# Hamilton-Jacobi Method for Mechanical Systems on Time Scales

Authors: Xiang-Hua Zhai; Yi Zhang
Journal: Complexity
 (2018)

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## Abstract

This paper presents the Hamilton-Jacobi method for integrating the equations of motion of mechanical systems on time scales. We give the criterion and four basic forms of canonical transformation on time scales. Also, various examples are given to illustrate the role played by a generating function in the canonical transformation. By choosing an appropriate generating function, we construct the Hamilton-Jacobi equation on time scales and prove the Jacobi theorem on time scales. An example for an Emden-Fowler type equation is discussed to show the application of the method.


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## Body

## 1. Introduction

As it is well known, the Hamilton-Jacobi equation [1–4], that is an important nonlinear partial differential equation, represents a reformulation of classical mechanics. In addition, the Hamilton-Jacobi method is very useful in integrating differential equations of motion for the holonomic mechanical systems [5–9], the nonholonomic mechanical systems [10, 11], and the nonconservative mechanical systems [12]. The differential equations that can be realized by Hamilton formularization [13–15] can also be solved by the Hamilton-Jacobi method [13, 14]. Moreover, the discrete analogues of the Hamilton-Jacobi equation have been studied by Elnatanov and Schiff [16] and Lall and West [17]. Ohsawa et al. [18] developed the discrete Hamilton-Jacobi theory within the framework of discrete Hamiltonian mechanics. Until now, the Hamilton-Jacobi theory is well understood both from the continuous and discrete points of view, and it has many important applications in optimal control problems, particle physics, fluid mechanics, quantum mechanics, and cosmology. In this paper, the Hamilton-Jacobi method for mechanical systems is studied in version of time scales.A time scale is an arbitrary nonempty closed subset of the real numbers. The calculus of time scales, which has recently attracted a lot of attention, was introduced by Hilger [19] in 1988 to unify continuous and discrete analysis. In terms of a unique formalism, this theory can deal with not only continuous and discrete analysis but also complex processes such as the control systems, impulsive dynamical systems, electromechanical systems, neural network, and economical systems [20–24]. The calculus of variations on time scales that was one of the popular topics has been well studied [25–35]. Inspired by these works, the Euler-Lagrange equations, the Hamilton canonical equations [36–38], and the Birkhoff’s equations [39] were established for mechanical systems on time scales. Besides, the Noether’s theorems [37–42] were established in finding conserved quantities for mechanical systems on time scales. However, it is worth mentioning that the famous Hamilton-Jacobi method also represents an important integration method. It is worth and necessary to study the Hamilton-Jacobi method on time scales.The outline of this paper is as follows: we first present the basic definitions and properties about the calculus of time scales in Section2. In Section 3, we give the Hamilton canonical equations on time scales. Section 4 focuses on the canonical transformation theory on time scales. In Section 5, we construct the Hamilton-Jacobi equations on time scales and prove the Jacobi theorem on time scales. Section 6 gives the conclusions and future directions of research.


