# Tailored high-contrast attosecond electron pulses for coherent excitation
and scattering

Sergey V. Yalunin<EMAIL_ADDRESS>Armin Feist  Claus Ropers<EMAIL_ADDRESS>1IV. Physical Institute, University of Göttingen, 37077 Göttingen, Germany
2Max Planck Institute for Biophysical Chemistry, 37077 Göttingen, Germany

###### Abstract

Temporally shaping the density of electron beams using light forms the basis
for a wide range of established and emerging technologies, including free-
electron lasers and attosecond electron microscopy. The modulation depth of
compressed electron pulses is a key figure of merit limiting applications. In
this work, we present an approach for generating background-free attosecond
electron pulse trains by sequential inelastic electron-light scattering.
Harnessing quantum interference in the fractional Talbot effect, we suppress
unwanted background density in electron compression by several orders of
magnitude. Our results will greatly enhance applications of coherent electron-
light scattering, such as stimulated cathodoluminescence and streaking.

Actively shaped free-electron beams offer the capability for enhanced sensing
and microscopy [1, 2, 3, 4, 5, 6, 7], with numerous applications employing
transverse shaping, incorporating tailored phase masks [8, 9, 10, 11, 12, 13],
quasi-static electromagnetic optical elements [14, 15, 16, 17] or scattering
at the ponderomotive potential [18, 19, 20]. In the time domain, inelastic
electron-light scattering (IELS) [21, 22, 23, 24, 25] leads to correlated gain
or loss of angular [26, 27] and linear momenta with energy [28, 29, 30].
Following dispersive propagation, inelastic interactions can lead to a
reshaping of the electron density into a train of attosecond pulses [31, 32,
33, 24]. These states have been experimentally prepared and characterized by
quantum state tomography [34] or streaking [35, 36, 37, 38].

Besides their direct use for attosecond streaking and spectroscopy, such
density-modulated electron beams are proposed for a multitude of applications,
e.g., to imprint an external phase onto cathodoluminescence (CL) [39, 40, 41,
42], to induce a microscopic polarization in two-level systems (TLS), or to
coherently build up mode amplitudes or local polarizations using independent
electrons [43, 44, 45, 46, 47, 48], thus promising a merger of electron
microscopy with coherent spectroscopy.

A key limitation for these efforts is the quality of compression and the
amount of uncompressed background density [34, 49, 36, 37, 46], sometimes
described by the classical multi-electron bunching factor for SASE-FELs [50,
51, 52, 53]. However, even for a pure single-electron state [54, 55, 56]
focused in the quantum regime [34], limited coherence arises [40, 42, 43, 44,
47].

In this Letter, we directly address this issue by devising an experimentally
feasible scheme to prepare essentially background-free attosecond electron
pulse trains, drastically enhancing the coherence in electron-light
interactions. Introducing a single-color, multi-plane phase shaping approach,
we predict tailored electron states representing excellent approximations to
point-like classical currents within the optical cycle. These states are
capable of producing CL with a near-unity degree of coherence, a maximized
microscopic polarization of few-level systems, and the enhanced coherent
build-up of mode and transition amplitudes using multiple independent
electrons.

Figure 1: Temporal electron pulse shaping and its prospective applications in
attosecond physics, including streaking and the interaction with
nanostructures and quantum systems.

The wavefunction shaping scheme is depicted in Fig. 1. It is based on
sequential temporal phase plates using IELS to modulate the electron momentum,
each followed by a drift stage to transfer this momentum modulation into a
train of attosecond density pulses [34, 36, 35]. As shown below, tailored
precompression removes most of the electron density background and greatly
enhances the coherence properties of a variety of excitations and scattering
processes in the sample plane [Fig. 1, bottom].

The Letter is organized as follows. First, we briefly present a quantum
mechanical approach to temporal phase plates and its classical limit. Second,
the generation of nearly background-free attosecond electron pulses using
sequential interactions is described. Finally, we discuss the consequences and
features of using such electron states in selected applications, including
streaking, CL and the excitation of two-level systems.

Multislice method for electron propagation.—There are various approaches to
theoretically describe inelastic electron-light scattering and propagation
[21, 22, 23, 24]. Here, we employ a multislice approach [57], in which forward
propagation of electrons along the $z$ axis is described by the effective
Schrödinger equation derived in the Supplemental Material 111See Supplemental
Material for derivation of Eq. (1), and simulated space-time evolution of the
probability density after inelastic scattering:

$i\hslash v\frac{\partial}{\partial
z}\Psi(\mathbf{r},t)=\left[H_{I}(\mathbf{r},t)+D-{\mathcal{E}}\right]\Psi(\mathbf{r},t),$
(1)

where $v$ is the mean electron velocity, $\mathcal{E}=i\hslash\partial_{t}$
the translation operator, and $H_{I}(\mathbf{r},t)\approx-
evA_{z}(\mathbf{r},t)$ the time-dependent scattering potential describing the
interaction with light [58]. The dispersion operator $D$ is given by

$D=\frac{{\mathcal{E}}^{2}}{2\gamma^{3}mv^{2}}-\frac{\hslash^{2}}{2\gamma
m}\nabla^{2}_{\perp},$ (2)

where $\gamma=1/\sqrt{1-v^{2}/c^{2}}$ is the Lorentz factor, $m$ the
electron’s rest mass, and $\nabla_{\perp}^{2}$ the Laplacian operator of the
transverse coordinates. The Fresnel operator $U(h)=\exp(-iDh/\hslash v)$
describes the propagation of the electron state from $z$ to $z+h$ without
interaction. In the presence of an electromagnetic field, the solution of Eq.
(1) can be obtained with the split-operator technique [59], using the Fresnel
operator, the propagator for $D=0$ [23, 30], and one of the higher-order
decomposition schemes [60, 61]. Since we are mainly interested in temporal or
longitudinal focusing, for simplicity, we assume that the interaction is
independent of $x,y$, separating the wavefunction into its temporal
$\psi(z,t-z/v)$ and transverse parts, where the latter is known analytically
for cylindrical beams [10, 62]. The advancement of the temporal part from $z$
to $z+h$ is obtained by the second-order expression:

$\psi(z+h,t)=U(h/2)e^{i\Phi(z,t)}U(h/2)\psi(z,t),$ (3)

where the phase function is given by

$\Phi(z,t)=-\frac{1}{\hslash v}\int_{z}^{z+h}H_{I}(z,t+z/v)dz.$ (4)

This procedure is efficiently implemented with a fast Fourier transform
algorithm. It is unitary and hence preserves the probability current
$j=v|\psi(z,t)|^{2}$ integrated over time [63]. It follows from Eq. (3) that a
scattering potential confined to an interval $(z,z+h)$ can be regarded as a
thin inelastic or temporal phase plate located at $z+h/2$, by analogy with
elastic phase plates [20, 64]. Thus, the forward propagation and temporal
aberrations can be reduced to a phase function $\Phi(t)$ describing the
temporal phase plate.

Classical limit.—We first outline the classical picture of temporal
aberrations (${\hslash\to 0}$), where $\hslash\Phi(t)$ has the meaning of a
classical action. Its time derivative defines the change of the electron’s
energy and velocity:

$\Delta v(t)=-\frac{\hslash\Phi^{\prime}(t)}{\gamma^{3}mv}.$ (5)

Suppose that electrons uniformly distributed in time traverse a temporal phase
plate at $z=0$ and gain a velocity change (5). The change of the probability
density with increasing $z$ is determined by the trajectories:

$t(z)=t-\Delta v(t)z/v^{2}.$ (6)

The attracting fixed points in this map, i.e., the zeros of $\Delta v(t_{0})$
with the time derivative $\Delta v^{\prime}(t_{0})>0$, correspond to paraxial
temporal foci. Paths concentrate and form a caustic near such points,
resulting in narrow peaks in the density [49]. For a time-harmonic phase
$\Phi(t)=2g\cos(\omega t)$ characterized by an effective interaction strength
$g$ (see an expression in [58]) and a frequency $\omega$, the attracting
points are given by $t_{0}=nT$, ${n\in\mathbb{Z}}$ with $T=2\pi/\omega$ being
the optical period. The paraxial focus lies at the distance $l_{f}$ from the
phase plate, with

$l_{f}=\frac{v^{2}}{\Delta v^{\prime}(t_{0})}=\frac{l_{T}}{8\pi g},\quad
l_{T}=\frac{4\pi m\gamma^{3}v^{3}}{\hslash\omega^{2}},$ (7)

where $l_{T}$ is the Talbot distance [65], which amounts to 200 mm for 120-keV
electrons and 800-nm light.

Figure 2: (a),(b) Evolution of the magnitude of $\psi(z,t)$ within one optical
period after single and triple interactions with 800-nm light. Calculation for
120-keV electrons and interaction strengths ${g_{1}=0.39}$, ${g_{2}=0.52}$,
${g=4}$. Solid and dotted black: Trajectories contributing in the central peak
and background. (c) Optimized rms duration for single, dual and triple
monochromatic interactions as a function of the main compression strength $g$
(and free variation of the precompression strengths). Dashed: Respective
durations when the main compression is a parabolic phase modulation. (d) Semi-
logarithmic plot of the density in the rms focus, illustrating a great
reduction in the background density using sequential focusing.

Temporal focusing by monochromatic light is imperfect since paraxial
trajectories do not converge into one point in the space-time diagram in Fig.
2(a), affecting the electron pulse duration. Another type of temporal
aberration stems from repelling fixed points, at which half of the electrons
are steered to a nearly homogeneous background in density [Fig. 2(a)]. The
latter does not improve with increasing $g$ because it depends only on the
product $gz$ (see Eq. (6)), and also phase-squeezed light does not reduce the
background [65]. In principle, both types of temporal aberrations could be
eliminated using multiple harmonics to approach a parabolic phase modulation
of the form $\Phi(t)=-g\omega^{2}s^{2}(t)$, where ${s(t)=(t+T/2\ \textrm{mod}\
T)-T/2}$ is the periodic saw-like function with zeros $nT$, $n\in\mathbb{Z}$.
Generalized electron beam shaping using multiple harmonics has recently been
theoretically considered [46], and two-color phase modulation experiments have
been performed in the context of attosecond focusing and quantum state
reconstruction [34]. However, superimposing an even larger number of harmonics
with controlled amplitude and phase may render this approach rather
impractical for attosecond focusing. Instead, as we show in the following,
sequential monochromatic interactions in separate planes represent an even
more powerful and experimentally tractable scheme to address temporal
aberrations.

Eliminating focusing aberrations by sequential scattering.—As described above,
the aberrations in temporal focusing are most severe for those parts of the
probability density furthest from the center of the cycle. We show here that
weak precompression stages of interaction strength less than unity can reshape
the density for much more efficient focusing [cf. Fig. 2(b)]. Specifically,
the fractional Talbot effect [66] offers a powerful way to suppress the
density near the repelling points, without adding further phase aberrations.
At one fourth of the Talbot distance, the wavefunction around the attractive
points is a superposition of the phase-modulated initial state
$\psi(0,t)\approx\psi_{0}(t_{0})\exp[2ig_{1}\cos(\omega t)]$ and its replica
shifted by half of the optical period [67, 68]:

$\psi(l_{T}/4,t)=\frac{\exp(-i\pi/4)}{\sqrt{2}}\left[\psi(0,t)+i\psi(0,t+T/2)\right].$
(8)

Importantly, due to destructive interference, the probability density

$\rho(l_{T}/4,t)\approx|\psi_{0}(t_{0})|^{2}\\{1+\sin\left[4g_{1}\cos(\omega
t)\right]\\}$ (9)

vanishes at the repelling points $t=t_{0}\pm T/2$ when $g_{1}=\pi/8$. While
such a modulation itself does not provide much temporal compression, a second
stronger temporal phase plate completes the focusing, but now with greatly
reduced background density.

