Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics

Abstract

The incompressibility constraint for fluid flow was imposed by Lagrange in the so-called Lagrangian variable description using his method of multipliers in the Lagrangian (variational) formulation. An alternative is the imposition of incompressibility in the Eulerian variable description by a generalization of Dirac’s constraint method using noncanonical Poisson brackets. Here it is shown how to impose the incompressibility constraint using Dirac’s method in terms of both the canonical Poisson brackets in the Lagrangian variable description and the noncanonical Poisson brackets in the Eulerian description, allowing for the advection of density. Both cases give the dynamics of infinite-dimensional geodesic flow on the group of volume preserving diffeomorphisms and explicit expressions for this dynamics in terms of the constraints and original variables is given. Because Lagrangian and Eulerian conservation laws are not identical, comparison of the various methods is made.

A geometric look at MHD and the Braginsky dynamo

“…For the incompressible case one drops the internal energy e(ρ, s) and in its place uses a Lagrange multiplier to enforce that the flow map conserves volumes, φ * µ = µ. The theory can be developed further from Lagrangian to Hamiltonian mechanics, using a non-canonical Poisson bracket as discussed in Morrison (1982Morrison ( , 1998, Morrison, Andreussi & Pegoraro (2020) and reviewed in Webb (2018), with extensions to relativistic MHD (D'Avignon, Morrison & Pegoraro, 2015) and two-fluid MHD (Lingam, Miloshevich & Morrison, 2016).…”

Cosmological Vlasov–Poisson equations for dark matter

“…The matter density contrast appearing in the Eq. ( 6) can be exactly expressed through a mass conservation law using the Dirac delta (3) D (Matsubara 2008;Taylor and Hamilton 1996;McDonald and Vlah 2018;Morrison et al 2020), here for the case when → 0 , initially, which, roughly speaking, may be interpreted as the continuum limit…”

Cosmological Vlasov-Poisson equations for dark matter: Recent developments and connections to selected plasma problems

“…Nonetheless, from here on we will mostly work with this simplified case of an Einstein-de Sitter model to avoid unnecessary cluttering of the expressions. The matter density contrast appearing in the equations ( 6) can be exactly expressed through a mass conservation law using the Dirac-delta 𝛿 (3) D [15,20,21,22], here for the case when 𝛿 → 0 initially,…”

“…where 𝝃 ★ = 𝜕 𝜏 𝝃 ★ (𝒒, 𝜏)| 𝜏=𝜏 ★ ; see [14,44,45,46] for complementary methods applied to the one-dimensional case. Note that the re-occurring terms W and M in (22) are a priori unknowns for times 𝜏 > 𝜏 ★ , since they are functions of the unknown post-shell-crossing displacement 𝝃 (cf. equations 14-15).…”

“…Turning back to the theoretical solution scheme for the Lagrangian master equations (22), the leading-order behaviour of the involved source terms shortly after shell-crossing time 𝜏 ★ is given by W (𝒙(𝒒, 𝜏)) W (𝒙 • (𝒒, 𝜏)) and M (𝒙(𝒒, 𝜏)) M (𝒙 • (𝒒, 𝜏)), which is, strictly speaking, exact at 𝜏 = 𝜏 ★ , and approximatively thereafter, since 𝒙 • (𝒒, 𝜏) = 𝒒 + 𝑛 max 𝑛=1 𝜻 (𝑛) (𝒒) 𝜏 𝑛 constitutes only a solution of Vlasov-Poisson until 𝜏 ★ .…”