## 2. Preliminaries on Time Scales

We begin by introducing the definitions and properties on time scales needed in the sequel. More discussions and proofs can be found in [21, 24].Atime scaleT is an arbitrary nonempty closed subset of the real numbers ℝ. For all t∈T, the forward jump operator σ:T→T is defined by σt=infs∈T:s>t, while the backward jump operator ρ:T→T is defined by ρt=sups∈T:s<t, with infϕ=supT and supϕ=infT.A pointt∈T is called right-dense, right-scattered, left-dense, and left-scattered if σt=t, σt>t, ρt=t and ρt<t, respectively. The graininess function μ:T→0,∞ is defined by μt=σt−t. If T has a left-scattered maximum M, then we define Tk=T−M, otherwise Tk=T.Letf:T→ℝ be a function. Then the delta derivative fΔt of f at a point t∈Tk is defined to be the number with the poverty that given any ε>0, there exists a neighborhood U of t (i.e., U=t−δ,t+δ∩T) for some δ>0 such that
(1)fσt−fs−fΔtσt−s≤εσt−s,foralls∈U.A functionf:T→ℝ is called rd-continuous if it is continuous at right-dense points in T and its left-sided limits exist (finite) at left-dense points in T. The set of rd-continuous functions can be denoted by Crd. The set of differentiable functions with rd-continuous derivative is denoted by Crd1.A functionF:T→ℝ is called an antiderivative of f:T→ℝ provided FΔt=ft holds for all t∈Tk. Then the indefinite integral of f is defined by ∫fτΔτ=Ft+C, where C is an arbitrary constant. The definite integral is defined by ∫abftΔt=Fb−Fa, for all a,b∈T.For delta differentiablef and g, the next formulae hold
(2)fσt=ft+μtfΔt,(3)fgΔt=fΔtgt+fσtgΔt=gΔtft+gσtfΔt,(4)∫abftgΔtΔt=fgb−fga−∫abfΔtgσtΔt,(5)∫abfνtνΔtΔt=∫νaνbft¯Δ¯t¯,where we abbreviate f∘σ by fσ, that is, fσt=fσt.Letf:Λn=T1×T2×⋯×Tn=t=t1,t2,…,tn:ti∈Ti→ℝ be a function. The partial delta derivative of f respect to ti∈Tik is defined as the limit
(6)limsi→tisi≠σitift1,t2,…,ti−1,σiti,ti+1,…,tn−ft1,t2,…,ti−1,si,ti+1,…,tnσiti−si=∂ftΔiti.Let the functionf:T1×T2→ℝ be σ1-completely delta differentiable at the point t0,s0. If the functions φ and ψ have delta derivatives at the point ξ0, then the composite function
(7)Fξ=fφξ,ψξforξ∈Thas a delta derivative at that point which is expressed by the formula
(8)FΔξ0=∂ft0,s0Δ1tφΔξ0+∂fσ1t0,s0Δ2sψΔξ0,where t0,s0=φξ0,ψξ0, σ1φξ0=φσξ0.Remark 1.
IfT=ℝ and Fξ=fξ,ψξ, then (8) means that
(9)FΔξ0=∂fξ0,s0Δ1t+∂fξ0,s0∂sψΔξ0.Remark 2.
If the functionf:T1×T2→ℝ be σ2-completely delta differentiable at the point t0,s0, if the functions ψ and φ have delta derivatives at the point ξ0, then the composite function
(10)Fξ=fψξ,φξforξ∈Thas a delta derivative at that point which is expressed by the formula
(11)FΔξ0=∂fs0,t0Δ1sψΔξ0+∂fσ2s0,t0Δ2tφΔξ0,where s0,t0=ψξ0,φξ0, σ2ψξ0=ψσξ0.
The formulae (8) and (11) constitute a dual pair in which the shift appears at two different places. This is a natural phenomenon in the calculus on time scales such as the product rule (3) has two forms.Let a functionf:T1×T2→ℝ have the mixed delta derivatives ∂2ft,s/Δ1tΔ2s and ∂2ft,s/Δ2sΔ1t in some neighborhood of the point t0,s0∈T1k×T2k. If these derivatives are continuous at the point t0,s0, then
(12)∂2ft,sΔ1tΔ2s=∂2ft,sΔ2sΔ1t.


## 3. Hamilton Canonical Equations on Time Scales

In this section, we present the Hamilton canonical equations on time scales that have been discussed in literatures [36–38].Assuming that the configuration of a mechanical system is determined by generalized coordinateq, the Lagrangian L is Lt,qσt,qΔt, where t∈T, q:a,bT→ℝ, q∈Crd1. The Hamilton action is
(13)Sq⋅=∫abLt,qσ,qΔΔt.The Hamilton principle on time scales can be expressed as(14)δS=0,which satisfies the relationships
(15)δqΔ=δqΔ,δqσ=δqσ,and the boundary conditions
(16)δq∣t=a=δq∣t=b=0.The generalized momentum and the Hamiltonian are(17)p=∂L∂qΔ,H=Hp,qσ,t=pqΔ−L.The Hamilton canonical equations on time scales can be derived from the principle above, that is,(18)qΔ=∂H∂p,pΔ=−∂H∂qσ.


