Quadratic first integrals of time-dependent dynamical systems of the form $\ddot{q}^{a}= -\Gamma^{a}_{bc}\dot{q}^{b} \dot{q}^{c} -\omega(t)Q^{a}(q)$

We consider the time-dependent dynamical system $\ddot{q}^{a}= -\Gamma_{bc}^{a}\dot{q}^{b}\dot{q}^{c}-\omega(t)Q^{a}(q)$ where $\omega(t)$ is a non-zero arbitrary function and the connection coefficients $\Gamma^{a}_{bc}$ are computed from the kinetic metric (kinetic energy) of the system. In order to determine the quadratic first integrals (QFIs) $I$ we assume that $I=K_{ab}\dot{q}^{a} \dot{q}^{b} +K_{a}\dot{q}^{a}+K$ where the unknown coefficients $K_{ab}, K_{a}, K$ are tensors depending on $t, q^{a}$ and impose the condition $\frac{dI}{dt}=0$. This condition leads to a system of partial differential equations (PDEs) involving the quantities $K_{ab}, K_{a}, K,$ $\omega(t)$ and $Q^{a}(q)$. From these PDEs, it follows that $K_{ab}$ is a Killing tensor (KT) of the kinetic metric. We use the KT $K_{ab}$ in two ways: a. We assume a general polynomial form in $t$ both for $K_{ab}$ and $K_{a}$; b. We express $K_{ab}$ in a basis of the KTs of order 2 of the kinetic metric assuming the coefficients to be functions of $t$. In both cases, this leads to a new system of PDEs whose solution requires that we specify either $\omega(t)$ or $Q^{a}(q)$. We consider first that $\omega(t)$ is a general polynomial in $t$ and find that in this case the dynamical system admits two independent QFIs which we collect in a Theorem. Next, we specify the quantities $Q^{a}(q)$ to be the generalized time-dependent Kepler potential $V=-\frac{\omega (t)}{r^{\nu}}$ and determine the functions $\omega(t)$ for which QFIs are admitted. We extend the discussion to the non-linear differential equation $\ddot{x}=-\omega(t)x^{\mu }+\phi (t)\dot{x}$ $(\mu \neq -1)$ and compute the relation between the coefficients $\omega(t), \phi(t)$ so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane-Emden equation.


Introduction
The equations of motion of a dynamical system define in the configuration space a Riemannian structure with the metric of the kinetic energy (kinetic metric). This metric is inherent in the structure of the dynamical system; therefore, we expect that it will determine the first integrals (FIs) of the system which are important in its evolution. On the other hand a metric is fixed by its symmetries, that is, the linear collineations: Killing vectors (KVs), homothetic vectors (HVs), conformal Killing vectors (CKVs), affine collineations (ACs), projective collineations (PCs); and the quadratic collineations: second order Killing tensors (KTs). The question then is how the FIs of the dynamical system and the geometric symmetries of the kinetic metric are related.
The standard way to determine the FIs of a differential equation is the use of Lie/Noether symmetries which applies to the point as well as the generalized Lie/Noether symmetries. The relation of the Lie/Noether symmetries with the symmetries of the kinetic metric has been considered mostly in the case of point symmetries for autonomous conservative dynamical systems moving in a Riemannian space. In particular, it has been shown (see e.g. [1], [2], [3], [4]) that the Lie point symmetries are generated by the special projective algebra of the kinetic metric whereas the Noether point symmetries are generated by the homothetic algebra of the kinetic metric, the latter being a subalgebra of the projective algebra. A recent clear statement of these results is discussed in [5].
In addition to the autonomous conservative systems this method has been applied to the time-dependent potentials W (t, q) = ω(t)V (q), that is, for equations of the formq a = −Γ a bcq bqc − ω(t)V ,a (q) (see e.g. [6], [7], [8], [9], [10], [11], [12]). In this case it has been shown that the Lie point symmetries, the Noether point symmetries and the associated FIs are computed in terms of the collineations of the kinetic metric plus a set of constraint conditions involving the time-dependent potential and the collineation vectors. These time-dependent potentials are important because (among others) they contain the time-dependent oscillator (see e.g. [8], [10], [13], [14], [15]) and the time-dependent Kepler potential (see e.g. [12], [16], [17], [18]). A further development in the same line is the extension of this method to time-dependent potentials W (t, q) with linear damping terms [12]. It has been shown that under a suitable time transformation the damping term can be removed and the problem reduces to a time-dependent potential of the form W (t, q) =ω(t)V (q) but with differentω(t). Finally the Lie/Noether method has been applied to the study of partial differential equations (PDEs) [4], [19], [20], [21].