Figure 3: (a),(b) Evolution of $\langle b^{n}\rangle$ (solid lines) after dual
(${g_{1}=0.44}$) and triple (${g_{1}=0.39}$, ${g_{2}=0.52}$) interactions with
the compression strength ${g=4}$. Dashed: The evolution after the first
interaction only. (c) Values of $\langle b^{n}\rangle$ obtained at the
distance where $\langle b\rangle$ reaches a maximum. (d),(e) Wigner
distributions in the focus, showing a localization of electrons in phase
space.

Depending on the specific application, several properties can be considered
for assessing the quality of attosecond focusing. A good measure of background
density is given by the root-mean-squared (rms) duration $\Delta t(z)$,

$\Delta t(z)=\sqrt{\langle s^{2}\rangle},\quad\langle
s^{2}\rangle=\frac{1}{C}\int_{-\infty}^{\infty}s^{2}(t)\rho(z,t)dt,$ (10)

where $\rho(z,t)=|\psi(z,t)|^{2}$ is the probability density, $C$ its
integral, and $s(t)$ the saw-like function. Figure 2(c) illustrates the
significant improvement of attosecond focusing in terms of the rms duration
upon adding one or two prefocusing stages. As a function of the main
compression strength $g$, $\Delta t(z)$ is evaluated at the respective minima
along $z$, for a single interaction, a parabolic phase profile, and the
optimized dual and triple (two weak and one strong) interactions. It is
evident that the single-interaction case is rather limited in terms of its
achievable rms duration, being minimized near $g\approx 0.45$ at $\Delta
t\approx 0.426$ fs. By an effective suppression of background density [see
Fig. 2(d)], the dual interaction substantially reduces $\Delta t$. Although
the most substantial absolute improvement is already obtained for the case of
a single precompression phase plate, adding additional interaction planes
leads to a further optimization, as evident from the results for a total of
three phase plates shown in Figs. 2(b)-2(d) (see also Ref. [58]). It is worth
noting that the idealized parabolic phase modulation applied after the
precompression stages leads to only minor additional improvements [see Fig.
2(c)], illustrating that the introduced prefocusing scheme indeed strongly
reduces aberrations and provides rms durations very close to the theoretical
minimum.

Applications of background-free pulses.—We note that the background-free
attosecond pulses produced by this scheme will be exceedingly useful in a
number of recently proposed schemes for coherent interactions of free
electrons with nanostructures and individual quantum systems [69, 70]. This
includes electron-mediated coherence transfer and the coherent build-up of
excitations for multiple subsequent electrons [40, 39, 42, 43, 44, 45, 46, 47,
48]. In each of the underlying processes, the modulation amplitude of the
electron density at the fundamental modulation frequency, or its harmonics,
plays a major role. This is quantified by the expectation values or moments
$\langle b^{n}\rangle$ of the ladder operator $b=\exp(i\omega t)$ [71, 44].
Different terminologies relating to these expectation values have been used,
including the degree of coherence [40], the coherence factor [65], or the
bunching factor [72, 52, 39, 53].

Figure 4: Field-driven streaking spectrograms simulated with the electron
pulses shown in Fig. 2(d). White lines: Temporal profile of the streak field
$-\hslash\Phi^{\prime}(\tau)$, where $\Phi(\tau)$ is given by Eq. (4). Yellow:
Calculated mean electron energy, $E(\tau)$.

The moments arising from a single interaction, $\langle
b^{n}\rangle=J_{n}[4g\sin(2\pi nz/l_{T})]$ (see also Ref. [44]), are limited
by the maxima of the Bessel functions $J_{n}$. Our sequential interaction
scheme results in a substantial and nearly simultaneous enhancement of
$\langle b^{n}\rangle$ above these values, as shown in Figures 3(a)-3(c).
Here, we numerically optimized the first moment $\langle b\rangle$ by a
variation of the precompression strengths $g_{1,2}$ and the respective
distances $d_{1,2}$ between interaction planes, for a fixed value of $g=4$.
For a dual interaction, the optimization yields a maximum value of $\langle
b\rangle_{max}\approx 0.97$ for $g_{1}\approx 0.44$ and $d_{1}\approx
0.24l_{T}$, well above the theoretical maximum $\langle
b\rangle_{max}=\max[J_{1}(z)]\approx 0.58$ for a single interaction, discussed
in Ref. [42, 44]. Note that the optimum distance $d_{1}$ is indeed very close
to the quarter Talbot distance. An additional precompression phase plate
$g_{2}$ leads to a further enhancement [Fig. 3(b)], with values of $\langle
b\rangle_{max}\approx 0.99$ and 0.998 for $g=4$ and $20$, respectively. The
enhancements of the higher moments are equally striking, reaching a nearly
Gaussian distribution spanning multiple orders for a triple interaction [Fig.
3(c)].

The background-free attosecond focusing has direct implications for electron-
driven radiative emission in the form of CL [73, 74, 75, 76, 77, 78, 79, 80].
Specifically, our results imply a degree-of-coherence of CL near unity [39,
40, 41, 42]. Similarly, due to the pronounced localization in time [see Fig.
2(d)] and the large moments of $b$, the background-free attosecond electron
pulses lead to an almost fully coherent excitation of TLSs with a transition
energy $\Delta E=\hslash\omega$ [43, 44, 45, 46, 47, 48]. The quantum state in
this case can be readily described using the Bloch model of the density matrix
[81]. For a TLS initially in the ground state, the excitation can be seen as a
“rotation” of the Bloch vector $\mathbf{a}=-\hat{\bf z}$ around a unit vector
$\mathbf{n}$ in the $x$-$y$ plane,
$\mathbf{a}(\theta)=\mathbf{a}\cos\theta+\langle
b\rangle(\mathbf{n}\times\mathbf{a})\sin\theta$, where $\theta$ is the
rotation angle. This angle can be determined through the transition
probability $P_{2}=\sin^{2}(\theta/2)$ and Eq. (11) in Ref. [43]. Unmodulated
electrons, or electrons with small $\langle b\rangle$, lead to a loss of the
partial information about the relative phase between the eigenstates of the
TLS, because of the large uncertainty in the transition time [82]. In
contrast, modulated electrons with $\langle b\rangle\approx 1$ retain this
information and lead to a purity of the final state of
$\textrm{Tr}(\rho^{2})=1+\frac{1}{2}(\langle b\rangle^{2}-1)\sin^{2}\theta$
near unity. Incidentally, the dipole moment of the final state is maximized.
Moreover, due to minimized quantum entanglement between the interacting
systems, subsequent excitations driven by independent electrons modulated by
the same reference wave can coherently build up either the cathodoluminescence
in a particular mode or the transition amplitude in a TLS [43, 44, 7], leading
to transition probabilities scaling with $N^{2}$ for small $\theta$, where $N$
is the number of electrons incident within the decoherence or dephasing time.

We note that the electron pulses produced here can be regarded as optimized
approximations to point-like, classical phase-space densities, as is clearly
seen by their Wigner function shown in Fig. 3(e). This makes them ideally
suited for temporal probing of periodic electromagnetic fields by means of
free-electron energy or angular streaking [25, 35, 36, 37, 38]. In energy
streaking, a time-periodic electric field
$F(\tau)=\sum_{n}c_{n}e^{in\omega\tau}$, or more precisely its spatial Fourier
components in Eq. (4), are mapped onto the energy domain via the expression:
$E(\tau)\sim\langle F(t+\tau)\rangle=\sum_{n}\langle b^{n}\rangle
c_{n}e^{in\omega\tau}$, where $E(\tau)$ is the displacement of the mean energy
of electrons, and $\tau$ the time delay between the electron pulse train and
the streak field. Of course, the rather limited moments $\langle b^{n}\rangle$
resulting from single-field focusing [see Fig. 3(c)] strongly affect the
performance of the streaking technique. Furthermore, as shown in Figs. 4(a)
and 4(c), the presence of a background in electron pulses produced with the
conventional scheme leads to pronounced spectral distortions. In contrast, the
background-free electron pulses produced with our sequential focusing scheme
lead to an almost perfect temporal representation of the streak field [see
Figs. 4(b) and 4(d)].

Possible experimental implementations of our focusing scheme can be realized
over a wide range of energies accessible in scanning and transmission electron
microscopes. The main design parameter to consider is the Talbot distance [Eq.
(7)], which scales with the cube of the electron velocity. Specifically, the
pre-compression distance $l_{T}/4$ at 800 nm optical wavelength reduces from
120 mm at 200 keV to 8.6 mm at 40 keV and 1 mm at 10 keV, dimensions for which
technical solutions should readily be found.

Conclusion.—In summary, we have theoretically demonstrated how the concept of
sequential scattering by light and the fractional Talbot effect can be applied
to generate high-contrast attosecond electron pulses. We achieve an efficient
correction of temporal aberrations, which leads to greatly enhanced coherence
properties in electron-light scattering as compared to conventional temporal
compression. Although the core of our generation scheme is a true quantum
effect, harnessing destructive interference of the wavefunction, the resulting
electron states exhibit an almost entirely positive Wigner function and
greatly improved localization in phase space. Such states closely represent
classical currents and produce practically fully coherent excitation and
radiation at the modulation frequency. Finally, we believe that similar
schemes may be applicable also for free-electron lasers operating in the
quantum regime [54, 55, 56].

###### Acknowledgements.

We thank K. E. Priebe, Th. Rittmann, H. Lourenço-Martin, O. Kfir, V. Di Giulio
and F. J. García de Abajo for insightful discussions. This work was funded by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -
432680300/SFB 1456 (project C01), 255652344/SPP 1840 (project ‘Kohärente
Wechselwirkungen starker optischer Nahfelder mit freien Elektronen’), the
Gottfried Wilhelm Leibniz program, and the European Union’s Horizon 2020
research and innovation programme under grant agreement No 101017720 (FET-
Proactive EBEAM).

