Generalized virial theorem for the Liénard-type systems

A geometrical description of the virial theorem (VT) of statistical mechanics is presented using the symplectic formalism. The character of the Clausius virial function is determined for second-order differential equations of the Liénard type. The explicit dependence of the virial function on the Jacobi last multiplier is illustrated. The latter displays a dual role, namely, as a position-dependent mass term and as an appropriate measure in the geometrical context.


Introduction
The virial theorem (VT) is an important theorem of classical mechanics which has been successfully applied in the last century to a number of relevant physics problems, mainly in astrophysics, cosmology, molecular physics, quantum mechanics and in statistical mechanics. In mechanics, it provides a general equation relating the average over time of the total kinetic energy, T , of a stable system consisting of N particles, bound by potential forces, with that of the total potential energy, V TOT , where angle brackets represent the average over time of the enclosed quantity. The word virial derives from vis, the latin word for force or energy and was introduced by Rudolf Clausius in 1870 [1]. The scalar virial theorem says that kinetic and potential energies must be in balance, whereas the tensor virial theorem [2,3] says that the kinetic and potential energies must be in balance in each separate direction. The scalar theorem is useful for estimating global average property while the tensor virial theorem is useful for relating the shapes of the system. Recently, Li et al [4] considered the hypervirial theorem (HVT) and obtained the hypervirial relations. The HVT and Hellmann-Feynman theorem are shown to provide a powerful method of generating perturbation expansions.
In this paper we consider geometrical mechanics approach and necessity of geometric treatment of VT [5,6] can be argued as follows. The standard VT is based on the transformation properties of kinetic and potential energies under dilations, and therefore is only valid for systems with R n as configuration space. In order to generalize the VT for other systems, the tools of geometric mechanics are used.

Standard approach of virial theorem
Clausius introduced the virial function for a one-particle system for studying the motion of a particle of mass m under the action of a force F . The time evolution of G is given by where we use Newton's second law, F = mẍ. On integrating this expression between t = 0 and t = T and dividing by the total time interval T we find where K(ẋ) = 1 2 m(ẋ ·ẋ) and A is the time average. By time average we mean If either the motion is periodic of period T or the possible values of the function G are bounded, then we take the limit T → ∞: If the potential V is homogeneous of degree k, Euler's theorem of homogeneous functions implies that x · ∇V = kV , thus we obtain by 2K(ẋ) = k V (x) . If E is the total energy, then For harmonic oscillator: and for Kepler equation

Symplectic form, Hamiltonian systems and virial theorem
The pair (M, ω) of a smooth manifold M with a symplectic form ω is called a symplectic manifold. A necessary condition for the existence of a symplectic form ω on M is that M should have even dimension 2n, so for all practical purposes one can imagine M as a generalized phase space. Throughout this paper we assume that M is a smooth manifold. Starting with a Hamiltonian function H , one produces a vector field X H as follows: The flow φ t commutes with X H for any w ∈ C ∞ (M) If we choose f as an observable by integrating both sides from 0 to T we obtain If the function remains bounded in its time evolution, taking the limit when T goes to infinity As an instance, when M = T * R 3 and G = x · p we obtain If the Hamiltonian is given by then Thus, taking the limit of T going to infinity we obtain 2K = x · ∇V .

Virial theorem for Henon-Heiles system
The Hamiltonian of the Henon-Heiles system is given by Three known integrable cases were found by the Painlevé method in [7], Fordy [8] employed the technique connecting the Hamiltonian formalisms of stationary and nonstationary flows to prove that these were the only three integrable cases of the system The bi-Hamiltonian properties have been exhibited only for the first case where the additional first integral is given by and other two cases do not have a second first integral. The virial function Thus, we obtain classical VT The asymptotic invariant polynomial P(x, y) converges rapidly to 2E in both regular and chaotic cases; the only apparent difference between the regular and chaotic cases is the somewhat larger oscillations in the latter case. The convergence of the virial integrals for chaotic orbits is complicated by sticky regions, where an orbit can spend long intervals exploring islands [9].

Conformal Hamiltonian systems and virial theorem
Let (M, ω) be a symplectic manifold, where M is a differentiable manifold endowed with a symplectic form ω. Consider a diffeomorphism φ such that φ * (ω) = kω, where 0 = k ∈ R are called non-strictly canonical transformations. Let the vector field be the generator of a one-parameter group of non-strictly canonical transformations φ * (ω) = k( )ω. Then, there exists a real number a = 0 such that Lie derivative of the dynamical vector field satisfies L ω = aω, with k and a related by k( ) = exp(a ). Note that, in [5] a dynamics given by a Hamiltonian vector field and a non-strictly canonical infinitesimal symmetry were considered. In this study, we are considering the dynamics given by a non-Hamiltonian vector field and also a Hamiltonian vector field X. Procedures are different but the result is almost the same.
If π: E → Q is a vector bundle, the vector field generating dilation along the fibres is However, if satisfies L ω = aω, then aX G + is a locally-Hamiltonian vector field because Let = X f − aX G , then the action of X on the Hamiltonian is given by If we assume that H = K(p) + V (q), we find In this case, the virial expression is given by

Application to Gierer-Meinhardt system
The dynamical vector field of the autocatalytic Gierer-Meinhardt (GM) system [10] satisfies L ω = (c + h)ω. Hence the flow is made of non-strictly canonical transformations with valence e (c+h)t . When we impose c + h = 0, becomes symplectic or Hamiltonian vector field. When restricted to the symplectic case, i.e., c = −h, the associated Hamiltonian is given by with standard Poisson brackets {x, y} = 1 and equations of motion are given bẏ When the manifold is R 2 with coordinates (x, y) and ω = dx ∧ dy, the vector fields whose flow is made of non-strictly canonical transformations arė where H: R 2 → R is the Hamiltonian. The flow has the property φ * ω = e κt ω, and so the symplectic inner product of any two tangent vectors contracts exponentially if κ < 0. If ω is a locally symplectic structure (LCS) then the two LCS ω and ω = e κt ω are conformally equivalent [11]. Thus, the general activator-inhibitor system can be expressed aṡ Given H ∈ C ∞ (M), the vector field (or X c H -usual notation) satisfies So, it turns out that Z = y∂ y .

Virial theorem and the Liénard II system
Given a second-order ordinary differential equation (ODE) we define the Jacobi last multiplier (JLM) M as a solution of the following ODE: In fact, as the dynamical vector field for (1) is (2) is the JLM equation (e.g., see [12,13]).
Assuming (1) to be derivable from the Euler-Lagrange equation of a Lagrangian L, one can show that the JLM is related to the Lagrangian by the following equation: Moreover, in each JLM, M defines in an essentially unique way (i.e., up to addition of a gauge term) a Lagrangian such that (3) holds.
In case of the Liénard II equation where , and one can show that the solution of the JLM is given by Furthermore, it follows from (3) that its Lagrangian is where the potential term The image under the Legendre transformation yields the conjugate momentum so that the final expression for the Hamiltonian is where it is explicitly denoted in terms of the last multiplier M(x) to highlight its role as a position-dependent mass term [14].

Another virial theorem
Given a differentiable function L on T Q we can construct a semibasic one-form θ L ∈ ∧ 1 (T Q) and an exact two-form ω L ∈ ∧ 2 (T Q), given by where S is the vertical endomorphism. The Lagrangian L is said to be regular, then ω L is symplectic. The energy function is The dynamics is given by the vector field X L ∈ X(T Q) such that i(X L )ω L = dE L . If a vector field X ∈ X(T Q) is the complete lift of a vector field Y ∈ X(Q), then Let Y be the vector field in R given by whose complete lift is given by When the vector field Y is such that then L X ω L = aω L and L X E L = aE L .
Such a vector field X is a symmetry of the dynamical vector X L , i.e., The important point is that the function G = i(X)θ L is such that X L (G) = aL and therefore we obtain the virial type result L = 0. In the particular case of a positiondependent mass Lagrangian the condition X(L) = aL implies that the unknown functions ξ and the potential V (x) are to be determined from the following relations: As an example, we consider the vector field X that is the complete lift of the vector field Y generating the dilations in Q = R 3 , given by and the Lagrangian of a harmonic oscillator with k = m, L = 1 2 mv · v − 1 2 mx · x. Here θ L (x, v) = mv dx, then ω L = m dx ∧ dv and the energy function is E L = 1 2 m(v 2 + x 2 ). Therefore, and consequently G = mxv, X(L) being i.e., the particular case a = 2. One can also check Thus VT provides X L G = 2 L = 0.
Liénard-type equation. When operates on G = m(x)ξ(x)v for Liénard II equation it yields eqs (11) and (12), where V (x)/m(x) = g(x). We can determine ξ(x) from (11) to be Using this expression, the potential turns out to be

Virial theorem in quantum world
Let H be the complexified Hilbert space of Lebesgue integrable functions on R 3 with Hermitian inner product The realification of the Hilbert space H (real Banach manifold) is endowed with a natural symplectic structure Given a self-adjoint linear operatorĤ we can define a real function h in H by Therefore, ifĤ is the Hamiltonian of a quantum system, the Schrödinger equation describing time evolution plays the role of Hamiltonian equations for the Hamiltonian dynamical system (H, ω, h), where h(u) = u,Ĥ u , the integral curves of X h satisfẏ Thus h(u) and −iĤ stand for Hamiltonian and Hamiltonian vector field for the quantum system.
Let us consider two Hamiltonian functions f (u) = u,F u and g(u) = u,Ĝu corresponding to two self-adjoint operatorsF andĜ. Suppose that the Hamiltonian of a quantum system is Let G be given by then we obtain the standard quantum virial u,p ·pu − u, (x·∇V (x))u = 0.

Application to quantum Liénard II
In this section, we apply the scheme to quantum Liénard II equation [15]. It has been shown in [16] that the appropriate Hilbert space for the description of position-dependent mass systems is H = L 2 (R, dμ) with dμ = √ m(x) dx. Here (dμ/dx) = √ m(x)(∂/∂x), is the (square root of the) Radon-Nikodym derivative [17] of the measure dμ with respect to the Lebesque measure dx.
Let us recall few basic definitions. Let (X, F ) be a measure space and m be a nonnegative Borel function. Note that is a measure satisfying ν(E) = 0 implies μ(E) = 0. We say μ is absolutely continuous with respect to ν and the function f is called the Radon-Nikodym derivative or density of μ with respect to ν and is denoted by dμ/dν. Symplectomorphism and Radon-Nikodym derivative. The momentum of the Liénard II is given by The transformation (p x , x) to (p μ , μ) satisfies p x dx = p μ dμ or dp x ∧ dx = dp μ ∧ dμ.
The important point is that there are no translations on the measure. If dμ = √ M(x)dx is a measure then the norm of the function ψ is given by The Hamiltonian operator is given bŷ We illustrate our construction using the free particle motion of the Mathews-Lakshmanan system [18] characterized by The Lagrangian function is invariant under the action of the vector field X = √ 1 + λx 2 (∂/∂x) such that the complete lift satisfies X c (L) = 0. This vector field can be seen as a Killing vector field for the metric g = (1 + λx 2 ) −1 dx 2 . It is clear that the natural measure in the real line is not invariant under such vector fields; instead, the only invariant measures are the multiples Thus the multiplier is M(x) = (1 + λx 2 ) −1 .
In order to apply the VT we introduce the generator of the dilationĜ = ξ(x)(∂/∂x) with the property The general solution of ξ(x) is given by

Discussion and outlook
All the classical cases discussed in this paper are finite-dimensional in nature. This can be generalized to infinite-dimensional framework. In 1970, Vlasov et al [19] proved that solutions of nonlinear Schrödinger equation iu t (x 1 , x 2 , t) + 1 2 ∇u + |u| 2 u = 0, u(x 1 , x 2 , t) = u 0 (x 1 , x 2 ) satisfy the so-called virial theorem (also called variance identity), given by We can derive this relation using the following quadratic form associated with a selfadjoint differential operator of first-order: where we have used i∂ t u = − 1 2 u + β|u| 2σ u.
Under the NLSE flow F (u) and (dF /dt) are called Morawetz and virial identity. The term virial identity comes from the analogy to the virial theorem in classical mechanics (for a rigorous proof, see [20]). In this paper, we have given a brief outline of the application of the generalized virial theorem. It would be interesting to study the virial theorem associated with Riemannian manifolds and other Hamiltonian partial differential equations.