## 4. Canonical Transformations on Time Scales

Now, we consider the transformation from the old variablesq, p to the new variables Q, P
(19)Q=Qp,q,t,P=Pp,q,t,and suppose that (19) is invertible. Undergoing the transformation, the canonical equations (18) on time scales will be transformed into other new equations. However, the new equations may not have the canonical form in general. Among the possible transformations, there is a class of transformations called canonical transformations whose properties are particularly useful in dynamics. That is, the canonical equations (18) are transformed into the new canonical equations
(20)QΔ=∂H∗∂P,PΔ=−∂H∗∂Qσ,where the new Hamiltonian H∗P,Qσ,t is
(21)H∗P,Qσ,t=PQΔ−L.The following criterion gives the condition under which (19) is canonical on time scales.Criterion 1.
If (19) satisfies the condition
(22)pqΔ−PQΔ+H∗−H=ΔFΔt,
then it is a canonical transformation on time scales, whereF is delta differentiable.Proof 1.
Formula (22) can be written as
(23)pΔq−PΔQ+H∗−HΔt=ΔF,suppose that q and p exist with the isochronal variations δq and δp. Then (23) becomes
(24)pδq−PδQ=δF.
Taking the delta derivative of (24) with respect to t, we obtain
(25)pΔδq+pσδqΔ−PΔδQ−PσδQΔ=δFΔ.
Taking the variation of (22), we have
(26)qΔδp+pδqΔ−QΔδP−PδQΔ+δH∗−δH=δFΔ.
By using (15) and subtracting (25) from (26) yields
(27)qΔδp−pΔδq−QΔδP−PΔδQ+δH∗−δH+p−pσδqΔ−P−PσδQΔ=0.
According to (3), (27) becomes
(28)qΔδp−pΔδqσ−QΔδP−PΔδQσ+δH∗−δH=0.
Taking note of (18), we obtain
(29)QΔδP−PΔδQσ−δH∗=0.
Therefore, (20) can be derived from (29).In fact, (22) is also the necessary condition of the canonical transformation on time scales. According to the Hamilton principle on time scales, if both the new variables and old variables satisfy
(30)δ∫abPQΔ−H∗P,Qσ,tΔt=0,(31)δ∫abpqΔ−Hp,qσ,tΔt=0,then (19) is canonical on time scales. From (30) and (31), the two integrands should satisfy the following relation
(32)pqΔ−Hp,qσ,t−PQΔ−H∗P,Qσ,t=ΔFΔt,where F can be an arbitrary function with new variables, old variables, and time.Because the canonical transformation completely depends on the choice of the arbitrary functionF, the function F can be called the generating function. Clearly, F and H∗ can be solved from (23), and the solution of the canonical transformation problem on time scales is not unique. In order to achieve the transformation between the two groups of canonical variables, the generating function F should contain both new variables and old variables [1]. With that case in mind, the four forms of the canonical transformations on time scales are discussed below, and some simple and important examples are given to illustrate the effect of the canonical transformation on time scales and the generating function.Case 1.
The generating function of the first kind has the form(33)F=F1Q,q,t,we have
(34)ΔF=∂F1Q,q,tΔ1QΔQ+∂F1Qσ,q,tΔ2qΔq+∂F1Qσ,qσ,tΔ3tΔt.
Substituting (34) into (23), we obtain
(35)−P+∂F1Q,q,tΔ1QΔQ+p−∂F1Qσ,q,tΔ2qΔq+H∗−H−∂F1Qσ,qσ,tΔ3tΔt=0.
From (35), we can find
(36)P=−∂F1Q,q,tΔ1Q,p=∂F1Qσ,q,tΔ2q,H∗=H+∂F1Qσ,qσ,tΔ3t.Example 1.
If the generating function has the form(37)F1=qQ,then the corresponding canonical transformation gives
(38)p=Qσ,P=−q,H∗=H.
The transformation (38) shows that the new generalized coordinates depend on the old generalized momentum while the new generalized momentum depends on the old generalized coordinates.Case 2.
The generating function of the second kind has the form(39)F=F2P,q,t−QP,combining (39) and (23), we have
(40)Qσ−∂F2P,q,tΔ1PΔP+p−∂F2Pσ,q,tΔ2qΔq+H∗−H−∂F2Pσ,qσ,tΔ3tΔt=0.
From (40), we obtain
(41)Qσ=∂F2P,q,tΔ1P,p=∂F2Pσ,q,tΔ2q,H∗=H+∂F2Pσ,qσ,tΔ3t.Example 2.
If the generating function has the form(42)F2=Pq,then the corresponding canonical transformation gives
(43)p=Pσ,Qσ=q,H∗=H.Case 3.
The generating function of the third kind has the form(44)F=F3Q,p,t+pq.
Similarly, we have(45)P=−∂F3Q,p,tΔ1Q,qσ=−∂F3Qσ,p,tΔ2p,H∗=H+∂F3Q,p,tΔ3t.Example 3.
If the generating function has the form(46)F3=−pQ,then the corresponding canonical transformation gives
(47)qσ=Qσ,P=p,H∗=H.
The transformation (47) shows that the new generalized coordinates and the new generalized momentum are the same with the old ones. Hence, the generating function (46) is corresponding to the identical transformation.Case 4.
The generating function of the fourth kind has the form(48)F=F4P,p,t+pq−PQ.
Similarly, we have(49)Qσ=∂F4P,p,tΔ1P,qσ=−∂F4Pσ,p,tΔ2p,H∗=H+∂F4Pσ,pσ,tΔ3t.Example 4.
If the generating function has the form(50)F4=pP,then the corresponding canonical transformation gives
(51)qσ=−Pσ,Qσ=p,H∗=H.
Compared with (38), (51) is actually the same with (38). Hence, different generating functions can correspond to the same canonical transformation.The canonical transformations expressed by the four kinds of generating functions above are only part of the canonical transformations on time scales. However, it is quite extensive. Ift is not explicitly contained in F1, F2, F3, and F4, then H∗=H, (23) can be written as
(52)pΔq−PΔQ=ΔF.This kind of transformation can be called contact transformation on time scales.