## References

  * Grillo  _et al._ [2017a] V. Grillo, T. R. Harvey, F. Venturi, J. S. Pierce, R. Balboni, F. Bouchard, G. Carlo Gazzadi, S. Frabboni, A. H. Tavabi, Z.-A. Li, R. E. Dunin-Borkowski, R. W. Boyd, B. J. McMorran, and E. Karimi, Observation of nanoscale magnetic fields using twisted electron beams, Nature Communications 8, 689 (2017a).
  * Polman  _et al._ [2019] A. Polman, M. Kociak, and F. J. García de Abajo, Electron-beam spectroscopy for nanophotonics, Nature Materials 18, 1158 (2019).
  * Lourenço-Martins  _et al._ [2021] H. Lourenço-Martins, D. Gérard, and M. Kociak, Optical polarization analogue in free electron beams, Nature Physics 10.1038/s41567-021-01163-w (2021), arXiv:2006.06284 .
  * Midgley and Dunin-Borkowski [2009] P. A. Midgley and R. E. Dunin-Borkowski, Electron tomography and holography in materials science, Nature Materials 8, 271 (2009).
  * Kruit  _et al._ [2016] P. Kruit, R. Hobbs, C.-S. Kim, Y. Yang, V. Manfrinato, J. Hammer, S. Thomas, P. Weber, B. Klopfer, C. Kohstall, T. Juffmann, M. Kasevich, P. Hommelhoff, and K. Berggren, Designs for a quantum electron microscope, Ultramicroscopy 164, 31 (2016).
  * Bliokh  _et al._ [2017] K. Bliokh, I. Ivanov, G. Guzzinati, L. Clark, R. Van Boxem, A. Béché, R. Juchtmans, M. Alonso, P. Schattschneider, F. Nori, and J. Verbeeck, Theory and applications of free-electron vortex states, Physics Reports 690, 1 (2017), arXiv:1703.06879 .
  * García de Abajo and Konečná [2021] F. J. García de Abajo and A. Konečná, Optical Modulation of Electron Beams in Free Space, Physical Review Letters 126, 123901 (2021), arXiv:2011.03884 .
  * Uchida and Tonomura [2010] M. Uchida and A. Tonomura, Generation of electron beams carrying orbital angular momentum, Nature 464, 737 (2010).
  * Verbeeck  _et al._ [2010] J. Verbeeck, H. Tian, and P. Schattschneider, Production and application of electron vortex beams, Nature 467, 301 (2010).
  * McMorran  _et al._ [2011] B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, Electron Vortex Beams with High Quanta of Orbital Angular Momentum, Science 331, 192 (2011).
  * Voloch-Bloch  _et al._ [2013] N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Generation of electron Airy beams, Nature 494, 331 (2013).
  * Grillo  _et al._ [2014] V. Grillo, G. Carlo Gazzadi, E. Karimi, E. Mafakheri, R. W. Boyd, and S. Frabboni, Highly efficient electron vortex beams generated by nanofabricated phase holograms, Applied Physics Letters 104, 043109 (2014).
  * Grillo  _et al._ [2017b] V. Grillo, A. H. Tavabi, F. Venturi, H. Larocque, R. Balboni, G. C. Gazzadi, S. Frabboni, P.-H. Lu, E. Mafakheri, F. Bouchard, R. E. Dunin-Borkowski, R. W. Boyd, M. P. J. Lavery, M. J. Padgett, and E. Karimi, Measuring the orbital angular momentum spectrum of an electron beam, Nature Communications 8, 15536 (2017b), arXiv:1609.09129 .
  * McMorran  _et al._ [2017] B. J. McMorran, T. R. Harvey, and M. P. J. Lavery, Efficient sorting of free electron orbital angular momentum, New Journal of Physics 19, 023053 (2017), arXiv:1609.09124 .
  * Harvey  _et al._ [2017] T. R. Harvey, V. Grillo, and B. J. McMorran, Stern-Gerlach-like approach to electron orbital angular momentum measurement, Physical Review A 95, 021801 (2017), arXiv:1606.03631 .
  * Verbeeck  _et al._ [2018] J. Verbeeck, A. Béché, K. Müller-Caspary, G. Guzzinati, M. A. Luong, and M. Den Hertog, Demonstration of a 2 × 2 programmable phase plate for electrons, Ultramicroscopy 190, 58 (2018), arXiv:1711.11373 .
  * Pozzi  _et al._ [2020] G. Pozzi, V. Grillo, P.-H. Lu, A. H. Tavabi, E. Karimi, and R. E. Dunin-Borkowski, Design of electrostatic phase elements for sorting the orbital angular momentum of electrons, Ultramicroscopy 208, 112861 (2020).
  * Kapitza and Dirac [1933] P. L. Kapitza and P. A. M. Dirac, The reflection of electrons from standing light waves, Mathematical Proceedings of the Cambridge Philosophical Society 29, 297 (1933).
  * Freimund  _et al._ [2001] D. L. Freimund, K. Aflatooni, and H. Batelaan, Observation of the Kapitza–Dirac effect, Nature 413, 142 (2001).
  * Schwartz  _et al._ [2019] O. Schwartz, J. J. Axelrod, S. L. Campbell, C. Turnbaugh, R. M. Glaeser, and H. Müller, Laser phase plate for transmission electron microscopy, Nature Methods 16, 1016 (2019).
  * Barwick  _et al._ [2009] B. Barwick, D. J. Flannigan, and A. H. Zewail, Photon-induced near-field electron microscopy, Nature 462, 902 (2009).
  * García de Abajo  _et al._ [2010] F. J. García de Abajo, A. Asenjo-Garcia, and M. Kociak, Multiphoton Absorption and Emission by Interaction of Swift Electrons with Evanescent Light Fields, Nano Letters 10, 1859 (2010).
  * Park  _et al._ [2010] S. T. Park, M. Lin, and A. H. Zewail, Photon-induced near-field electron microscopy (PINEM): Theoretical and experimental, New Journal of Physics 12, 123028 (2010).
  * Feist  _et al._ [2015] A. Feist, K. E. Echternkamp, J. Schauss, S. V. Yalunin, S. Schäfer, and C. Ropers, Quantum coherent optical phase modulation in an ultrafast transmission electron microscope, Nature 521, 200 (2015).
  * Kirchner  _et al._ [2014] F. O. Kirchner, A. Gliserin, F. Krausz, and P. Baum, Laser streaking of free electrons at 25 keV, Nature Photonics 8, 52 (2014).
  * Cai  _et al._ [2018] W. Cai, O. Reinhardt, I. Kaminer, and F. J. G. de Abajo, Efficient orbital angular momentum transfer between plasmons and free electrons, Physical Review B 98, 045424 (2018).
  * Vanacore  _et al._ [2018] G. M. Vanacore, I. Madan, G. Berruto, K. Wang, E. Pomarico, R. J. Lamb, D. McGrouther, I. Kaminer, B. Barwick, F. J. García de Abajo, and F. Carbone, Attosecond coherent control of free-electron wave functions using semi-infinite light fields, Nature Communications 9, 2694 (2018), arXiv:1712.08441 .
  * García de Abajo  _et al._ [2016] F. J. García de Abajo, B. Barwick, and F. Carbone, Electron diffraction by plasmon waves, Physical Review B 94, 041404 (2016), arXiv:1603.07551 .
  * Vanacore  _et al._ [2019] G. M. Vanacore, G. Berruto, I. Madan, E. Pomarico, P. Biagioni, R. J. Lamb, D. McGrouther, O. Reinhardt, I. Kaminer, B. Barwick, H. Larocque, V. Grillo, E. Karimi, F. J. García de Abajo, and F. Carbone, Ultrafast generation and control of an electron vortex beam via chiral plasmonic near fields, Nature Materials 18, 573 (2019).
  * Feist  _et al._ [2020] A. Feist, S. V. Yalunin, S. Schäfer, and C. Ropers, High-purity free-electron momentum states prepared by three-dimensional optical phase modulation, Physical Review Research 2, 043227 (2020).
  * Sears  _et al._ [2008] C. M. S. Sears, E. Colby, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, Production and characterization of attosecond electron bunch trains, Physical Review Special Topics - Accelerators and Beams 11, 061301 (2008).
  * Hilbert  _et al._ [2009] S. A. Hilbert, C. Uiterwaal, B. Barwick, H. Batelaan, and A. H. Zewail, Temporal lenses for attosecond and femtosecond electron pulses, Proceedings of the National Academy of Sciences 106, 10558 (2009).
  * Baum and Zewail [2009] P. Baum and A. H. Zewail, 4D attosecond imaging with free electrons: Diffraction methods and potential applications, Chemical Physics 366, 2 (2009).
  * Priebe  _et al._ [2017] K. E. Priebe, C. Rathje, S. V. Yalunin, T. Hohage, A. Feist, S. Schäfer, and C. Ropers, Attosecond electron pulse trains and quantum state reconstruction in ultrafast transmission electron microscopy, Nature Photonics 11, 793 (2017).
  * Kozák  _et al._ [2018] M. Kozák, N. Schönenberger, and P. Hommelhoff, Ponderomotive Generation and Detection of Attosecond Free-Electron Pulse Trains, Physical Review Letters 120, 103203 (2018).
  * Morimoto and Baum [2018] Y. Morimoto and P. Baum, Diffraction and microscopy with attosecond electron pulse trains, Nature Physics 14, 252 (2018).
  * Schönenberger  _et al._ [2019] N. Schönenberger, A. Mittelbach, P. Yousefi, J. McNeur, U. Niedermayer, and P. Hommelhoff, Generation and Characterization of Attosecond Microbunched Electron Pulse Trains via Dielectric Laser Acceleration, Physical Review Letters 123, 264803 (2019).
  * Black  _et al._ [2019] D. S. Black, U. Niedermayer, Y. Miao, Z. Zhao, O. Solgaard, R. L. Byer, and K. J. Leedle, Net Acceleration and Direct Measurement of Attosecond Electron Pulses in a Silicon Dielectric Laser Accelerator, Physical Review Letters 123, 264802 (2019).
  * Pan and Gover [2019] Y. Pan and A. Gover, Spontaneous and stimulated emissions of a preformed quantum free-electron wave function, Physical Review A 99, 052107 (2019).
  * Kfir  _et al._ [2021] O. Kfir, V. Di Giulio, F. J. G. de Abajo, and C. Ropers, Optical coherence transfer mediated by free electrons, Science Advances 7, eabf6380 (2021).
  * Karnieli  _et al._ [2021] A. Karnieli, N. Rivera, A. Arie, and I. Kaminer, The coherence of light is fundamentally tied to the quantum coherence of the emitting particle, Science Advances 7, eabf8096 (2021), arXiv:2011.00623 .
  * Di Giulio  _et al._ [2021] V. Di Giulio, O. Kfir, C. Ropers, and F. J. García de Abajo, Modulation of Cathodoluminescence Emission by Interference with External Light, ACS Nano 15, 7290 (2021), arXiv:2101.07748 .
  * Gover and Yariv [2020] A. Gover and A. Yariv, Free-Electron–Bound-Electron Resonant Interaction, Physical Review Letters 124, 064801 (2020).
  * Zhao  _et al._ [2020] Z. Zhao, X.-Q. Sun, and S. Fan, Quantum entanglement and modulation enhancement of free-electron-bound-electron interaction, arXiv:2010.11396 [physics, physics:quant-ph] (2020), arXiv:2010.11396 [physics, physics:quant-ph] .
  * Rätzel  _et al._ [2020] D. Rätzel, D. Hartley, O. Schwartz, and P. Haslinger, A Quantum Klystron – Controlling Quantum Systems with Modulated Electron Beams, arXiv:2004.10168 [physics, physics:quant-ph] (2020), arXiv:2004.10168 [physics, physics:quant-ph] .
  * Reinhardt and Kaminer [2020] O. Reinhardt and I. Kaminer, Theory of Shaping Electron Wavepackets with Light, ACS Photonics 7, 2859 (2020).
  * Gover  _et al._ [2020] A. Gover, B. Zhang, D. Ran, R. Ianconescu, A. Friedman, J. Scheuer, and A. Yariv, Resonant Interaction of Modulation-correlated Quantum Electron Wavepackets with Bound Electron States, arXiv:2010.15756 [quant-ph] (2020), arXiv:2010.15756 [quant-ph] .
  * Morimoto  _et al._ [2021] Y. Morimoto, P. Hommelhoff, and L. B. Madsen, Coherent scattering of an optically modulated electron beam by atoms, Physical Review A 103, 043110 (2021).
  * Baum [2017] P. Baum, Quantum dynamics of attosecond electron pulse compression, Journal of Applied Physics 122, 223105 (2017).
  * Feng  _et al._ [2014] C. Feng, H. Deng, D. Wang, and Z. Zhao, Phase-merging enhanced harmonic generation free-electron laser, New Journal of Physics 16, 043021 (2014).
  * Hirschmugl  _et al._ [1991] C. J. Hirschmugl, M. Sagurton, and G. P. Williams, Multiparticle coherence calculations for synchrotron-radiation emission, Physical Review A 44, 1316 (1991).
  * Gover  _et al._ [2019] A. Gover, R. Ianconescu, A. Friedman, C. Emma, N. Sudar, P. Musumeci, and C. Pellegrini, Superradiant and stimulated-superradiant emission of bunched electron beams, Reviews of Modern Physics 91, 035003 (2019).
  * Deng  _et al._ [2021] X. Deng, A. Chao, J. Feikes, A. Hoehl, W. Huang, R. Klein, A. Kruschinski, J. Li, A. Matveenko, Y. Petenev, M. Ries, C. Tang, and L. Yan, Experimental demonstration of the mechanism of steady-state microbunching, Nature 590, 576 (2021).
  * Bonifacio  _et al._ [2009] R. Bonifacio, N. Piovella, and G. Robb, The quantum free electron laser: A new source of coherent, short-wavelength radiation: The quantum free electron laser, Fortschritte der Physik 57, 1041 (2009).
  * Kling  _et al._ [2015] P. Kling, E. Giese, R. Endrich, P. Preiss, R. Sauerbrey, and W. P. Schleich, What defines the quantum regime of the free-electron laser?, New Journal of Physics 17, 123019 (2015).
  * Carmesin  _et al._ [2020] C. M. Carmesin, P. Kling, E. Giese, R. Sauerbrey, and W. P. Schleich, Quantum and classical phase-space dynamics of a free-electron laser, Physical Review Research 2, 023027 (2020).
  * Cowley and Moodie [1957] J. M. Cowley and A. F. Moodie, The scattering of electrons by atoms and crystals. I. A new theoretical approach, Acta Crystallographica 10, 609 (1957).
  * Note [1] See Supplemental Material for derivation of Eq. (1), and simulated space-time evolution of the probability density after inelastic scattering.
  * Fleck  _et al._ [1976] J. A. Fleck, J. R. Morris, and M. D. Feit, Time-dependent propagation of high energy laser beams through the atmosphere, Applied physics 10, 129 (1976).
  * Yoshida [1990] H. Yoshida, Construction of higher order symplectic integrators, Physics Letters A 150, 262 (1990).
  * Suzuki [1992] M. Suzuki, General theory of higher-order decomposition of exponential operators and symplectic integrators, Physics Letters A 165, 387 (1992).
  * Lubk [2018] A. Lubk, Paraxial Quantum Mechanics, in  _Advances in Imaging and Electron Physics_, Vol. 206 (Elsevier, 2018) pp. 15–58.
  * Reimer and Kohl [2008] L. Reimer and H. Kohl,  _Transmission Electron Microscopy_, Vol. 36 (Springer New York, New York, NY, 2008).
  * Konečná and de Abajo [2020] A. Konečná and F. J. G. de Abajo, Electron Beam Aberration Correction Using Optical Near Fields, Physical Review Letters 125, 030801 (2020).
  * Di Giulio and García de Abajo [2020] V. Di Giulio and F. J. García de Abajo, Free-electron shaping using quantum light, Optica 7, 1820 (2020), arXiv:2008.00957 .
  * Berry and Klein [1996] M. V. Berry and S. Klein, Integer, fractional and fractal Talbot effects, Journal of Modern Optics 43, 2139 (1996).
  * Guigay [1971] J. Guigay, On Fresnel Diffraction by One-dimensional Periodic Objects, with Application to Structure Determination of Phase Objects, Optica Acta: International Journal of Optics 18, 677 (1971).
  * Cloetens  _et al._ [1997] P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, Fractional Talbot imaging of phase gratings with hard x rays, Optics Letters 22, 1059 (1997).
  * Di Giulio  _et al._ [2019] V. Di Giulio, M. Kociak, and F. J. G. de Abajo, Probing quantum optical excitations with fast electrons, Optica 6, 1524 (2019), arXiv:1905.06887 .
  * Ben Hayun  _et al._ [2021] A. Ben Hayun, O. Reinhardt, J. Nemirovsky, A. Karnieli, N. Rivera, and I. Kaminer, Shaping quantum photonic states using free electrons, Science Advances 7, eabe4270 (2021).
  * Feist  _et al._ [2017] A. Feist, N. Bach, N. Rubiano da Silva, T. Danz, M. Möller, K. E. Priebe, T. Domröse, J. G. Gatzmann, S. Rost, J. Schauss, S. Strauch, R. Bormann, M. Sivis, S. Schäfer, and C. Ropers, Ultrafast transmission electron microscopy using a laser-driven field emitter: Femtosecond resolution with a high coherence electron beam, Ultramicroscopy 176, 63 (2017), arXiv:1611.05022 .
  * Schroeder  _et al._ [2001] C. B. Schroeder, C. Pellegrini, and P. Chen, Quantum effects in high-gain free-electron lasers, Physical Review E 64, 056502 (2001).
  * Bendaña  _et al._ [2011] X. Bendaña, A. Polman, and F. J. García de Abajo, Single-Photon Generation by Electron Beams, Nano Letters 11, 5099 (2011).
  * Lin  _et al._ [2018] X. Lin, S. Easo, Y. Shen, H. Chen, B. Zhang, J. D. Joannopoulos, M. Soljačić, and I. Kaminer, Controlling Cherenkov angles with resonance transition radiation, Nature Physics 14, 816 (2018).
  * Mignuzzi  _et al._ [2018] S. Mignuzzi, M. Mota, T. Coenen, Y. Li, A. P. Mihai, P. K. Petrov, R. F. M. Oulton, S. A. Maier, and R. Sapienza, Energy–Momentum Cathodoluminescence Spectroscopy of Dielectric Nanostructures, ACS Photonics 5, 1381 (2018).
  * Talebi  _et al._ [2019] N. Talebi, S. Meuret, S. Guo, M. Hentschel, A. Polman, H. Giessen, and P. A. van Aken, Merging transformation optics with electron-driven photon sources, Nature Communications 10, 599 (2019).
  * Schilder  _et al._ [2020] N. J. Schilder, H. Agrawal, E. C. Garnett, and A. Polman, Phase-Resolved Surface Plasmon Scattering Probed by Cathodoluminescence Holography, ACS Photonics 7, 1476 (2020).
  * van Nielen  _et al._ [2020] N. van Nielen, M. Hentschel, N. Schilder, H. Giessen, A. Polman, and N. Talebi, Electrons Generate Self-Complementary Broadband Vortex Light Beams Using Chiral Photon Sieves, Nano Letters 20, 5975 (2020).
  * Christopher  _et al._ [2020] J. Christopher, M. Taleb, A. Maity, M. Hentschel, H. Giessen, and N. Talebi, Electron-driven photon sources for correlative electron-photon spectroscopy with electron microscopes, Nanophotonics 9, 4381 (2020).
  * Remez  _et al._ [2019] R. Remez, A. Karnieli, S. Trajtenberg-Mills, N. Shapira, I. Kaminer, Y. Lereah, and A. Arie, Observing the Quantum Wave Nature of Free Electrons through Spontaneous Emission, Physical Review Letters 123, 060401 (2019).
  * Scully and Zubairy [1997] M. O. Scully and M. S. Zubairy,  _Quantum Optics_, 1st ed. (Cambridge University Press, 1997).
  * García de Abajo and Di Giulio [2021] F. J. García de Abajo and V. Di Giulio, Optical Excitations with Electron Beams: Challenges and Opportunities, ACS Photonics 8, 945 (2021), arXiv:2010.13510 .


