CHARACTERIZATION OF PARTIAL HAMILTONIAN OPERATORS AND RELATED FIRST INTEGRALS

. We focus on partial Hamiltonian systems for the characterization of their operators and related ﬁrst integrals. Firstly, it is shown that if an operator is a partial Hamiltonian operator which yields a ﬁrst integral, then so does its evolutionary representative. Secondly, extra operator conditions are provided for a partial Hamiltonian operator in evolutionary form to yield a ﬁrst integral. Thirdly, characterization of partial Hamiltonian operators and related ﬁrst integral conditions are provided for the partial Hamiltonian system. Applications to mechanics are presented to illustrate the theory.


1.
Introduction. The classical correspondence between symmetries and first integrals for the canonical Hamiltonian equations has been vigorously studied by several authors including Marsden and Weinstein [9], Kozlov [4], Olver [14], Dorodnitsyn and Kozlov [2] and Mahomed and Roberts [7]). The link between Hamiltonian symmetries and their first integrals was first promulgated by the Italian mathematician Levi-Civita [6] (see the insightful translation by Saccomandi and Vitolo [15]). A Hamiltonian symmetry in evolutionary form (see, e.g. [14]) determines a first integral of the canonical Hamilton equations up to a time-dependent function. Dorodnitsyn and Kozlov [2] alluded to some apparent disadvantages of this approach. They [2] looked at the Hamiltonian symmetries which are not restricted to phase space and also transform the time variable t. A direct method to construct first integrals for suitable gauge terms that require integration was proposed by [2]. The two approaches [6,14,2] utilized for the construction of a first integral once the symmetry is known fails to unveil the first integral uniquely. In the first approach as in [14], the first integrals are obtained up to a time dependent function whereas in the second as in [2], the first integrals are established up to a suitable gauge term. Recently, Mahomed and Roberts [7] showed how one can determine the first integral uniquely by providing an extra integrability condition on the first integral for the first method and giving the integrability conditions on the gauge term for the second method. It is shown, as a consequence (see [7]) that both the methods are in fact equivalent.
The partial Hamiltonian approach occurs naturally in economic growth theory. Naz et al [10,11] developed a partial Hamiltonian approach in order to construct first integrals of partial Hamilton systems arising in economic growth theory. However, this was established up to some gauge terms. Also Naz [12] has presented some applications of partial Hamiltonian approach in mechanics and other areas of applied mathematics. A partial Lagrangian method was initiated in Kara et al [3]. Following up on this the partial or discount free Lagrangian approach [13] has been developed to derive the first integrals and closed-form solutions in economic growth theory by using a Lagrangian formulation that naturally arises in this context.
In this work, we focus on the characterization of partial Hamiltonian operators and their related first integrals in a unique manner. This adds to the literature in multiple ways. Firstly, it is shown that if an operator is a partial Hamiltonian operator that yields a first integral, then so does its evolutionary representative. Secondly, extra operator conditions are provided for a partial Hamiltonian operator in evolutionary form to yield a first integral. Thirdly, characterization of partial Hamiltonian operators and related first integral conditions are provided for the partial Hamiltonian systems.
The layout of the paper is as follows. Preliminaries on known aspects of the partial Hamiltonian method are presented in Section 2. In Section 3, the formulas are provided in order to find the partial Hamiltonian operators in evolutionary form and the first integrals of a partial Hamiltonian system in order to result in a first integral. Moreover, extra conditions that give rise to a first integral are provided for the partial Hamiltonian operator in evolutionary form. Characterization of Hamiltonian operators and related first integral conditions for partial Hamiltonian systems are given in Section 4. In Section 5, applications taken from mechanics are presented to show the effectiveness of the approach proposed here. Finally, conclusions are presented in Section 6.