## 5. Hamilton-Jacobi Method on Time Scales

The form of Hamiltonian will be simpler if we choose an appropriate generating function, such asH∗=0. If H∗=0, (20) becomes
(53)QΔ=∂H∗∂P=0,PΔ=−∂H∗∂Qσ=0.By integrating, we have(54)Q=α,P=β,where α and β are constants and α=ασ, β=βσ.In order to achieve the purpose above depends on the generating function we choose. According to the relationship between the new Hamiltonian and the old Hamiltonian(55)H∗=H+∂FΔt.Thus, the generating function should satisfy
(56)H+∂FΔt=0.If we chooseF=F2P,q,t−QP and use symbol S take the place of F2, then we can obtain the following equation
(57)Hp,qσ,t+∂SPσ,qσ,tΔ3t=0.SubstitutingP=β into S, we have
(58)S=Sβ,q,t.Obviously,β can be determined by initial conditions. Hence, the transformation relation corresponding to the function S=F2P,q,t can be written as
(59)α=∂Sβ,q,tΔ1β,p=∂Sβ,q,tΔ2q.Substitutingp into (57), we obtain
(60)H∂Sβ,q,tΔ2q,qσ,t+∂Sβ,qσ,tΔ3t=0.Equation (60) can be called the Hamilton-Jacobi equation on time scales.Theorem 1.
If functionSβ,q,t is a complete integral of the Hamilton-Jacobi equation (60) on time scales, that is, S∈Crd2, β is the constant of integration, and
(61)det∂2Sβ,q,tΔ1βΔ2q≠0,and it satisfies the Hamilton-Jacobi equation (60) on time scales, then the first integrals of the canonical equations (18) on time scales are determined by
(62)α=∂Sβ,q,tΔ1β,(63)p=∂Sβ,q,tΔ2q.Proof 2.
First, we show that (62) is the first integral of the system on time scales, namely,
(64)∂2Sβ,q,tΔ1βΔ2qqΔ+∂2Sβ,qσ,tΔ1βΔ3t=0.
Taking note of (18) and (64), we have
(65)∂2Sβ,q,tΔ1βΔ2q∂HSβ,q,t/Δ2q,qσ,t∂Sβ,q,t/Δ2q+∂2Sβ,qσ,tΔ1βΔ3t=0.
Taking the partial delta derivative of (60) with respect to β and noting (9), we have
(66)∂H∂Sβ,q,t/Δ2q,qσ,t∂∂Sβ,q,t/Δ2q∂2Sβ,q,tΔ2qΔ1β+∂2Sβ,qσ,tΔ3tΔ1β=0.
Notice that (66) coincides with (65). Therefore, (64) is an identity.
Similarly, we verify that (63) is also the first integral. Actually, taking the delta derivative of (63) with respect to t, we have
(67)pΔ=∂2Sβ,q,tΔ2qΔ2qqΔ+∂2Sβ,qσ,tΔ2qσΔ3t.
Then (67) becomes
(68)−pΔ+∂2Sβ,q,tΔ2qΔ2q∂HSβ,q,t/Δ2q,qσ,t∂Sβ,q,t/Δ2q+∂2Sβ,qσ,tΔ2qσΔ3t=0.
Taking the partial delta derivative of (60) with respect to q yields
(69)∂H∂Sβ,q,t/Δ2q,qσ,t∂∂Sβ,q,t/Δ2q∂2Sβ,q,tΔ2qΔ2q+∂H∂qσ+∂2Sβ,qσ,tΔ3tΔ2qσ=0.
ConsideringS∈Crd2 and (18), we know that (69) coincides with (68), that is, (68) is also an identity, completing the proof.Example 5.
The Emden-Fowler type equations have significant applications in gas dynamics, fluid mechanics, relativistic mechanics, and nuclear physics and also play an important role in the study of symmetries and dynamical inverse problems.Different kinds of Emden-Fowler type equations in version of time scales have been established and many achievements have been made [43–45].Now, we investigate the integrals of an Emden-Fowler type equation on time scales. Assume that the Hamiltonian of the system on time scales has the following form(70)H=12p21tσt+am+2tm+2σtqσ2m+4,where a and m are constant. The Hamilton canonical equations of system on time scales are