# The Principle Bundle Structure
of
Continuum Mechanics

Stefano Stramigioli,
University of Twente, The Netherlands

###### Abstract

In this paper it is shown that the structure of the configuration space of any
continua is what is called in differential geometry a principle bundle Frankel
(2011). A principal bundle is a structure in which all points of the manifold
(each configuration in this case) can be naturally projected to a manifold
called the base manifold, which in our case represents pure deformations. All
configurations projecting to the same point on the base manifold (same
deformation) are called fibers. Each of these fibers is then isomorphic to the
Lie group $\mathfrak{se(3)}$ representing pure rigid body motions.
Furthermore, it is possible to define what is called a connection and this
allows to split any continua motion in a rigid body sub-motion and a
deformable one in a completely coordinate free way. As a consequence of that
it is then possible to properly define a pure deformation space on which an
elastic energy can be defined. This will be shown using screw theory Ball
(1900), which is vastly used in the analysis of rigid body mechanisms but is
not normally used to analyse continua. Beside the just mentioned result, screw
theory will also be used to relate concepts like helicity and enstrophy to
screw theory concepts.

††journal: Journal of Geometry and Physics

##  1 Introduction

The study of continuous matter is of fundamental importance in all branches of
engineering mechanics Abraham and Marsden (1994), spanning from solid
deformations Jerrold E. Marsden (1983) to fluids Chorin and Marsden (1993). A
lot of interest is present also about proper coordinate free analysis
descriptions of continua, like for example the excellent recent work of Kolev
and Desmorat (2021).

On the other hand in rigid body mechanics, the theory of screws, Ball (1900)
is vastly used to describe the motion of mechanisms of interconnected rigid
bodies. A nice example of its use can for example be found in the analysis of
dynamic balance by de Jong et al. (2019). The theory of screws is based on two
theorems: Mozzi’s theorem, for which an hystorical perspective can be found in
Ceccarelli (2000)111In the literature of screw theory Chasles theory (1830) is
often cited but Mozzi’s presented the same result in 1763. and Poinsot’s
theorem. Mozzi’s theorem states that any instantaneous rigid body motion can
be interpreted as the action of a rotation around an axis in space
superimposed to a possible motion along the same axis, proportional to a
scalar called the pitch which indicates the length traveled along the line for
a full rotation along the line, like the pitch of a “workshop” screw.
Poinsot’s theorem is a dual theorem which basically says that any systems of
forces applied to a rigid body, would have the same effect of a single force
applied along a line in space and superimposed to a possible torque around the
same line, also relating the two by a scalar pitch. These two theorems where
then used in Ball (1900) to create a theory based on the geometry of lines, or
better screws, for the analysis of rigid body kinematics. An extensive nice
treatment of the mathematics and its relation to Lie groups and Clifford’s
algebras can be found in Selig (2005), and specifically to Lie algebras in
Stramigioli et al. (2002). In this work it will be shown that it is possible
to associate to each velocity and vorticity in a certain point of a continua,
an infinitesimal (associated to the infinitesimal volume at the point) screw
describing them at the same time. Such an infinitesimal screw (relating to the
infinitesimal volume element with velocity $v$ and vorticity $dv$), as in the
rigid body case, will be characterised by an axis and a pitch, but in this
case the motion of the point will be considered acted upon by the
infinitesimal screw. It will be shown that the concepts of helicity and
enstrophy, well known in fluid dynamics, will get a very geometric
interpretation in the context of infinitesimal screws.

