Euler-Poisson equations

The Euler-Poisson Equations Describing the Motion of a Heavy Rigid Body around a Fixed Point... 

These equations are called the Poisson equations. The complete system of all six differential equations is called the Euler-Poisson equations. Their integration is a rather nontrivial problem. 

It is seen that if the solutions of the Euler-Poisson equations... 

These integrals are written down in the coordinates of the moving system (x, y, z). Hence, for a complete integration of the Euler-Poisson equations it suffices to find two more independent first integrals. 

We omit here the explicit expression because we will not need it hereafter. Thus, the general case is reduced to the case of an equation for two variables, which is integrated using the above technique. Now go back to the analysis of the Euler-Poisson equations. Integration of these equations is apparently equivalent to integration of the system... 

Therefore, the equation for M is certainly satisfied if we set Af = 1. Thus, we have found one particular solution of this equation, and M =1 may be considered to be precisely the last multiplier of the Euler-Poisson equations. 

In all three cases (Euler, Lagrange,Kovalevskaya) the problem reduces to quadratures. The general solution of the Euler-Poisson equations is expressed in the Euler and Lagrange cases in terms of elliptic functions, and in the Kovalevskaya case in terms of hyperelliptic functions (see, for instance, [156]). 

Comparing these equations with those of Subsection 1.1, we see that they are exactly the Euler-Poisson equations of motion of a heavy rigid body with a fixed point. Since h is a self- conjugate operator, in a certain orthogonal reper connected with the rotating body, it is then reduced to a diagonal form and then its matrix becomes... 

Again, it is always possible to define a gauge where the 00 part of the Einstein equations reduces exactly to the Poisson equation, but at the cost of drastically modifying the mass conservation and the Euler equations. 

Vishik, S. V., and Dolzhansky, F. V. "The analogues of Euler-Poisson and magnetic hydrodynamic equations connected with Lie groups. Dokl. Akad. Nauk SSSR 238 (1978), No. 5, 1032-1035. 

It has been observed by [27, 24] that the equations of motion of a free rigid body are subject to reduction. (For a detailed discussion of this interesting topic, see [23].) This leads to an unconstrained Lie-Poisson system which is directly solvable by splitting, i.e. the Euler equations in the angular momenta ... 

Theorem 4.2.6. Let p G G be a linear operator, on a semisimple Lie algebra G, self-conjugate with respect to the Killing form. The Euler equation X = [X, pX] is Hamiltonian simultaneously with respect to both Poisson brackets (the element a is a covector of general position), and, a if and only if p [Pg.217]

Proposition 4.2.3. The Euler equations deSned by the self-conjugate operator C G G such that C = Ci Q C2, Ci K K,C2 P P will be Hamiltonian simultaneously with respect to the Poisson brackets of Lie algebras dual in the sense of Cartan if and only if the operator C2 is scalar. 

Compatible Poisson brackets on Lie algebras were analyzed in the paper by Reyman [117], where such brackets appeared from infinite-dimensional graduated Lie algebras and were applied to the study of the various generalizations of Toda chains. In the same paper [117], Reyman pointed out the Hamiltonian property of the Euler equations for the shifts of invariants of semisimple Lie algebras indicated earlier in the paper [247]. 

Application of Hamilton s method in the calculus of variations to an electrostatic field model of a suspension of charged colloidal particles with selection of a Lagrangian to yield Poisson s law as the Euler equation has led to the following results (1)... 

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