2.
Preliminaries. Again as in many papers on the Hamiltonian formalism we let t be the independent variable (which is usually the time) and (q, p) = (q 1 , ..., q n , p 1 , ..., p n ) the phase space coordinates. In applications to equations of mechanics, q 1 , ..., q n are taken as the position coordinates and p 1 , ..., p n are called the conjugate momenta. The following notions and results are adapted from [14,2,7,10,11]. Definition 1. The Euler operator δ/δq i and the variational operator δ/δp i are and where is the total derivative operator with respect to t. As usual the summation convention is implemented for repeated indices here and in the sequel. The variables t, q, p are considered independent and only connected by the differential relationsṗ CHARACTERIZATION OF PARTIAL HAMILTONIAN OPERATORS 725 A partial Hamiltonian system satisfies (see [10]) where we view Γ i as a nonzero function of t, p i , q i in general and H is a partial Hamiltonian function (see [10,11]). In the event Γ i = 0 for all i, then we have the usual canonical Hamiltonian system. The partial Hamiltonian system naturally arises in economics (see, e.g. [1]).
The operator X which is a generator of Hamiltonian symmetry (see e.g. [14]) is This operator can also be written in characteristic form as whereη i andζ i are the characteristic functions given bȳ The operator (7) has evolutionary representative [14] X or is called the canonical form of the operator X. The operators (7) and (9) are equivalent since X −X = ξD t . The first prolongation of the operator X is given by Definition 2. A first integral I of the partial Hamiltonian system (5) is obtained from the relation where I = I(t, q, p), which is satisfied on all solutions to (5).
We also have the case that which is called the characteristic form of the conservation law (11) with the functions Q i and R i , i = 1, . . . , n, the associated characteristic functions. When the condition (11) is satisfied on the solutions to (5), I is referred to as a first integral of the system (5). Hamiltonian symmetries in evolutionary or canonical form have been considered before (see, e.g. [14]). Furthermore, Hamiltonian symmetries have also been studied in [2] and the Hamiltonian version of Noether's theorem is provided therein. Naz et al [10] developed a partial Hamiltonian approach to derive first integrals of partial Hamiltonian systems arising in economic growth theory. Naz [12] has amply illustrated that the formulas for the partial Hamiltonian approach can well be invoked to deduce first integrals of many partial Hamiltonian systems other than those in economics.
Definition 3 (see [10]). An operator X of the form as given in (6) is said to be a partial Hamiltonian operator corresponding to a partial Hamiltonian H(t, q, p) if there exists a function B(t, q, p) such that is satisfied on all solutions to system (5).
The following theorem is essential for the construction of first integrals for our system (5).
Theorem 1 (see [10]). The first integral corresponding to the system (5) associated with a partial Hamiltonian operator X is determined from for some B(t, p, q) which is a gauge-like function.
If Γ i = 0 and B = B(t, p, q), then this formula (14) is valid for an invariant Hamiltonian action up to divergence [2] as well.
3. Partial Hamiltonian operators in evolutionary form. In this section, we present the partial Hamiltonian operators (see [10]) in evolutionary form. The partial Hamiltonian identity is provided and is utilized to derive the operator determining equations in evolutionary form.
The operator X as in (6) is a generator of symmetry of the partial Hamiltonian system (5) if are satisfied on the system (5). The proof of this is rather simple. The action of the generator X on the first equation of system (5) easily gives which straightforwardly yields (15). The action of the generator X on the second equation of system (5) results in and this directly provides (16).

Lemma 1.
If operator X defined as in (6), is a partial Hamiltonian operator that yields a first integral, then so does the evolutionary representativeX = X − ξD t .
Proof. If X is a partial Hamiltonian operator yielding a first integral, then it satisfies the condition (13). Introducing X =X + ξD t , ζ i =ζ i + ξṗ i and η i =η i + ξq i in equation (13), we haveζ holds for any smooth functions H(t, q, p) and suitable functionsB and Γ i .
Proof. This identity easily follows from direct computations.
Theorem 2. IfX is a partial Hamiltonian operator in evolutionary form, thenX satisfies the following conditions on system (5).

4.
Characterization of operators and related first integral conditions. In this section, we provide a characterization of the Hamiltonian operator which directly corresponds to a first integral and as a consequence obtain the extra conditions on the first integral.
Theorem 3. Necessary and sufficient conditions that the operator X of the form (7), yields a first integral of system (5) is that the characteristicsη i andζ i of X are also the characteristics of the conservation law of system (5) and additionally satisfies the relations as well as the operator conditions (21) and (22).
Proof. Any conservation law of the partial Hamiltonian system (5) can be written in the equivalent characteristic form as for suitable multipliers Q i and R i yet to be determined. First, we act with the Euler operator δ/δq j , j = 1, . . . , n on equation (30) and this yields is δD t I/δq j = 0 which must hold for all t, q and p. This immediately results in where δ i j is the Kronecker delta. Separating the terms containsq i andṗ i from equation (31) gives and the remaining terms of equation (31) then results in Equation (34) with the aid of (32) and (33) reduces to This is precisely one half of the operator conditions, viz.
which hold on the solutions of (5), if and only if following characteristics are chosen Equations (32) and (33), with the aid of (37), provide the required conditions (27) and (28). The equation (30) takes the following form To complete the proof of the second part, the action of the variational derivative operator with respect to p, δ/δp j , j = 1, . . . , n on (38) results in After expansion and setting the coefficients ofṗ i andq i to zero yield The rest of the terms in (39) arē Equation (42), with the aid of (40) and (41) yield the second half of the operator conditions of (5) which are satisfied on the solutions to (5). Note that equation (41) is the same as what we deduced in the first part of the proof after interchanging i and j due to symmetry in the indices. This completes the proof. ∂η ∂q and Theorem 4. For each partial Hamiltonian operator X =X + ξD t which satisfies the conditions (27), (28) and (29), there corresponds a first integral I which is determined uniquely (up to an ignorable constant) from Proof. If the partial Hamiltonian generator X satisfies (27), (28), 29) and operator conditions (21), 22), then holds. The expansion of the left hand side of (48) and equating it to the right hand side of (48) straightforwardly gives the results (47) . This completes the proof.