Considering that each of these screws are equivalent to elements of a Lie
algebra Stramigioli et al. (2002), they can be summed or integrated. Such an
operation will reveal the superimposed motion that each point will generate:
if all motions would follow the same rigid body motion, the resultant would be
such a motion, but in case of non rigid motions, the resultant will show the
“main rigid body motion” of the continuum. This construction will be presented
precisely and it will be shown that, thanks to this, it will be possible to
uniquely decompose, in a completely coordinate free way, the rigid component
of a continuum motion and the resting pure deformation. This will be done by
the introduction of what is called a differential geometric connection Frankel
(2011) in the principal bundle of motions.

During the paper, to stress the coordinate invariance of the approach,
different terms will be used as synonymous like geometric or intrinsic.

The paper will start in Sec.2 with the general framework formulation of the
analysis of a continua. In Sec.3 it will be shown that the configuration space
of a continuum has indeed a principle bundle structure whose fibers are
isomorphic to $SE(3)$. In Sec. 4 the geometry and basics of screws theory will
be reviewed to then in Sec.5 use this insight to define the connection on the
principle bundle introduced in Sec.3. Finally, in Sec.6 the results will be
framed in the context of a geometric description of potential energy of the
deformation, only using an energy formulation to then draw some concluding
remarks in Sec.7.

##### Notation

A compact, orientable, $n$-dimensional Riemannian manifold $M$ with (possibly
empty) boundary $\partial M$ models the spatial container of a non
relativistic continuum mechanical system. It possesses a metric field $g$ and
induced Levi-Civita connection $\nabla$ . The space of vector fields on $M$ is
defined as the space of sections of the tangent bundle $TM$, that will be
denoted by $\Gamma({TM})$. The space of differential $p$-forms is denoted by
$\Omega^{p}(M)$ and we also refer to $0$-forms as functions, $1$-forms as co-
vector fields and $n$-forms as top-forms. For any $v\in\Gamma(TM)$, we use the
standard definitions for the interior product by
$\iota_{v}:\Omega^{p}(M)\to\Omega^{p-1}(M)$ and the Lie derivative operator
$\mathcal{L}_{v}$ acting on tensor fields of any valence. The Hodge star
operator $\star:\Omega^{p}(M)\to\Omega^{n-p}(M)$, the volume form $\mu=\star
1$, as well as the ”musical” operators $\flat:\Gamma(TM)\to\Omega^{1}(M)$ and
$\\#:\Omega^{1}(M)\to\Gamma(TM)$, which respectively transform vector fields
to 1-forms and vice versa, are all uniquely induced by the Riemannian metric
in the standard way. To complement a purely coordinate-free notation, we will
represent tensorial quantities of interest in local coordinates, denoted by
$x$, when considered insightful, using Einstein summation convention.

##  2 The Continuum Mechanics Setting

The goal of this paper is to show the intrinsic structures which appear in the
description of the kinematics of a continua. For this reason, there will be no
use of coordinates. The setting which will be used, using the approach of Noll
(1978) is the one of a continua moving in space and the various entities of
the problem will be introduced next.

###  2.1 Matter and Space

We consider the motion of matter in space where matter is mathematically
modeled as a manifold $\mathcal{M}$ of dimension $n$ to which some properties
may be associated via sections of a proper bundle. For example, in the case of
a purely mechanical description, we can consider
$\mu_{m}\in\Omega^{n}(\mathcal{M})$ as the mass top form describing the mass
distribution, where we have indicated with $\Omega^{n}(\mathcal{M})$ the
sections of the $n$ alternating bundle to which $n$-forms on $\mathcal{M}$
belong. Any other property could be defined in the same way. What is important
to realise is that such properties are all those properties which are
conserved for physical reasons and will be advected by the motion because
strictly connected to the matter and matter only, independently of any motion
it takes in the space. This could be for example also electrical charge. On
the other hand the scalar density $\rho$ is a quantity which is not
conserved/advected because its definition does not depend only on the matter,
but on the relation between the mass topform $\mu_{m}$ and a representation of
the volume form of the space.

We consider space as an $n$ dimensional Riemannian manifold $(\mathcal{S},g)$
where with $g\in\text{sym}_{+}(T_{2}^{0}\mathcal{S})$ we indicate the positive
definite symmetric metric field on $\mathcal{S}$ with induced volume form
$\mu\in\Omega^{n}(\mathcal{S})$.

###  2.2 The configuration space

We can now put matter into space with an embedding of the form:

$e:\mathcal{M}\rightarrow\mathcal{S}$

which will describe in a certain moment where each of the points of matter
will be situated in space. Each of such objects will define therefore a
configuration of the body in space and therefore the set $\mathcal{C}$ of all
proper embedding of $\mathcal{M}$ into $\mathcal{S}$ will describe the
configuration space. It is possible to see that, inheriting the differential
structures of $\mathcal{M}$ and $\mathcal{S}$, $\mathcal{C}$ is an infinite
dimensional differentiable manifold. From now on we will consider furthermore
the elements of $\mathcal{C}$ to be bijections to simplify the exposition of
what follows. This is always possible if we generalise the concept of matter
to also vacuum matter representing those parts of space where no mass is
present but a more precise description can be used Noll (1978); Kolev and
Desmorat (2021).

###  2.3 Motions

We can now define a motion as a smooth curve in $\mathcal{C}$ of the type

$c:I\rightarrow\mathcal{C}\;:t\mapsto c_{t}$

which associates to a real time value $t$ in the interval $I$ the
configuration $c(t)$ at that instant. We can then see that geometrically
$\dot{c}(t):=\frac{d}{ds}c(s)|_{s=t}\in T_{c}\mathcal{C}$, and this allows to
define the following map:

$E_{c}:T_{c}\mathcal{C}\rightarrow\Gamma(T\mathcal{S});v\mapsto(x\mapsto(x,(v\circ
c^{-1})(x))$

which associates to a configuration $c$ and its velocity $\dot{c}$ a complete
vector field in the space $c(\mathcal{M})$. Due to the hypothesis of the
bijective nature of elements in $\mathcal{C}$ we can than also take the
pullback $c^{*}$ of a vector field and we have the complete picture relating
the Lagrangian (configuration) velocity to the Eulerian vectorfield (space)
and the Convective vector field (matter):

${\underbrace{\Gamma(T\mathcal{M})}_{\text{convective}}}$${\underbrace{T_{c}\mathcal{C}}_{\text{Lagrangian}}}$${\underbrace{\Gamma(T\mathcal{S})}_{\text{Eulerian}}}$$\scriptstyle{(c^{*}\circ
E_{c})}$$\scriptstyle{E_{c}}$

##  3 The Principle Bundle structure of continuum motion

We can now consider the manifold $\text{Diff}(\mathcal{S})$ of diffeomorphic
auto-morphisms of $\mathcal{S}$ and considering that $\mathcal{S}$ has a
metric, we can consider the isometries within $\text{Diff}(\mathcal{S})$ which
will be indicated as $G\subset\text{Diff}(\mathcal{S})$. It is possible to see
that $G$ is a Lie group and in the case that $(\mathcal{S},g)$ is flat and
therefore Eucledian, $G$ would correspond to the Special Eucledian group
$SE(n)$. Keeping it general, we can consider $G$ as the group of isometric
right actions on $\mathcal{S}$:

$\lhd:\mathcal{S}\times G\rightarrow\mathcal{S}:(p,h)\mapsto p\lhd h$

with the corresponding infinitesimal generator of the corresponding Lie
algebra $\mathfrak{g}$ for which $\forall T\in\mathfrak{g}$ we can define:

$V_{T}:\mathcal{S}\rightarrow T\mathcal{S}:p,\mapsto\frac{d}{ds}(p\lhd
e^{sT})|_{s=0}$

for which the isometric property should be satisfied $\forall
T\in\mathfrak{g}$:

$\mathcal{L}_{V_{T}}g=0.$

We can also define an equivalent relation in $\mathcal{C}$. Considering
$c_{1},c_{2}\in\mathcal{C}$:

$c_{1}\sim c_{2}\Leftrightarrow\exists h\in
G\;\text{s.t.}\;c_{1}(\mathcal{M})=c_{2}(\mathcal{M})\lhd h$

and it is straight forward to see that such an equivalence relation satisfies
the needed properties for an equivalence relation. We can then define the pure
deformation space as $\mathcal{D}:=\mathcal{C}/\sim$ which can be recognised
as the space of orbits for the group $G$. This shows that $\mathcal{C}$ has
the following fiber bundle structure:

$\pi:\mathcal{C}\rightarrow\mathcal{D}:c\rightarrow[c]$

where $[c]$ indicates the equivalence class in $\mathcal{D}$ to which $c$
belongs. It is possible to finally check that the right action $\lhd$ is free
and therefore $\mathcal{C}\rightarrow\mathcal{D}$ is a principle bundle on
which $G$ acts.

By the previous construction it is possible to see if a motion expressed by an
element $v_{c}(c)\in T_{c}\mathcal{C}$ is a pure rigid body motion by
observing if it lies in the vertical space
$\text{Ver}(c):=\text{Ker}(\pi_{*}(c))$, the kernel of $\pi_{*}$ in $c$, which
is isomorphic to $\mathfrak{g}$. On the other hand, without a horizontal space
$\text{Hor}(c)$ complementary to $\text{Ver}(c)$ such that we have
$T_{c}\mathcal{C}=\text{Ver}(c)\oplus\text{Hor}(c)\;\forall c\in\mathcal{C}$,
it is not possible to uniquely decompose any general vector in a vertical and
horizontal component . For a principal bundle, the definition of a horizontal
differential distribution $\text{Hor}(c)$ is done with a connection which is a
Lie algebra valued (in our case $\mathfrak{g}$) one form. As it will be shown
in this paper it is possible to define such a connection in a very precise
way, but before doing so, we will need to restrict to the case of the 3
dimensional space and to the Lie group of positive isometries $SE(3)$, the
Special Eucledian group with corresponding Lie algebra $\mathfrak{se(3)}$.

##  4 The geometry of $\mathfrak{se(3)}$ and screws

Without the intention to be exhaustive about a theory which is well known and
documented Ball (1900); Selig (2005); Stramigioli and Bruyninckx (2001), a
short introduction of screw theory will now be given.

