Equivalence of energy methods in stability theory
Abstract
We will prove the equivalence of three methods, the so called energy methods, in order to establish the stability of an equilibrium point for a dynamical system. We will illustrate by examples that this result simplifies enormously the amount of computations especially when the stability cannot be decided with one of the three methods.
MSC: 37C10, 37C75.
Keywords: dynamical systems, stability theory.
1 Introduction.
Let be a smooth manifold and
(1.1) 
be a dynamical system on given by the vector field and suppose is an equilibrium state for (1.1), i.e. . The problem of nonlinear stability of equilibrium states is a very old one and the most know and remarkable results were obtained by Lyapunov [5]. They are based on finding what is called a Lyapunov function such that:


, for

, where is the derivative of along the trajectories of (1.1).
In practice it is sometimes very difficult to find such a function. In many situations one can use constants of motion as Lyapunov functions, i.e. functions such that . This was extensively used in the context of HamiltonPoisson systems where the Hamiltonian and the Casimirs of the Poisson structure are constants of motion. The methods for studying stability using constants of motion are the so called energy methods. The most general results using this methods for establishing stability can be found in [9] and [6]. Since in the present paper we are discussing local nonlinear stability we can replace, by considering a coordinate chart around the equilibrium , the manifold with , where is the dimension of .
In 1965 Arnold [1] gives the following criteria for determining nonlinear stability for an equilibrium point of (1.1).
Theorem 1.1
Later, in 1985, Holm, Marsden, Ratiu and Weinstein [4] give another method for establishing stability of an HamiltonPoisson system, the so called EnergyCasimir method.
Theorem 1.2
The above result has also an infinite dimensional analogue for HamiltonPoison systems on Banach spaces, see [4].
Studying the stability of relative equilibria, in 1998, Ortega and Ratiu [7] obtain, as a corollary of their results about stability of relative equilibria, the following theorem.
Theorem 1.3
The aim of our paper is to prove the equivalence of these three methods. This shows that when is an equilibrium point for (1.1) and we choose as a set of constants of motion, if we conclude stability of with one of the methods, then the other two will also give stability of . Thus we can choose the most convenient method from the computational point of view. Since computations can become cumbersome in some examples it is important to know that if we cannot conclude stability of using the set of constants of motion with one of the methods, then we cannot conclude stability of by applying the other two methods using the same set of constants of motion.
2 Equivalence of the three methods
In order for the paper to be selfcontained we will start by proving Arnold’s result on stability since in his original paper [1] the proof was omitted. In order to do this we need the following preliminary results which will play a crucial role in all that follows.
We will begin by establishing the notations and conventions to be used throughout this paper. A vector will be considered as a column vector or a matrix. Its transpose will be a row vector or a matrix.
Let be a real valued function. The gradient of at a point is defined as the column vector
If is a vector valued map, then it will be represented as a column vector of its component functions , namely
If , then we introduce the notation
where is a matrix which has as columns the gradient vectors . Note that the transpose matrix is the Jacobian matrix of at the point .
Let be a real valued function and . We will use the following notations,
For the proof of Theorem 1.1 we will need the following result that can be found in references [2] and [8] .
Proposition 2.1
Let be a symmetric matrix and a positive semidefinite symmetric matrix. We assume that
for all , satisfying . Then there exists a scalar such that
Proof. We will prove by contradiction. Then for every integer , there exists a vector with such that:
(2.1) 
The sequence is bounded and consequently it has a subsequence, that we will denote also by , converging to a vector with . Taking the limit in (2.1) we obtain
(2.2) 
Since
the inequality (2.2) implies that converges to zero and hence .
It follows from the hypothesis that and this contradicts (2.2).
Let be an equilibrium point for the dynamic (1.1) and let be a set of constants of notion for the dynamic (1.1). We define the following quadratic form,
(2.3) 
We have the following characterization for the vector subspace defined in Theorem 1.1.
Lemma 2.1
Proof.
The hypothesis (i) implies that . As a consequence of the hypothesis (ii) and Proposition 2.1 we can find such that and thus for in a small neighborhood of the equilibrium point .
Let us define now the function by the following relation,
It is easy to see that is a Lyapunov function and consequently via Lyapunov’s theorem the equilibrium state is nonlinear stable.
The proofs of Theorem 1.2 and Theorem 1.3 can be found in the original papers [4] and [7]. They are also based on finding a corresponding Lyapunov function.
Now we will prove the main result of this paper.
Theorem 2.2
Proof. ”” Assume that the hypotheses of Theorem 1.1 hold. Consider the following functions , , for and arbitrary for the moment, and given in Theorem 1.1. As in the proof of Theorem 1.1, the conditions (i) and (ii) of Theorem 1.1 imply the conditions (i) and (ii) of Theorem 1.2 for a certain given by Proposition 2.1.
”” This is obvious since positive or negative definiteness on the whole space implies positive or negative definiteness on the subspace .
””. Assume that the hypotheses of Theorem 1.3 hold. Let for . It is obvious that condition (i) of Theorem 1.3 implies condition (i) of Theorem 1.1. Also because some of ’s might be zero we have the inclusion . Then
for any .