(71)qΔ=∂H∂p=ptσt,pΔ=−∂H∂qσ=−atm+2σtqσ2m+3.Whena=1, m=1, and T=ℝ, the equations lead to the classical Emden equation
(72)tq¨+2q̇+tq5=0.The Hamilton-Jacobi equation on time scales is given by(73)∂Sβ,qσ,tΔ3t+121tt+h∂Sβ,q,tΔ2q2+am+2tm+2t+hqσ2m+4=0.Equation (73) is hard to solve by the method of variable separation. Now, assume m=−3 and t∈T=hℤ, h>0, then σt=t+h, μt=h. Thus, (73) becomes
(74)∂Sβ,qσ,tΔ3t+12tt+h∂Sβ,q,tΔ2q2−aqσ−2=0.By using the method of variable separation, we set the complete integral as(75)S=S1t+S2q.Substituting (75) into (74), we have
(76)2tt+h∂S1tΔ3t=aqσ−2−∂S2qΔ2q2=β.By integrating (76), we obtain
(77)S1=−β2t,S2=∫aq+hqΔ−2−β1/2Δ2q,S=−β2t+∫aq+hqΔ−2−β1/2Δ2q,and
(78)α=∂Sβ,q,tΔ1β=−12t−12∫aq+hqΔ−2−β−1/2Δ2q,p=∂Sβ,q,tΔ2q=∂S2qΔ2q=∫aqt+h+hqΔt+h−2−β1/2−aqt+hqΔt−2−β1/2qt+h−qtΔ2q.In this example, we gave an Emden-Fowler type of equation (71) on time scales which was a specific case of the general equation in [44]. By applying Theorem 1, we found the integrals of (71) when T=hℤ.The Emden model on time scales contains not only continuous and discrete cases but also other more general cases. More potential applications for the Emden model on time scales in symmetries, dynamical inverse problems and oscillations, and control are worth looking forward to.


## 6. Conclusions

In this paper, we presented the Hamilton-Jacobi method for mechanical systems on time scales. We started from the canonical transformation theory on time scales. Four examples were given to show the role played by a generating function in the canonical transformation. Then we constructed the Hamilton-Jacobi equation (60) on time scales and proved the Jacobi theorem on time scales. We illustrated how this method works for dynamical equations on time scales. The results of this paper are more general. The continuous and discrete analogues of the Hamilton-Jacobi method are special cases of this paper. Moreover, this method can also be applied to the constrained mechanical systems on time scales, the symplectic dynamical systems on time scales, and the impulsive dynamical systems on time scales.From a geometrical point of view, the Hamilton-Jacobi theory on time scales remains an interesting open question. Further works about finding the integral of dynamical equations on time scales are still worth doing, for example, the Poisson theory on time scales and the Lie symmetry theory on time scales.



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