In the modeling of rigid body motions, the role of $SE(3)$ is fundamental
because it is isomorphic to the configuration space of a rigid body motions as
explained in the previous section. Infinitesimal rigid body motions
(generators) can then be represented using elements of the corresponding Lie
algebra $\mathfrak{se(3)}$ which are in mechanics called instantaneous twists.
In screw-theory Ball (1900) the kinematics of rigid-body mechanisms is
described using the geometry of lines and screws in the Euclidian space. The
latter are geometric lines with the extra information of an orientation of the
line, a scalar magnitude and an extra scalar quantity $\lambda$ called the
pitch of the screw. This is based on Mozzi’s theorem which basically says that
any rigid body instantaneous motion can be expressed as a rotation around an
axis in the Euclidean space, plus a translation along the same axis. The
magnitude of the screw will then indicate the speed of rotation around this
axis. The relation between the rotation speed and the translation speed is
given by what is called the pitch as it happens in a mechanical screw. As
shown in Stramigioli et al. (2002); Selig (2005), the description of rigid
body motions using Lie groups and screws are equivalent. This means that we
can properly interpret an abstract element of the Lie algebra
$\mathfrak{se(3)}$ as a screw in the Eucledian space acted upon $SE(3)$. Due
to this fact, it is geometrically meaningful to take linear combinations of
screws using the vector structure of $\mathfrak{se(3)}$ and this will be one
of the core insights used: after introducing the concepts of screws, we will
see that it is possible to associate an infinitesimal screw to any motion of
the continua and then sum/integrate in the continua. Furthermore, being
$\mathfrak{se(3)}$ not only a vector space, but also a Lie algebra, it
possesses a skew symmetric operation which corresponds in screw theory to what
is called the cross product of screws.

One of the advantages of the screw interpretation is that the infinitesimal
motion can be associated to a geometrical object living in the same space
where the motion is happening, which is not the case with the more abstract
element of a Lie algebra which belongs to a different space. The screw
construction is possible by first individualising the straight line which is
kept invariant by the infinitesimal motion under consideration, as expressed
by Mozzi’s theorem, and then identifying the other scalar information which
defines the screw, the pitch $\lambda$.

###  4.1 Screw representation

We can uniquely identify a screw with one vector $\omega\in T_{p}\mathcal{S}$,
representing rotation, and a vector $v\in T_{p}\mathcal{S}$ Stramigioli and
Bruyninckx (2001). Properly speaking this geometry uses what are called line
bounded vectors for the geometry of lines but this distinction is not needed
for our purpose. We can then define what is called a twist as:

$T:=\begin{pmatrix}\omega\\\ v\end{pmatrix}$ (1)

As said, it can also be seen that this ”double vector” composed by two three
dimensional vectors is a proper representation of an element of
$\mathfrak{se(3)}$ as it will be constructed hereafter, and therefore we can
sum such elements Stramigioli (2001). In order to see this, the first needed
step is to decompose $v$ along $\omega$ ($v_{\omega}$) and a part belonging to
the orthogonal complement ($v_{\omega}^{\perp}$) and this can be done using
the inner product $g$ in the case in which $\omega\neq 0$:

$v=v_{\omega}+v_{\omega}^{\perp}.$

The case in which $\omega=0$ represents the degenerate case of a “pure
translation” which in this setting can be properly framed as a rotation with
an axis at infinity using the concept of improper lines of projecting geometry
Harvey Lipkin (1985).

####  4.1.1 Moment of the line and screw axis

In the non degenerate case, the information about $\omega$ and
$v_{\omega}^{\perp}$ uniquely identifies the axis of the screw. This is done
by identifying the unique axis in space for which a rotation around such an
axis with speed $\omega$ would result in the velocity $v_{\omega}^{\perp}$. It
is possible to see that the axis of the rotation is oriented along the
$\omega$ direction and it passes through a point which is situated at

$r:=\frac{\omega\times v}{<\omega,\omega>}$ (2)

from the point $p$, the base point of the tangent space $T_{p}\mathcal{S}$
where $\omega$ and $v$ live. The operator $\times$ in this cases indicates the
vector product of vectors in 3D Euclidean space. It is clear that by
construction, the component $v_{\omega}$ plays no role in the definition of
$r$ because it is aligned with $\omega$ and therefore results in a vanishing
vector product of parallel vectors.

####  4.1.2 The Pitch of the screw

The pitch of the screw is then defined as the unique positive scalar $\lambda$
for which:

$v_{\omega}=\lambda\omega$

holds. It is easy to see that we can easily calculate $\lambda$ as:

$\lambda=\frac{<\omega,v>}{<\omega,\omega>}$ (3)

In this case it is possible to see that $v_{\omega}^{\perp}$ plays no role
because by construction it is perpendicular to $\omega$. This shows that
$v_{\omega}$ is the component of velocity which is related to the translation
along the axis of the screw and it is different than zero if and only if the
pitch $\lambda$ is different than zero which is the case if and only if the
motion is not a pure rotation.

It is possible to see that if the motion of the body is rigid, the previous
analysis in each other point $q\neq p$ belonging to the rigid body, would
result in $\bar{\omega},\bar{v}\in T_{q}\mathcal{S}$ which would give the same
axis in the Euclidean space, and the same $\lambda$. Ultimately this results
in an expression of Mozzi’s theorem:

$\begin{pmatrix}\omega\\\ v\end{pmatrix}=\underbrace{\begin{pmatrix}\omega\\\
r\times\omega\end{pmatrix}}_{\text{rotation}}+\begin{pmatrix}0\\\
\lambda\omega\end{pmatrix}$ (4)

which clearly gives an expression of the all motion as built up from $\omega$
indicating the direction and speed of the motion, $r$ indicating where the
axis of the screw lies in Euclidian space, and the scalar $\lambda$. It is
important to notice that $r\times\omega$ which is called the moment of the
line depends on the geometrical position of the line. A point of the axis of
the screw can be reached by moving along $r$ from $p$. This is a properly
defined statement because of the flatness of the space and the existence of a
metric which allows such a construction.

####  4.1.3 Summing screws and the Adjoint representation of $SE(3)$

In the previous construction, we have given a way to associate to a motion a
screw which is identified by a line with a magnitude plus a pitch. In order to
sum these objects, we could do it geometrically as it is done in screw theory
by a geometric operation of lines, or use the fact that a screw is
correspondent to an element of $\mathfrak{se(3)}$ and use its vector
structure. For what will be explained later, we want to be able to calculate
an integral which is a generalisation of a sum and the second method using
$\mathfrak{se(3)}$ is much more convenient.

For the Lie group $SE(3)$ of positive isometries of the Euclidean space of
which $\mathfrak{se(3)}$ is the algebra, we can consider what is called the
Adjoint representation of the group. If we take $h,k\in SE(3)$, we can
consider what is called the conjucation automorphism $\Psi_{k}(h):=khk^{-1}$
and we can then define

$Ad_{k}:\mathfrak{se(3)}\rightarrow\mathfrak{se(3)};T\mapsto(\Psi_{k})_{*}T$
(5)

which basically describes how elements of $\mathfrak{se(3)}$ change by
changing the observer by $k$. In our case this would mean how the screw
changes in the Euclidean space by moving the observer by $k$ which more
precisely would mean that if consider $q=k(p)$ where $p,q$ are points in the
Euclidian space, $\omega_{p},v_{p}\in T_{p}\mathcal{S}$, representing a motion
as a screw in $p$, $\omega_{q},v_{q}\in T_{q}\mathcal{S}$ representing the
same screw (motion) in $q$, and $k\in SE(3)$, then we have that Selig (2005):

$\begin{pmatrix}\omega_{q}\\\
v_{q}\end{pmatrix}=Ad_{k}\begin{pmatrix}\omega_{p}\\\ v_{p}\end{pmatrix}$ (6)

Thanks to the $Ad_{k}$ map, it is now possible to meaningfully define sum of
multiple screws by first transforming them to the same tangent space:

$\begin{pmatrix}\omega_{q}\\\
v_{q}\end{pmatrix}=\sum_{i}Ad_{k_{i}}\begin{pmatrix}\omega_{p_{i}}\\\
v_{p_{i}}\end{pmatrix}$ (7)

where $p_{i}\in T_{p_{i}}\mathcal{S}$ and $q=k_{i}(p_{i})$. This will be the
main construction which will be extended to integration in the next section.

##  5 The Pure Deformation Connection

As explained in the previous section, the screw resulting by the previous
analysis, will be the same if calculated in any moving point of a rigid body.
In case of a general moving continua, this will clearly not be the case in
general, but thanks to the fact that such screws, are elements of the Lie
algebra $\mathfrak{se(3)}$, it is completely meaningful to sum/integrate their
contributions in order to calculate the “total screw” of the volume which has
been used for the integration. In case the motion would be rigid, this would
correspond to the screw of a rigid body as explained, but in case that the
motion will not be rigid, this will result in an expression of a
$\mathfrak{se(3)}$ valued 1-form which paired with the vector field of
motions, would result after integration in a screw. Such an object by
construction is what is called a differential geometric connection which can
then be used to define a horizontal space in the principal bundle.

###  5.1 Screw interpretation of flow, vorticity, entropy and helicity

If we considered a rotating rigid disc with an angular velocity of $\omega$,
it is easy to see that all points on the disc would have a tangential velocity
$v$ equal to the angular velocity times their distance from the axis.
Furthermore, the corresponding vorticity, which is expressed in vector
calculus with the rotational $\nabla\times v$, would be constant on the all
disk and equal to $2\omega$ because, as shown in Chorin and Marsden (1993), we
have:

$\omega=\frac{1}{2}\nabla\times v$

Writing this expression using exterior calculus, we then have that

$\omega=\frac{1}{2}(dv^{\flat})^{\sharp}$

which would mean that eq.(6) would be satisfied $\forall p,q$ on the rigid
body.

If for a general continum media, we consider a covariant representation of
both the velocity field $v\in\Gamma(T\mathcal{S})$ as a one form field
$v^{\flat}\in\Omega^{1}(\mathcal{S})$ obtained by lowering its contravariant
index with the metric $g$, and then the vorticity as its exterior derivative
$dv^{\flat}\in\Omega^{2}(\mathcal{S})$, we can then transform the two-form
$dv^{\flat}$ to the one-form $\star dv^{\flat}$ using the Hodge star operator
$\star$ based on the metric $g$ and finally rise the index using the metric
$g$ which we indicate with the standard $\sharp$ notation. We can then see
with reference to (1) that the infinitesimal (to be integrated on the volume)
screw can then be defined as:

$\delta T:=\begin{pmatrix}\frac{1}{2}(\star dv^{\flat})^{\sharp}\\\
v\end{pmatrix}.$ (8)

This is because, as just stated, you have a clear correspondence
$\omega\leftrightarrow\frac{1}{2}(\star dv^{\flat})^{\sharp}$ of the vorticity
as the local angular velocity.

A number of new concepts can then be introduced, like the enstropy topform:

$e:=dv^{\flat}\wedge\star dv^{\flat}\in\Omega^{3}(\mathcal{S})$ (9)

which plays the role of measuring the norm of the vorticity once integrated on
the volume, and the helicity topform:

$h:=v^{\flat}\wedge dv^{\flat}\in\Omega^{3}(\mathcal{S}).$ (10)

With analogy to eq.(2) we can then define the ray one form:

$r:=\frac{\iota_{v}dv^{\flat}}{\star e}\in\Omega^{1}(\mathcal{S})$ (11)

and with analogy to eq.(3) we can define the pitch function

$\lambda:=\frac{\star h}{\star e}\in\Omega^{0}(\mathcal{S})$ (12)

As the enstropy is used as a normalisation factor corresponding to the
$<\omega,\omega>$ in eq.(2) and eq.(3), it is interesting to observe that the
helicity is proportional to the defined pitch form which is exactly the
intuition of the concept (and corresponding naming) of helicity: with no
pitch, there would be no helical motion of the fluid and the helicity would
vanish.

