Euler equilibrium points

The curve of constant ratio g/v represent the natural prolongation of Hill s curves with continuity between the corresponding Figures as Figures 3-5. The two minimums correspond to g/v = 1 at the equilateral Lagrangian points and the three collinear saddle points are the usual Euler equilibrium points. 

The eigenvalues are ico. The reason a method like Euler s method can never perform well for molecular dynamics is that molecular dynamics is a Hamiltonian system and at the bottoms of basins on the energy surface, which correspond to stable centers, we expect all the eigenvalues of a local linearization of the problem to be purely imaginary. The stability condition always fails to hold and, for Euler s method, z grows exponentially rapidly away from the equilibrium point. 

As an alternative, consider the Backward Euler method q + = q + hp +i, p +i = p — hq +i. Then it is easily shown that both eigenvalues of the matrix lie in the interior of the unit disk, and hence all solutions of the recurrence relation tend to the origin with increasing n (the origin is an attractive equilibrium point). In this case, all densities evolve toward the Dirac distribution centered at the origin (5[ ](5 p]. The only (distributional) solution of Cjp = 0 is again 3[( ]5[p], which in this case is attractive. 

The nonlinear study of bifurcations of the elastic equilibrium of a straight bar involves, in a way, a change of the physical point of view, mostly due to the mathematical difficulties related to the direct approach of the Bernoulli-Euler (B.-E.) equation. This aspect gave rise to various models describing the same phenomenon, such as Kirchhoff s pendulum analogy [1], as well as to different methods of calculus, such as Thompson s potential energy method [2], [3]. In this paper, we use the linear equivalence method (LEM) to a B.-E. type model, thus deducing an approach for the critical and postcritical behavior of the cantilever bar. 

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