Geometric mechanics at eleven parameters: the sixteen as a top, a fluid and a plasma
Abstract
Souriau's mechanics makes a classical body a coadjoint orbit of its symmetry group, and the eleven-parameter relativity closes the eleven quantities of a body into the sixteen entries of an element of u(2, 2). This paper states the mechanics of the sixteen at three sizes. As a top, the coadjoint orbits of U (2, 2) are counted by the Jordan type of the element: twelve-dimensional for a regular one, ten, eight, six and zero on the degenerate ones, six for the twistor elements. Through the moment of a body they are twelve for a massive spinning body, eight and nilpotent for a spinless one, ten for a massless body described with a point and six for Penrose's twistor. The Lie-Poisson equations with a linear Hamiltonian are the flows of the charges on the moment. With Manakov's inertia they are an integrable top with a Lax pair, whose rotations are linearly stable with closed-form frequencies and nonlinearly stable on an open set of inertias by the energy-Casimir method. Its boosts are unstable, the runaway of a non-compact orbit that no bound of the algebra limits. As a fluid, the fluid element of the eleven is a semidirect-product Lie-Poisson system with the thermal element as its conjugate. The conformal fluid has for equilibria the one-parameter subgroups of the eleven, of three types, elliptic, hyperbolic and parabolic, with the conformal soliton, the Milne expansion and the static fluid as representatives and mixed classes beside them. All fifteen conformal currents are conserved on every conformal solution. Spin hydrodynamics is the Lie-Poisson fluid of the sixteen with a closure on the Lorentz block, and the polarisation of hyperons a memory of hadronisation or its local value according to a hadronic rate. The quantised vortices of a rotating superfluid are the discrete form of the rotation multiplier inside the light cylinder, with the vortex counts of the pulsars. The Navier-Stokes equation is the dissipation that leaves the orbits, against the double bracket that keeps them, with its relativistic form, the floor that the bound puts under a collision time, and its cascade as a scaling of Euler's equation that is not the dilatation of the eleven. As a plasma, the Vlasov-Maxwell system is a Lie-Poisson system whose Gibbs state on the orbit exists for the multipliers the orbit admits. Taylor's relaxation at fixed magnetic helicity is read as an analogue of the helicity block, and the chiral magnetic and vortical effects of the quarkgluon plasma as the responses of that block for the quarks. The gluons cannot carry that block, their helicity being no conserved charge, and free radiation cannot carry it as a potential. The paper claims no new equation of motion, no transport coefficient and no number; every number it computes is known. What it claims is the u(2, 2) instance of a known mechanics and the vocabulary of the thermal element on it.
Related
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BibTeX
@misc{reynaud2026geometric,
title = {{Geometric mechanics at eleven parameters: the sixteen as a top, a fluid and a plasma}},
author = {Patrick B. Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05766030v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric mechanics at eleven parameters: the sixteen as a top, a fluid and a plasma
AU - Patrick B. Reynaud
PY - 2026
DA - 2026/09/27
PB - HAL
UR - https://hal.science/hal-05766030v2
AB - Souriau's mechanics makes a classical body a coadjoint orbit of its symmetry group, and the eleven-parameter relativity closes the eleven quantities of a body into the sixteen entries of an element of u(2, 2). This paper states the mechanics of the sixteen at three sizes. As a top, the coadjoint orbits of U (2, 2) are counted by the Jordan type of the element: twelve-dimensional for a regular one, ten, eight, six and zero on the degenerate ones, six for the twistor elements. Through the moment of a body they are twelve for a massive spinning body, eight and nilpotent for a spinless one, ten for a massless body described with a point and six for Penrose's twistor. The Lie-Poisson equations with a linear Hamiltonian are the flows of the charges on the moment. With Manakov's inertia they are an integrable top with a Lax pair, whose rotations are linearly stable with closed-form frequencies and nonlinearly stable on an open set of inertias by the energy-Casimir method. Its boosts are unstable, the runaway of a non-compact orbit that no bound of the algebra limits. As a fluid, the fluid element of the eleven is a semidirect-product Lie-Poisson system with the thermal element as its conjugate. The conformal fluid has for equilibria the one-parameter subgroups of the eleven, of three types, elliptic, hyperbolic and parabolic, with the conformal soliton, the Milne expansion and the static fluid as representatives and mixed classes beside them. All fifteen conformal currents are conserved on every conformal solution. Spin hydrodynamics is the Lie-Poisson fluid of the sixteen with a closure on the Lorentz block, and the polarisation of hyperons a memory of hadronisation or its local value according to a hadronic rate. The quantised vortices of a rotating superfluid are the discrete form of the rotation multiplier inside the light cylinder, with the vortex counts of the pulsars. The Navier-Stokes equation is the dissipation that leaves the orbits, against the double bracket that keeps them, with its relativistic form, the floor that the bound puts under a collision time, and its cascade as a scaling of Euler's equation that is not the dilatation of the eleven. As a plasma, the Vlasov-Maxwell system is a Lie-Poisson system whose Gibbs state on the orbit exists for the multipliers the orbit admits. Taylor's relaxation at fixed magnetic helicity is read as an analogue of the helicity block, and the chiral magnetic and vortical effects of the quarkgluon plasma as the responses of that block for the quarks. The gluons cannot carry that block, their helicity being no conserved charge, and free radiation cannot carry it as a potential. The paper claims no new equation of motion, no transport coefficient and no number; every number it computes is known. What it claims is the u(2, 2) instance of a known mechanics and the vocabulary of the thermal element on it.
ER -