######  Theorem 1 (Screw Field Decomposition).

Given any continuum vector field $v$ with corresponding covariant
representation $v^{\flat}$ and respective vorticity $dv^{\flat}$, the
following decomposition holds:

$\begin{pmatrix}{\star dv^{\flat}}\\\ \star
v^{\flat}\end{pmatrix}=\begin{pmatrix}\star dv^{\flat}\\\ r\wedge\star
dv^{\flat}\end{pmatrix}+\begin{pmatrix}0\\\ \lambda dv^{\flat}\end{pmatrix}$
(13)

###### Proof.

We can start by writing the identity:

$\underbrace{\star(dv^{\flat}\wedge\star dv^{\flat})}_{\star
e}v^{\flat}=\underbrace{\star(dv^{\flat}\wedge\star
dv^{\flat})v^{\flat}-\star(v^{\flat}\wedge dv^{\flat})\star
dv^{\flat}}_{\alpha}+\underbrace{\star(v^{b}\wedge dv^{\flat})}_{\star h}\star
dv^{\flat}$ (14)

We can then further process the term $\alpha$. Using the identity
$\iota_{v}\alpha=\star(v^{\flat}\wedge\star\alpha)$ in the following we
obtain:

$\alpha=\star\iota_{v}(dv^{b}\wedge\star dv^{\flat})-(\iota_{v}\star
dv^{\flat})\star dv^{\flat}$ (15)

and considering that the scalar function commute with the Hodge

$\alpha=\star(\iota_{v}(dv^{\flat}\wedge\star dv^{\flat})-(\iota_{v}\star
dv^{\flat})dv^{\flat})$ (16)

and using the distributive property of the $\iota_{v}(\alpha\wedge\beta)$ we
obtain:

$\alpha=\star(\iota_{v}(dv^{\flat})\wedge\star dv^{b}).$ (17)

Finally, dividing the original equation for the scalar $\star e$, and then
taking the Hodge $\star$ on both sides of the equation the result is obtained

∎

Taking the Hodge and then the sharp of the second component and the sharp of
the first component, we have an expression for the infinitesimal twist of
eq.(8) as:

$\begin{pmatrix}({\star dv^{\flat}})^{\sharp}\\\
v\end{pmatrix}=\begin{pmatrix}({\star dv^{\flat}})^{\sharp}\\\
(\star(r\wedge\star dv^{\flat})))^{\sharp}\end{pmatrix}+\begin{pmatrix}0\\\
\lambda(\star dv^{\flat})^{\sharp}\end{pmatrix}$ (18)

which can be manipulated recognising being equal to

$\begin{pmatrix}\frac{1}{2}({\star dv^{\flat}})^{\sharp}\\\
v\end{pmatrix}=\begin{pmatrix}\frac{1}{2}({\star dv^{\flat}})^{\sharp}\\\
r^{\sharp}\times({\star dv^{\flat}})^{\sharp}\end{pmatrix}+\begin{pmatrix}0\\\
\lambda(\star dv^{\flat})^{\sharp}.\end{pmatrix}$ (19)

This expression gives a construction to identify geometrically exactly the
same screw interpretation of a screw for an infinitesimal element. It is
therefore possible for a continuum to define a screw field for which we have a
pitch field which is equal to the ratio of the density of the helicity and
entropy as introduced in eq.(3) and with reference to eq.(11) a ray vector
field as

$r^{\sharp}=\begin{pmatrix}\frac{\iota_{v}dv^{\flat}}{\star
e}\end{pmatrix}^{\sharp}.$

###  5.2 The Principal Bundle Connection

It is now possible to define the integral of the various contributions of the
continuum, but in order to do so, we first need to map the various
contributions to a common reference as explained in sec.4.1.3 via the $Ad_{k}$
where $k$ is an element of $SE(3)$ which is defined based on the location of
the point $p$ where the infinitesimal screw is considered. To clarify the
construction, consider to choose an ortho-normal frame bundle
($<e_{i},e_{j}>=\delta_{ij}$) with $e_{i}(p)\in T_{p}\mathcal{S}$ such that
$\text{span}\\{e_{i}(p)\\}=T_{p}\mathcal{S}$. This can be used for example to
express:

$v(p)=v^{i}(p)e_{i}(p)\qquad({\star dv^{\flat}})^{\sharp}(p)=(({\star
dv^{\flat}})^{\sharp})^{j}(p)e_{j}(p)$

where Einstein implicit summation is used. If we define the six dimensional
base of the screws (and therefore $\mathfrak{se(3)}$) at the point $p$ with:

$s_{i}(p):=\begin{pmatrix}e_{i}(p)\\\ 0_{3}(p)\end{pmatrix}\forall i=1..3\quad
s_{j}:=\begin{pmatrix}0_{3}(p)\\\ e_{(j-3)}(p)\end{pmatrix}\forall
j=4..6\quad$

where with $0_{3}(p)$ we indicate the null vector of $T_{p}\mathcal{S}$, we
could see that, due to the fact that screws are isomorphic to elements of
$\mathfrak{se(3)}$, $s_{1},..s_{6}$ is actually a basis of $\mathfrak{se(3)}$
which is created based on the choice of the point $p$ and the $e_{i}(p)\in
T_{p}\mathcal{S}$. Following this reasoning, if we would choose a different
point $q$ with $e_{i}(q)\in T_{q}\mathcal{S}$, and then define in the same way
the basis built on the new point and basis vectors, there would be an element
of $k_{p\rightarrow q}\in SE(3)$ such that $s_{i}(q)=Ad_{k_{p\rightarrow
q}}s_{i}(p)\;\forall i=1..6$. This means that if we have a general screw
$T\in\mathfrak{se(3)}$ we can express it using a frame attached to either $p$
or $q$ using two different frames in $p$ and $q$ as just introduced, and in
such a situation we can associate two numeric representations $T^{i}(p)$ or
$T^{i}(q)$ such that:

$T=T^{i}(p)s_{i}(p)=T^{j}(q)s_{j}(q)$

We can then define for each base $s_{i}(p)$ the canonical dual base
$s^{i}(p)\in\mathfrak{se^{*}(3)}$ such that $s^{i}(p)s_{j}(p)=\delta^{i}_{j}$.
Using the canonical dual base it is then possible for any element
$T\in\mathfrak{se(3)}$ and any base $s_{i}(p)$ to calculate the components for
which $T=T^{i}(p)s_{i}(p)$ as:

$T^{i}(p)=s^{i}(p)(T).$

####  5.2.1 The connection

It is now possible to define the principal bundle connection which can be used
to define the horizontal space. Consider the volume form
$\mu\in\Omega^{3}(\mathcal{S})$ which can be directly defined using the metric
$g$ as indicated previously. Consider a volume
$\bar{\mathcal{S}}\subset\mathcal{S}$ for which we want to describe the
decomposition between rigid body motion and pure deformation of the continuum
media. This could be for example the image of a solid via an embedding $c$. If
we consider a configuration velocity $v_{c}\in T_{c}\mathcal{C}$, this would
induce, as discussed previously, a vectorfield
$v=E_{c}(v_{c})\in\Gamma(T\mathcal{S})$. We can then rewrite
$\frac{1}{2}(\star dv^{\flat})^{\sharp}$ as a composition of operations as
$\frac{1}{2}(g^{-1}\circ\star\circ d\circ g)(v)$ to then define the following
$\mathfrak{se(3)}$ valued one form on $\mathcal{C}$:

$\nu_{\bar{S}}:T\mathcal{C}\rightarrow\mathfrak{se(3)}:\\\
v_{c}\mapsto\frac{1}{V}s_{i}(q)\int_{\bar{S}}s^{i}(q)\begin{pmatrix}Ad_{k_{p\rightarrow
q}}\begin{pmatrix}\frac{1}{2}(g^{-1}\circ\star\circ d\circ g)\\\
\text{Id}\end{pmatrix}E_{c}(v_{c})(p)\end{pmatrix}\otimes\mu(p)$ (20)

where we define the total volume:

$V:=\int_{\bar{\mathcal{S}}}\mu$

What eq.(20) actually does, is to take for each point $p$ the vector
$v(p)=E_{c}(v_{c})(p)$ and then build the infinitesimal screw using the form
introduced in eq.(8) which is a screw expressed in $p$. The result is than
mapped to any common frame in $q$ where the components can be extracted via
$s^{i}(q)$ to be then integrated. In this way six scalar integrals are
calculated which represent the integral of the components of the expression of
the infinitesimal screws in $q$ expressed with the bases $s_{i}(q)$. Outside
the integral, using the bases elements, the six scalar values are then
composed to the result in $\mathfrak{se(3)}$ and normalised by the volume
considered. If as an example we would consider a rigid body motion of the
continuum, the result of the integral would be the volume times the constant
screw representing the rigid body motion in $s_{i}(q)$ for which the external
division for the volume $V$ and multiplication for the base elements would
return the abstract screw representing the rigid body motion.

##  6 Implications of the presented structure

The connection $\nu_{\bar{S}}$ allows now to define in an intrinsic way a
Horizontal space for a volume $\bar{S}$ as:

$\text{Hor}_{{}_{\bar{S}}}(c):=\\{v_{c}\in
T_{c}\mathcal{C}\;s.t.\;\nu_{\bar{S}}(v_{c})=0\\}$

which allows now to define a unique decomposition of the velocities in
configuration space:

$T_{c}\mathcal{C}=\mathfrak{se(3)}\oplus\text{Hor}_{\bar{S}}(c).$

It interesting to relate the introduced decomposition to an important
identity:

$\nabla v^{\flat}=\frac{1}{2}\mathcal{L}_{v}g-\frac{1}{2}dv^{\flat}$

which should be interpreted as an equality in the $T^{0}_{2}\mathcal{S}$
tensor fields. The equality basically says that the covariant differential of
the one form representation of the vector field can be decomposed in a
symmetric part equal to $\frac{1}{2}L_{v}g$ which represents what is called
the rate of strain and an anti-symmetric part $\frac{1}{2}dv^{\flat}$ which
represents half of the vorticity. Locally, if the symmetric part is equal to
zero, it means that the local motion is not deforming the metric and
corresponds to a local rigid body motion. In such a case the all covariant
differential will correspond to the vorticity which represents the
infinitesimal rotation part of the screw corresponding to the first three
components. This expression gives therefore via $\frac{1}{2}L_{v}g$ a local
measure of the rate of strain and it excludes all translations which are
happening as a collection of the all continuum. The proposed connection gives
instead a way to decompose the global motion. Furthermore, the construction
presented for the connection can be modified to use a mass top-form
$\bar{\mu}_{m}:=c_{*}\mu_{m}\in\Omega^{n}(\mathcal{S})$ rather than the volume
top-form $\mu$. In such a case, the integrated infinitesimal screws
representing velocities, would be also multiplied by the density and the
volume (the mass form) which would result in the dimension of momenta rather
than velocities. This would result in the dual of $\mathfrak{se(3)}$
representing the total screw moment Stramigioli and Bruyninckx (2001)
belonging to $\mathfrak{se^{*}(3)}$. This will be the content of a future
paper showing the geodesic motion of the rigid part of a continuum.