If we take the second summand will be zero and consequently condition (ii) of Theorem 1.3 implies condition (ii) of Theorem 1.1.
In all of the three methods the stability is decided when a certain matrix is positive or negative definite. Consequently, Arnold’s method seems to be the most economical since it requires definiteness of a smaller matrix than the other two methods.
Next we will discus the situation in which condition (i) of Theorem 1.1 is not satisfied. Or equivalently, when the vectors , are linear independent. Consequently, in a small neighborhood of they generate an integrable distribution whose leaves are the level sets of the map . Eventually after shrinking all the points in are regular points for . There exists a diffeomorphism , where is a small neighborhood of in . Because are constants of motion for the dynamic (1.1) we obtain , where . If are coordinates induced by on from a set of coordinates around then the equations of motion corresponding to the vector field are
(2.4) 
Moreover, and is an equilibrium point for the dynamic generated by the vector field . The above system can be regarded as a bifurcation problem with the bifurcation parameter. We have the following result.
Theorem 2.3
This result was used in [3] for the stability problem of Ishii’s equation. Given the conditions of the above theorem it is enough to study the stability of a dynamical system that has fewer variables. Nevertheless, the problem is not free of difficulties since one has to find a set of adapted coordinates around for the local fibration generated by the map .
3 Examples
3.1 The free rigid body
Theorem 2.2 asserts that if stability is obtained with one of the methods, then it can be obtained with the other two as well. Indeed, let us consider the Euler momentum equations:
where . Then is an equilibrium point and
We study the stability of , by using Arnold’s method. Let , then iff . Also
and
. This shows that , is nonlinear stable.
Next, we will prove the same stability result using the other two methods. We begin with the EnergyCasimir method. Let . The first variation is given by
Then is equivalent with
is positive definite iff .
We can take
For OrtegaRatiu’s method we can take the same constant of motion used for applying Arnold’s method, i.e. .
3.2 Lorenz five component model
We will show in this example that if the stability of an equilibrium point cannot be decided with one of the three methods then it cannot be decided with the other two either. This is what Theorem 2.2 is predicting. It simplifies enormously the computations in the sense that if we do the computations using one of the methods and obtain that the stability cannot be decided, then it is useless to do the computations using the other two methods and the same set of constants of motion.
To illustrate this, we will take the example of Lorenz five component model. The equations are
where , , is an equilibrium point and , and are constants of motion.
We try to apply Arnold’s method. Take . Then is impossible for any . We have another possibility for choosing a constant of motion. Let . Then iff . Also
and
It is easy to see that is not definite.
Now we try to apply the EnergyCasimir method. Let . Then is impossible for any . We take the other possibility, namely . Then we have
Consequently iff . Also
which is not definite.
Finally we will try to apply OrtegaRatiu’s method. Let . We have that iff and then . Also
and consequently is not definite for any choice of .
References
 [1] V. Arnold; Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid, Doklady, tome 162, no 5 (1965), 773777.
 [2] D. Bertsekas; Constrained optimization and Lagrange multiplier metods, Academic Press, 1982.
 [3] P. Birtea, M. Puta ; On Ishii’s equation , C. R. Math. Acad. Sci. Paris, vol 341, no 2 (2005), 107111.
 [4] D. Holm, J. Marsden, T. Ratiu, A. Weinstein; Nonlinear stability of fluid and plasma equilibria, Physics Reports, vol 123, no 1 and 2 (1985), 1116.
 [5] A.M. Lyapunov; Probléme Générale de la Stabilité du Mouvement, Kharkov 1892. French translation in AA. Fac. Sci. Univ. Toulouse 9, 1907; reproduced in Ann. Math. Studies, 17, Princeton University Press 1949.
 [6] J.P. Ortega, V. PlanasBielsa, T. Ratiu; Asymptotic and Lyapunov stability of constrained and Poisson equilibria, J. Differential Equations, 214 (2005), 92127.
 [7] J.P. Ortega, T. Ratiu; Nonlinear stability of singular relative periodic orbits in Hamiltonian systems with symmetry, J. Geom. Phys., 32 (1999), 160188.
 [8] G.W. Patrick; Relative equilibria in hamiltonian systems: the dynamic interpretation of nonlinear stability on a reduced phase space, J. Geom. Phys., 9 (1992), 111119.
 [9] G.W. Patrick, M. Roberts, C. Wulff; Stability of Poisson equilibria and Hamiltonian relative equilibria by energy methods, Arch. Rational Mech. Anal., 174 (2004), 36–52.
P. Birtea
Departamentul de Matematică, Universitatea de Vest,
RO–1900 Timişoara, Romania.
Département de Mathématiques de Besançon, Université de
FrancheComté, UFR des Sciences et Techniques, 16 route de Gray,
F–25030 Besançon cédex,
France.
Email:
M. Puta
Departamentul de Matematică, Universitatea de Vest,
RO–1900 Timişoara, Romania.
Email: