Project supported by the National Natural Science Foundation of China (Grant No. 11502071) and the Special Research Project of Beijing Information Science and Technology University, China.
Abstract
We investigate the perturbation to discrete conformal invariance and the adiabatic invariants of Lagrangian systems. A variational algorithm is proposed for a system subjected to the perturbation quantities. The discrete determining equations of the perturbations to conformal invariance are established. For perturbed Lagrangian systems, the condition of the existence of adiabatic invariant is derived from the discrete perturbation to conformal invariance. The numerical simulations demonstrate that the variational algorithm has the higher precision and the longer time stability than the standard numerical method.
The adiabatic perturbation theory is utilized to give an approximate description of the dynamics of a system containing fast and slow variables.[1] If systems have no exact invariants, scholars focus on adiabatic (approximate) invariants of the perturbation systems.[2,3] Kruskal[4] and many others in later years[5,6] dealt with adiabatic invariants of the Hamiltonian systems. Djukić[7] and Cveticanin[8] developed the theory leading to adiabatic invariants of non-conservative systems which cannot be described by Hamiltonian canonical equations. The theory of the symmetries and conserved quantities gives a simpler method to solve the equations of dynamical systems.[9] The perturbation to the symmetries and adiabatic invariants develops rapidly because of the close relations between the integrability and the variations. As a whole, there are three types of perturbations to symmetries: the perturbation to Noether symmetries,[10–12] the perturbation to Lie symmetries,[13,14] and the perturbation to Mei symmetries.[15,16] Conformal invariance is built on the scale invariance, the translation invariance, the rotational invariance, and a variety of interactions.[17–21] Perturbation to conformal invariance is a modern approach to find the adiabatic invariants for dynamical systems. The solutions of the determining equations of the perturbation to conformal invariance should satisfy both the conformal invariance and the additional restriction equations. Although it can provide a new integral method, there has been no report on the perturbation to conformal invariance and the adiabatic invariants of dynamical systems.
The discrete equations serve as fundamental mathematical models in mechanics. The discrete symmetries offer the exact solutions for the discrete equations of dynamical systems through the discrete Noether theorem.[22] The Lie symmetries[23,24] and the Mei symmetries[25] also provide the exact invariants of the systems. Theory of perturbations to these three types of symmetries yields methods of getting the solutions of the difference equations. However, theory of perturbation to conformal invariance has not been proposed to solve discrete dynamical systems. The present work focuses on the perturbation to conformal invariance and the adiabatic invariants for discrete Lagrangian systems. Section 2 presents the discrete Lagrangian equations according to the continuous variables discretization. The definition of conformal invariance of the discrete Lagrangian systems is reviewed. The discrete determining equations of conformal invariance are established. Section 3 defines conformal invariance for discrete Lagrangian systems subjected to the perturbation quantities. Section 4 derives exact invariants directly from the conformal invariance of the systems without perturbation. The discrete perturbed Lagrangian systems and the z-th order adiabatic invariants are obtained based on the symmetrical transformations of the initial systems. Section 5 presents a numerical example to demonstrate the application of the proposed approach.
2. Conformal invariance of discrete Lagrangian systems
Consider the space of sequence (t, q), a mechanical system whose configurations are determined by one independent variable t and dependent variable q = (q1, q2,…,qn). Let us rewrite a finite-difference functional in the form
with spacing h+ = t+ − t, and . Let us take a variation of the difference functional (1) along some curve qs = ψs (t) at some point (t,qs). The variation will affect only two terms in the sum of Eq. (1),
with spacing h− = t − t−, and . Thus, we obtain the following expression for the total variation of the difference functional:
For Eq. (2), we have
with and . Substitute Eqs. (4) and (5) into Eq. (3) yields
The total variation of the difference functional (1) satisfies ΔL = 0. Considering the independence of Δqs and Δt, we have
where , is the right difference derivative in the lattices space . Expressions (7) and (8) are the equations for the difference Lagrangian systems as was reported in Ref. [26]. To consider the discrete Lagrangian equations on the uniform difference mesh h+ = h− = h, equations (7) and (8) become
Equations (9) and (10) require the discrete equations of the Lagrangian systems on the uniform difference mesh. Note that equations (9) are the first order difference equations, which are the difference Lagrangian equations. Equation (10) defines the lattice on which the Lagrangian equations are discretized, which is the energy equation. In the continuous limit, the lattice equation disappears, and both of them become the differential Euler equation.
The variation of the functional (1) along the orbit of a group is generated by the operator
where ξ00 = ξ00 (t,qs) and ηs0 = ηs0 (t,qs). The prolongation of the Lie group operator to the neighboring points and is as follows:
where , , , and .
The conformal invariance and conserved quantities for discrete Lagrangian systems under the infinitesimal transformation of Lie groups have been studied in Ref. [27]. Here we introduce this straightforward method of the conformal invariance of Mei symmetry when the perturbation exists. Mei symmetry (the form invariance) is a new type of symmetry theory proposed by Mei and his collaborators, relating to the form invariance of differential equations under dynamical transformations.[28,29] For the Lagrangian , the determining equations of Mei symmetry of the discrete Lagrangian systems are
The expanded forms of the determining equations of conformal invariance are given as follows.
3. Perturbation to conformal invariance of the discrete Lagrangian systems
Suppose that the discrete Lagrange systems are perturbed by small quantities
The total variation of the difference functional (1) satisfies
where δqs is the discrete virtual placement satisfying
Substituting Eqs. (21) and (6) into Eq. (20), we have
We obtain the following expression for the variation of the difference functional:
The equations of the discrete perturbed Lagrangian systems on the uniform difference mesh become
For the discrete perturbed Lagrangian systems, the original symmetries and the invariants of the systems may vary because of the action of εWs,d. Assume the variation is a perturbation based on the symmetrical transformation of the initial system, then ξ0 = ξ0 (t,qs) and ηs = ηs(t,qs), which denote the new generators after being perturbed and can be expressed as
The prolongation of the Lie group operators for the discrete perturbed system can be written as
where
According to the definition of the Mei symmetry, the functions Ls,d and become and . Expanding the functions and , substituting them into Eqs. (25) and (26), and ignoring high order small quantities, we can obtain the determining equations of Mei symmetry of the discrete Lagrangian systems
In the following definitions and theorems, the generators are the conformal invariance of Mei symmetry because the generators’ vector (29) is the Mei symmetry firstly.
4. Adiabatic invariants of discrete Lagrangian systems
The Noether theorems state the relationship between the conservation laws and the symmetries of the dynamical systems.[9–11] For the discrete Lagrangian systems, the discrete analogues of Noether theorem for difference equations were reported in Refs. [26] and [30]. For this perturbed system on uniform lattice, the new generators (27) and (28) satisfy
After being perturbed, the gauge function comes into
Substituting Eqs. (27), (28), and (40) into Eq. (39), we have
when m = 0, the condition Ws,d = 0. If the left-hand side of Eq. (41) equals zero, i.e.,
which is called the discrete Noether identity. Here the right-hand side of Eq. (41) equals zero, based on Eqs. (25) and (26), the discrete versions of the conservation law are
The discrete equations (43) are called the difference version of Noether conservation laws associated with the perturbed Lagrangian systems.
The discrete Noether identity (42) is the sufficient condition for obtaining adiabatic invariants (43). We need to have the appropriate Lie group operators to satisfy them, as well as make sure that the adiabatic invariants are non-trivial.
The definition of discrete adiabatic invariants for the Lagrangian systems is obtained as follows.
Based on Theorem 4 and Definition 3, Noether theorems for the discrete Lagrangian systems subjected to the perturbation quantities are obtained as follows.
5. An example
As motivation for the present work, consider a classical mechanical system with one degree of freedom, whose motion is given by the Newton equation
The integration of Eq. (50) provides the time dependence of the coordinate x(t) of a mass m = 1 particle in the potential V(x). Suppose the systems are subjected to the perturbation forces Fd = εW = −ε(dx/dt)2. The Lagrangian function is . In the case of the independent variable t = (t,t+,t++,…) and depended variables q = (q,q+,q++,…), the discrete force is .
When the systems do not have the perturbation forces, the difference Euler–Lagrangian equation can be written as Eqs. (9) and (10). The prolongation of the Lie group operators for the discrete system is , the conformal factor is according Theorem 4, the conserved quantity is , which is the conservation of angular momentum. Equations (9) and (10) describe the motion of the system away from the impact point by implicitly defining a map (q−,q) ↦ (q,q+) and are also the variational numerical algorithm. In order to implement the variational integrator for the harmonic oscillator, equation (9) can be written as . By taking b = ∂Ld/∂q for each k, this equation is simply , together with the next update b+ = −∂Ld/∂ q+. Solving the implicit equation for q+ and then evaluating the explicit equation that gives b+, we update (q,b) to (q+,b+). More numerical algorithms show that the discrete variational methods have good performances both on the accuracy and long-term tracking.[31,32] In Fig. 1, we can obtain more precise numerical results than those from the standard RK4 for this free harmonic oscillator under the initial conditions (q1, b1) = (0.01, 0.01). The variational algorithms demonstrate the high precision and the long time stability with comparisons to the standard method.
Fig. 1. Error in the energy for free harmonic oscillator versus time.
When adding the perturbation quantities, the difference Euler–Lagrangian equation of the systems can be written as
The difference equations (51) and (52) are also the discrete variational algorithms for this system. Then we can check the variational invariance of Eqs. (51) and (52) with regard to the symmetry of Lagrangian equation
One can readily verify the invariance of the functions Ls,d and εWs,d, that is
The conformal factor is . We have the solution of Eq. (42) with the prolongation of the Lie group operators . From the adiabatic invariants (43), we have
with the gauge function .
We implement the discrete equations (51) and (52) as numerical integrators of the perturbed Lagrangian systems. Figure 2 shows the adiabatic invariants (56) with different coefficients of ε under the initial conditions (q1, b1) = (0.01, 8). As we go forward in time, the perturbed parameter ε keeps on increasing, leading to a decrease in accuracy.
Fig. 2. Adiabatic invariants in Eq. (56) versus time.
6. Conclusions
We construct the conformal invariance perturbation and the adiabatic invariants of the discrete Lagrangian systems. The discrete equations of the perturbed Lagrangian systems are obtained utilizing the discrete variational approach. The discrete conformal invariances of both the free Lagrangian systems and the perturbed ones are proposed respectively. The adiabatic invariants of the discrete perturbed Lagrangian systems are obtained based the discrete Noether theorem of the systems. The discrete Noether theorem of the perturbed Lagrangian version gives the conditions of finding first integrals when their symmetries are known. The discrete conformal invariance and the numerical computation of the systems demonstrate the following points. (i) The conformal invariance proposes one way of searching the solutions of discrete perturbed Lagrangian mechanical systems. (ii) The variational algorithm is a reasonable structure-preserving scheme for analyzing the dynamical behaviors of the perturbed mechanical systems.