Applications.
We take examples from the literature to illustrate our results. 1. It it is always opportune to start with the simple harmonic oscillator equation which has partial Hamiltonian H = p 2 /2 and partial Hamiltonian systeṁ Here Γ = −q and Theorem 2 results in D tζ +η = 0, whereη = η − ξp andζ = ζ + ξq. If one confines ξ and η to be independent of p, then by solving (50) one ends up with eight operators which are of point type in (t, q) space. Only the following five X 1 = ∂ t , X 2 = sin 2t∂ t + q cos 2t∂ q − (p cos 2t + 2q sin 2t)∂ p , X 3 = cos 2t∂ t − q sin 2t∂ q + (p sin 2t − 2q cos 2t)∂ p , X 4 = sin t∂ q + cos t∂ p , X 5 = cos t∂ q − sin t∂ p (51) satisfy Theorem 3 which gives only the first condition as n = 1 here which is Just to mention that although X = q∂ q +p∂ p is an operator that identically satisfies (50), it does not satisfy condition (52) and thus does not provide a first integral. Important to note that the operators (51) are also point symmetries of our system (49) due to Γ being independent of p. These operators arise in reference [7] as well. However, we stress that here we have taken a partial Hamiltonian system with partial Hamiltonian H = p 2 /2. It turns out that for the n-dimensional harmonic oscillator system one has the similar results. The partial Hamiltonian in this case is and now the partial Hamiltonian system iṡ with Γ i = −q i . The partial Hamiltonian operators which satisfy Theorem 3 are again (n 2 + 3n + 6)/2 operators which are as stated in [7] (see Example 3.). Let us now demonstrate how one can compute a first integral via Theorem 4 by knowledge of an operator which satisfies Theorem 4. This is manifestly straightforward. We consider the operator X 2 in (51). For this operatorη = q cos 2t − p sin 2t and ζ = −p cos 2t − q sin 2t. Therefore the relations of Theorem 4 become I q = p cos 2t + q sin 2t, I p = q cos 2t − p sin 2t, I t = q(q cos 2t − p sin 2t) − p(p cos 2t + q sin 2t).
can observe this from X 2 already in Example 3. Hence the operator X which is a point operator in (t, r, θ, p 1 , p 2 ) space here will not be a point type operator in a Lagrangian context but rather a generalized operator with derivative dependency. We moreover notice that an integral can have more than one operator associated with it. For the angular momentum type integral above there are many operators which will gives rise to it by Theorem 4. These examples amply suffice in illustrating the main results as encapsulated in Theorems 2, 3 and 4.
6. Conclusions. We have focused on the partial Hamiltonian systems in terms of characterization of their operators and related first integrals. Firstly, it is shown that if an operator is a partial Hamiltonian operator which yields a first integral, then so does its evolutionary representative. Secondly, extra operator conditions are provided for a partial Hamiltonian operator in evolutionary form to yield a first integral. Thirdly, characterization of partial Hamiltonian operators and related first integral conditions are provided for the partial Hamiltonian system.
We have then applied this theory to familiar examples of mechanics starting with a linear paradigm. Then the modified Emden equation and generalized Ermakov system in polar form.
The approach we have developed here also shows that if one has any first integral that arises from a partial Hamiltonian system, then there is at least one operator which corresponds to this first integral which is the inverse problem for partial Hamiltonian systems. We illustrated the result with the angular momentum type integral of the generalized Ermakov system. Here we had many operators associated with this integral. This result provides a correspondence between first integrals and partial Hamiltonian operators for systems that have partial Hamiltonian formulation.