Ehrenfest regularization of Hamiltonian systems

Abstract

Imagine a freely rotating rigid body. The body has three principal axes of rotation. It follows from mathematical analysis of the evolution equations that pure rotations around the major and minor axes are stable while rotation around the middle axis is unstable. However, only rotation around the major axis (with highest moment of inertia) is stable in physical reality (as demonstrated by the unexpected change of rotation of the Explorer 1 probe). We propose a general method of Ehrenfest regularization of Hamiltonian equations by which the reversible Hamiltonian equations are equipped with irreversible terms constructed from the Hamiltonian dynamics itself. The method is demonstrated on harmonic oscillator, rigid body motion (solving the problem of stable minor axis rotation), ideal fluid mechanics and kinetic theory. In particular, the regularization can be seen as a birth of irreversibility and dissipation. In addition, we discuss and propose discretizations of the Ehrenfest regularized evolution equations such that key model characteristics (behavior of energy and entropy) are valid in the numerical scheme as well.

Multiscale Thermodynamics

“…This new requirement plays now a very important role in determining the potentials appearing in ( 29 ). The precise meaning of maximally invariant (or alternatively “quasi-invariant”) used in ( 30 ) as well as the meaning of “appropriately projected” used in ( 33 ) below remains still a part of the pattern recognition analysis of the upper time-evolution that has to be investigated [ 2 , 19 , 20 , 21 , 22 ].…”

“…The output of the reduction is the lower thermodynamics relation: obtained from ( 29 ) in the same way as ( 21 ) is obtained from ( 19 ) (see more in Section 2.2 ) and the vector field (compare with ( 20 )): which, if appropriately projected on the tangent space of the manifold and pushed forward on by the mapping ( 28 ), becomes the vector field generating the time-evolution at the lower level l . In this paper, we limit ourselves only to recalling the main idea behind the Chapman and Enskog analysis (see more in Section 4.1 and in [ 2 , 18 , 19 , 20 , 21 , 22 , 23 ]).…”

“… that is appropriately projected on the tangent space of the manifold (see more details in [ 2 , 19 , 20 , 21 , 22 ]). The reducing entropy is obtained as follows.…”

“…In the context of the reduction of kinetic theory to hydrodynamics, is the Boltzmann entropy. This type of the Chapman–Enskog sequence of reducing entropies that is induced by the sequence of the Chapman–Enskog reduced vector fields was discussed in [ 2 , 20 , 21 , 22 ].…”

Dynamic and Renormalization-Group Extensions of the Landau Theory of Critical Phenomena

“…The reducing time evolution can be either the time evolution taking place in and approaching an invariant (or in most cases a quasi-invariant) manifold that represents in the state space used on the lower level or it can be the time evolution of vector fields taking the vector field generating the upper time evolution to the vector field generating the lower time evolution. The former viewpoint is discussed for example in [ 11 , 12 , 13 , 14 ]. In this paper we follow the second route, discussed in [ 15 ], since on this route we can directly transpose the 2-level equilibrium thermodynamics introduced in the previous section to 2-level rate-thermodynamics.…”