Action principles for hydro- and thermo-dynamics
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or Special Issue Dedicated to the Memory of V.G. Kadyshevsky
Action Principles for Hydro- and Thermo-dynamics1 Christian Fronsdal Department of Physics, University of California los Angeles, USA e-mail: [email protected] Received October 16, 2016
Abstract—A Hamiltonian formulation of hydrodynamics is well known for the case of purely irrotational flows and it now exists in a more general case as well. The minimal extension of the action principle is obtained from two axioms: 1. That the number of independent degrees of freedom be 4, as in standard hydrodynamics. 2. That the equation of continuity must be one of the Euler-Lagrange equations, and that it allows for vorticity. Applications include: 1. Couette flow with a new criterion for the breakdown of laminar motion. 2. A rotating source for Einstein’s equation that respects the Bianchi identity. 3. A new approach to the electromagnetism of fluids. 4. A rigorous virial theorem for fluids. 5. A critique of the current state of the theory of atmospheres. DOI: 10.1134/S1063779617020046
1. INTRODUCTION Here is a quotation from John von Neumann. Lamenting the fact that “hydrodynamical problems, which ought to be considered relatively simple, offer altogether disproportionate difficulties”, he says “the true technical reason appears to be that variational methods have … hardly been introduced in hydrodynamics”. And he adds: “It is well known that they could be introduced, but what I would like to stress is that they have not been used to any practically important scale for calculations in that field” (von Neumann, 1945). As to the last remark, we now know why that is so. A bit of history will help us appreciate the importance of Action Principles. 1.1. Bernoulli and Maupertui Hydrodynamics owes much to Bernoulli, including the Bernoulli equation of hydrodynamics, better known in the limited form of the hydrostatic condition D (v ) (1.1) + p + f = 0, ρ Dt in which ρ is a density, v a velocity, p the pressure and f is any external force. Bernoulli was the mentor of Maupertui. 1 The article is published in the original.
Maupertui’s accomplishments were many, but here we are interested in his formulation of dynamics as a Principle of Least Action. Already in 1741 he had shown that a particle at rest in a field of force derived from a potential finds itself at a minimum of the potential. This relates to other phenomena where minima play a role and it was not a radical idea. But three years later he was able to extend the concept to a statement about bodies in motion. This was highly non-triv ial, as it involved the expression mv 2 2 for the kinetic energy, that had not yet been recognized as having any particular significance. More important, his new action principle is a statement about motion, about the path taken by the system with the passing of time. As to why we admire this formulation of dynamics it is surely, in part, for the same reason that excited Maupertui to lyrical and even religious heights: the sheer beauty of the concept. Aesthetic co
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