###  6.1 The pure deformation space and elastic energy

Thanks to the recognition of the principal structure of the configuration
space of a continua, it is possible to recognise the base space $\mathcal{D}$
as the space expressing the pure deformations of the matter. Suppose to be in
a configuration $c\in\mathcal{C}$ which corresponds to a deformation
$d=\pi(c)\in\mathcal{D}$ using the projection of the bundle. For an
infinitesimal motion $\delta c\in T_{c}\mathcal{C}$ it is now possible to
uniquely decompose it in an horizontal and vertical components:

$\delta c=\delta r+\delta d\quad\delta r\in\text{Ver}(c),\delta
d\in\text{Hor}(c)$

in which the vertical space is six dimensional and the horizontal is infinite
dimensional. If $\delta d=0$ we have a pure rigid motion and in such a case,
the infinitesimal vector field $\delta v=E_{c}(\delta c)\in T\mathcal{S}$ is
such that $\mathcal{L}_{\delta v}g=0$ everywhere. In all other cases, a
component $\delta d\neq 0$ will result in a motion for which
$(\mathcal{L}_{\delta v}g)(p)\neq 0$ for some $p\in\mathcal{S}$. Furthermore,
any finite motion starting in $c_{1}\in\mathcal{C}$ and ending in
$c_{2}\in\mathcal{C}$ for which the deformation is the same
$(d:=\pi(c_{1})=\pi(c_{2}))$, clearly represents configurations in which the
internal deformation $d$ is the same because by construction there would be a
rigid motion bringing $c_{1}$ to $c_{2}$. This means that a completely
coordinate invariant way of expressing potential energy of the deformation can
be described by defining an energy function:

$H:\mathcal{D}\rightarrow\mathbb{R}$ (21)

which associates to any internal state $d\in\mathcal{D}$ of the continua an
energy function. Remembering that $\mathcal{D}$ is an infinite dimensional
manifold and an expression for which $H$ can be specified can only be given
via an integral expression whose variational derivative would represent the
dual stress field to the deformation represented by the variation. To make
this statement more precise, we can use the Lie derivative identity of the
pull back:

$c^{*}(\mathcal{L}_{v}g)=\mathcal{L}_{(c^{*}v)}c^{*}g$

which gives an expression of the pullback of the rate of strain field in
$\mathcal{S}$ to the rate of strain field in $\mathcal{M}$. Considering the
non-singularity of the pull back $c^{*}$ due to the diffeomorphic hypothesis
and the positive definiteness of $g$, this implies that $c^{*}(L_{v}g)$ is an
equivalent description on $\mathcal{M}$ of the rate of strain
$\mathcal{L}_{v}g$ on $\mathcal{S}$. Furthermore, a material infinitesimal
deformation $\delta d\in T_{c}\mathcal{C}$ can be associated uniquely to an
infinitesimal vector field $\delta\bar{d}=E_{c}(\delta d)\in T\mathcal{S}$ and
to an infinitesimal vector field $c^{*}(\delta\bar{d})\in T\mathcal{M}.$ This
shows that, given a configuration $c\in\mathcal{C}$, and a deformation $\delta
d$, we can uniquely identify $\mathcal{L}_{c^{*}\delta\bar{d}}c^{*}g$.
Furthermore, $\mathcal{L}_{c^{*}\delta\bar{d}}c^{*}g$ can be recognised being
geometrically part of the tangent space of
$\text{sym}_{+}(T^{0}_{2}(\mathcal{M}))$, the space of positive definite
$(0,2)$ tensors of metrics at $c^{*}g$. This shows therefore that

$\mathcal{D}\simeq\Gamma(\text{sym}_{+}(T^{0}_{2}(\mathcal{M})))$

and a curve in $\mathcal{D}$ resulting in a $\mathcal{L}_{v}g\neq 0$ will
result in a corresponding $\mathcal{L}_{(c^{*}\delta\bar{d})}g_{m}\neq 0$
which will change the $\text{sym}_{+}(T^{0}_{2}(\mathcal{M}))$ tensor field
$g_{m}:=c^{*}g$. This gives a constructive way to define the energy function
(21) as:

$H:\Gamma(\text{sym}_{+}(T^{0}_{2}(\mathcal{M})))\rightarrow\mathbb{R}:\bar{g}\mapsto\int_{\mathcal{M}}\mathfrak{H}(\bar{g})$
(22)

where for each point $m\in\mathcal{M}$, $\mathfrak{H}$ associates to
$\bar{g}(m)=c^{*}(g)(m)\in\text{sym}_{+}(T^{0}_{2}(\mathcal{M}))$ a value
$\mathfrak{H}(\bar{g})(m)\in\Omega^{n}_{p}(\mathcal{M})$. It is now possible
to take the variational derivative of $H$ by considering the following
differential expression:

$H(g+\delta g)-H(g)=<\frac{\delta H}{\delta g}|\delta g>+O^{2}(\delta g)$ (23)

where $<|>$ is a proper dual pairing. In order to define this pairing in a
meaningful, geometric way, we can first observe that being $\delta g$ a
symmetric $(0,2)$ tensor, if we contract one of its indices with the inverse
of the metric $g$ we would get a linear operator:

$\delta\bar{g}_{i}^{j}=(g^{-1})^{jl}\delta g_{li}.$

Being $\delta\bar{g}$ a liner operator we can do two things: first we can
consider the eigen-spaces of $\delta\bar{g}$ which will indicate the principal
direction of deformations, and second, we can interpret it as a vector valued
one form:

$\delta\bar{g}(m)\in
T^{1}_{m}\mathcal{M}\otimes\Omega^{1}_{m}(\mathcal{M})\quad m\in\mathcal{M}.$

This last consideration gives a very useful expression of the variational
derivative of eq.(23):

$H(g+\delta g)-H(g)=\int_{\mathcal{M}}\mathfrak{T}\dot{\wedge}\delta\bar{g}$
(24)

where $\mathfrak{T}\in\Omega^{1}(\mathcal{M})\otimes\Omega^{n-1}(\mathcal{M})$
is a one form-valued $(n-1)$ form which can be naturally paired using
$\dot{\wedge}$ with $\delta\bar{g}$ and is defined as:

$a_{1}\otimes a_{2}\in T^{1}\mathcal{M}\otimes\Omega^{1}(\mathcal{M}),\qquad
b_{1}\otimes b_{2}\in\Omega^{1}(\mathcal{M})\otimes\Omega^{n-2}(\mathcal{M}),$
$\Rightarrow(a_{1}\otimes a_{2})\dot{\wedge}(b_{1}\otimes
b_{2}):=b_{1}(a_{1})a_{2}\wedge b_{2}\in\Omega^{n}(\mathcal{M})$

which can be therefore integrated on the all matter space to give a scalar
expression of the infinitesimal energy variation due to the deformation
$\delta\bar{g}$. $\mathfrak{T}$ is the stress which is a direct consequence of
the definition of the energy function $H$ and is naturally a one form
(representing an infinitesimal force) valued $(n-1)$ form (representing and
infinitesimal surface), which can associate to any infinitesimal surface the
corresponding infinitesimal force.

##  7 Conclusions

In this paper it has been shown that the infinite dimensional space of
configurations of a continuum has the structure of a principle bundle for
which the base space $\mathcal{D}=\text{sym}_{+}(T_{2}^{0}(\mathcal{M}))$ of
pure deformations is the space of sections of the metric on the matter space
on which a potential energy function can be defined intrinsically to describe
nonlinear elasticity.

Furthermore, it has been shown that from the velocity vector field together
with its vorticity a $\mathfrak{se(3)}$ valued vector field of screws can be
defined which, if properly integrated can be used to define a geometric
connection which allows to decompose the vector field in a rigid body motion
and a pure deformation. Last but not least the relevance of concepts like
helicity and enstropy have been defined and shown during this construction.

This work is an important step in defining a formulation of open and
interconnectable nonlinear elasticity which will be needed to create open
models of deformable bodies as needed in the ERC project portwings
(www.portwings.eu)

## Acknowledgement

The author would like to acknowledge the financial support of the European
commission ERC Grant agreement ID: 787675 of the project www.Portwings.eu and
the fruitful discussion and feedback from Dr. Federico Califano and Dr. Ramy
Rashad Hashem.

## References

  * Frankel (2011) T. Frankel, The Geometry of Physics, Cambridge University Press, 2011. doi:10.1017/CBO9781139061377.
  * Ball (1900) R. S. Ball, A Treatise on the Theory of Screws, Cambridge at the University Press, 1900\.
  * Abraham and Marsden (1994) R. Abraham, J. E. Marsden, Foundations of Mechanics, ii ed., addison, 1994.
  * Jerrold E. Marsden (1983) T. J. R. H. Jerrold E. Marsden, Mathematical foundations of elasticity, 1983.
  * Chorin and Marsden (1993) A. J. Chorin, J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, volume 4, Springer New York, New York, NY, 1993. doi:10.1007/978-1-4612-0883-9.
  * Kolev and Desmorat (2021) B. Kolev, R. Desmorat, Objective rates as covariant derivatives on the manifold of Riemannian metrics, 2021. URL: https://arxiv.org/abs/2106.01126. doi:10.48550/ARXIV.2106.01126.
  * de Jong et al. (2019) J. de Jong, J. van Dijk, J. Herder,  A screw based methodology for instantaneous dynamic balance,  Mechanism and Machine Theory 141 (2019) 267–282. doi:10.1016/j.mechmachtheory.2019.07.014.
  * Ceccarelli (2000) M. Ceccarelli,  Screw axis defined by Giulio Mozzi in 1763 and early studies on helicoidal motion,  Mechanism and Machine Theory 35 (2000) 761–770. doi:10.1016/S0094-114X(99)00046-4.
  * Selig (2005) J. Selig, Geometric Fundamentals of Robotics, Springer New York, New York, NY, 2005. doi:10.1007/b138859.
  * Stramigioli et al. (2002) S. Stramigioli, B. Maschke, C. Bidard,  On the Geometry of rigid-body motions: the relation between Lie groups and screws,  Journal of Mechanical Engineering Science 216 (2002) 13–23.
  * Noll (1978) W. Noll,  A General Framework for Problems in the- Statics of Finite Elasticity,  in: G. M. De La Penha, L. A. J. Medeiros (Eds.), Contemporary Developments in Continuum Mechanics and Partial Differential Equations, volume 30 of North-Holland Mathematics Studies, North-Holland, 1978, pp. 363–387. URL: https://www.sciencedirect.com/science/article/pii/S0304020808708727. doi:https://doi.org/10.1016/S0304-0208(08)70872-7.
  * Stramigioli and Bruyninckx (2001) S. Stramigioli, H. Bruyninckx, Geometry and Screw Theory for Robotics, 2001\.
  * Stramigioli (2001) S. Stramigioli, Modeling and IPC control of interactive mechanical systems — A coordinate-free approach, volume 266 of Lecture Notes in Control and Information Sciences, Springer London, London, 2001. URL: http://www.springerlink.com/index/10.1007/BFb0110400. doi:10.1007/BFb0110400.
  * Harvey Lipkin (1985) Harvey Lipkin, Geometry and mappings of screws with applications to the hybrid control of robotic manipulators, Ph.D. thesis, University of Florida, 1985.
  * Stramigioli and Bruyninckx (2001) S. Stramigioli, H. Bruyninckx,  Geometry of dynamic and higher-order kinematic screws,  in: Robotics and Automation, 2001. Proceedings 2001 ICRA. IEEE International Conference on, volume 4, IEEE Robotics and Automation Society, Seoul, Korea, 2001, pp. 3344–3349. doi:10.1109/ROBOT.2001.933134.