\\ Q & = & {Mbb}\displaystyle \frac{{\rm{d}}\omega \cos \beta }{2{g}{\rm{d}}{t}}+M({aa}-{cc})\displaystyle \frac{{\omega }^{2}\cos \alpha \cos \gamma }{2g}\\ R & = & {Mcc}\displaystyle \frac{{\rm{d}}\omega \cos \gamma }{2{g}{\rm{d}}{t}}+M({bb}-{aa})\displaystyle \frac{{\omega }^{2}\cos \alpha \cos \beta }{2g}\end{eqnarray} \tag{ 23 }$$

which are the known 'Euler equations' for a rigid body, referred to as principal inertia axes, and with the angular velocity components in terms of the angles α, β, γ, which are the angles subtended by the rotation axes with the principal ones fixed in the body. It could be said that these are the Euler angles, although actually they are usually defined by applying the rotation operator to the axes fixed on the body, so that each angle is related to the angular velocities of rotation known as precession, nutation and spin.

In modern language, these equations  are presented as:

$$\begin{eqnarray}{N}_{1} & = & {{I}_{1}}_{1}{\dot{\omega }}_{1}+({{I}_{3}}_{3}-{I}_{22}){\omega }_{2}{\omega }_{3}\\ {N}_{2} & = & {{I}_{2}}_{2}{\dot{\omega }}_{2}+({{I}_{1}}_{1}-{{I}_{3}}_{3}){\omega }_{1}{\omega }_{3}\\ {N}_{3} & = & {{I}_{3}}_{3}{\dot{\omega }}_{3}+({I}_{22}-{{I}_{1}}_{1}){\omega }_{1}{\omega }_{2}\end{eqnarray} \tag{ 24 }$$

where 1, 2, 3 are the principal axes of inertia fixed to the body; the components of angular velocity in this system are ${\boldsymbol{\omega }}=({\omega }_{1},{\omega }_{2},{\omega }_{3})$, the torque is ${\bf{N}}=({N}_{1},{N}_{2},{N}_{3})$ and the diagonal elements of the inertia tensor are I₁₁, I₂₂ and I₃₃.

Once Euler established the moment of moment idea, of an instantaneous rotation axis, Euler equations and principal axes of inertia, it could be said that his contribution to the mechanics of a  rigid body was complete, and it would seem that he also thought it himself, because he devoted himself to preparing a work that, judging by his title, represents a compendium of his work in the mechanics of rigid bodies, Theoria motus corporum rigidorum seu solidorum, appeared in 1765. Nevertheless, this voluminous work did not result in an adequate compendium, insomuch as Euler disaggregated it himself in excessive detail without clarifying what he posed before, nor stating anything else¹⁸ .

A real contribution to the mechanics of rigid bodies is found in his Nova methodus motu corporum rigidorum determinandi (A new method for generating the motion of a rigid body) in which, for the first time, Euler posed, explicitly, one beside the other, the fundamental laws of mechanics (Euler 1776, p 224):

$$\begin{eqnarray*}{\rm{I}}.\ \ \ \ \displaystyle \int {\rm{d}}{M}\left(\displaystyle \frac{{\rm{\text{dd}}}{x}}{{{\rm{d}}{t}}^{2}}\right) & = & {iP}\qquad \qquad \mathrm{IV}.\ \displaystyle \int {z}{\rm{d}}{M}\left(\displaystyle \frac{{\rm{d}}{\rm{d}}{y}}{{{\rm{d}}{t}}^{2}}\right)-\displaystyle \int {y}{\rm{d}}{M}\left(\displaystyle \frac{{\rm{d}}{\rm{d}}{z}}{{{\rm{d}}{t}}^{2}}\right)={iS}\\ \mathrm{II}.\ \ \displaystyle \int {\rm{d}}{M}\left(\displaystyle \frac{{\rm{d}}{\rm{d}}{y}}{{{\rm{d}}{t}}^{2}}\right) & = & {iQ}\qquad \qquad {\rm{V}}.\ \ \displaystyle \int {x}{\rm{d}}{M}\left(\displaystyle \frac{{\rm{d}}{\rm{d}}{z}}{{{\rm{d}}{t}}^{2}}\right)-\displaystyle \int {z}{\rm{d}}{M}\left(\displaystyle \frac{{\rm{\text{dd}}}{x}}{{{\rm{d}}{t}}^{2}}\right)={iT}\\ \mathrm{III}.\ \displaystyle \int {\rm{d}}{M}\left(\displaystyle \frac{{\rm{\text{dd}}}{x}}{{{\rm{d}}{t}}^{2}}\right) & = & {iR}\qquad \qquad \mathrm{VI}.\ \displaystyle \int {y}{\rm{d}}{M}\left(\displaystyle \frac{{\rm{\text{dd}}}{x}}{{{\rm{d}}{t}}^{2}}\right)-\displaystyle \int {z}{\rm{d}}{M}\left(\displaystyle \frac{{\rm{d}}{\rm{d}}{y}}{{{\rm{d}}{t}}^{2}}\right)={iU}\end{eqnarray*}$$

where dM is the infinitesimal mass element of the body; x, y, z, the position of the body in cartesian coordinates; P, Q, R the resultant of the external forces in each of the axes directions; i is a constant chosen depending on the used units; and S, T, U are the moments of the forces in the three directions x, y, z, respectively. In these equations, Euler did not provide the vector nature of S, T, U, i.e. did not consider them as the components of a unique physical entity (Borrelli 2011) and, although it could be alleged that in these equations the rotation equation of movement is already implicit, i.e. ${\bf{N}}=\tfrac{{\rm{d}}{\bf{L}}}{{\rm{d}}{t}}$ (with N the torque and L the angular momentum), Euler did not consider the time evolution of these equations.

The great contribution of Euler in his Nova methodus was to set these equations as general laws, independently of the specific problem, i.e. that they are valid for all bodies and all kinds of movement.

As Maltese thought, in this manuscript Euler is able to separate the description of the inertial properties from the application of the mechanics principles, inaugurating '... what we now call the Newtonian tradition in mechanics in its modern form' (Maltese 2000, p 340).

Mechanics in general, and rigid body mechanics in particular, was one of the topics that was of most relevant interest to Euler during his life. This interest allowed him to build, in a non-linear way—and in many cases in an apparently disordered one—the fundamental concepts of the mechanics of rigid bodies, which can be tracked in different works over 35 years, trying to solve the problems relative to the writing, presentation and publication dates.

Leaving this creative labyrinth, finally we find absolute clarity about the Eulerian thought that illuminated the 18th century culture, with a comparable brightness.

JJHG acknowledges partial support projects 20160105 and 20160576, as well as an EDI grant, all provided by SIP/IPN.