Besides the aforementioned Lie/Noether method there is a different method which computes the FIs in terms of the collineations of the kinetic metric without using Lie symmetries. This method we shall apply in this paper. It has as follows.
One assumes the generic quadratic first integral (QFI) to be of the form 1 where the coefficients K ab , K a , K are tensors depending on the coordinates t, q a and imposes the condition dI dt = 0. Using again the equations of motion to replace the quantitiesq a whenever they appear, this condition leads to a system of PDEs involving the unknown quantities K ab , K a , K and the dynamical elements, i.e. the potential and the generalized forces of the system. The solution of this system of PDEs provides the QFIs (1). For future reference we shall call this method the direct method .
The system of PDEs consists of two parts: a. The geometric part which is independent of the dynamical quantities; and b. the dynamical part which contains the scalar K and the dynamical quantities. The main conclusion of the geometric part is that the tensor K ab is a KT of the kinetic metric whereas the vector K a is related to the linear collineations of that metric. The dynamical part involves the scalar K which is determined by a set of constraint conditions which involve K ab , K a , K, the potential and the generalized forces. Once K is computed one gets the corresponding QFI I.
The direct method can always be related to the Noether symmetries. Indeed assuming that the system has a regular Lagrangian (which is always the case since we assume that there exists the kinetic energy) it can be shown by using the inverse Noether theorem (see [22] and section II in [23]) that to each QFI I one determines an associated gauged generalized Noether symmetry with generator η a = −2K abq b − K a and Noether function f = −K abq aqb + K whose Noether integral is the considered QFI. Therefore we conclude that all QFIs of the form (1) are Noetherian, provided the Lagrangian is regular, that is, the dynamical equations can be solved in terms ofq a .
Moreover, this method has been employed in the literature (see [17], [24], [25], [26]) both for autonomous and time-dependent dynamical systems. A recent account of this method in the case of autonomous conservative systems together with relevant references can be found in [27]. This approach being geometric is powerful and convenient because with minimal calculations it allows the computation of the FIs by using known results from differential geometry.
The purpose of the present work is to apply the direct method to compute the QFIs of time-dependent equations of the formq a = −Γ a bcq bqc − ω(t)Q a (q). Because many well-known dynamical systems fall in this category we intend to recover in a direct single approach all the known results derived from the Lie/Noether symmetry method, which are scattered in a large number of papers.
As explained above, the solution of the system requires that the tensor K ab is a KT of the kinetic metric. In general, the computation of the KTs of a metric is a major task. However for spaces of constant curvature this problem has been solved (see [28], [29], [30]). Therefore, in this paper, we restrict our discussion to Euclidean spaces only. Since the KT K ab is a function of t, q a we suggest two procedures of work: a. The polynomial method; b. the basis method.
In the polynomial method, one assumes a general polynomial form in the variable t both for the KT K ab and the vector K a and replaces in the equations of the relevant system. In the basis method, one computes first a basis of the KTs of order 2 of the kinetic metric and then expresses in this basis the KT K ab with the coefficients to be functions of t. The vector K a and the FIs follow from the solution of the system. Both methods are suitable for autonomous dynamical systems but for time-dependent systems it appears that the basis method is preferable.
Concerning the quantities ω(t) and Q a (q), again, there are two ways to proceed. a) Consider a general form for the function ω(t) and let the quantities Q a unspecified. In this case the quantities Q a act as constraints. b) Specify the quantities Q a and determine for which functions ω(t) the resulting dynamical system admits QFIs.
In the following we shall consider both the polynomial method and the basis method, starting from the former. As a first application, we assume the KT K ab = N (t)γ ab where N (t) is an arbitrary function and show that we recover all the point Noether integrals found in [12]. As a second application, we assume that ω(t) = b 0 + b 1 t + ... + b ℓ t ℓ with b ℓ = 0 and ℓ ≥ 1 whereas the quantities Q a are unspecified. We find that in this case the system admits two families of independent QFIs as stated in Theorem 1.
Subsequently, we consider the basis method. This is carried out in two steps. In the first step, we assume that we know a basis {C (N )ab (q)} of the space of KTs of the kinetic metric and require that K ab has the form In the second ste,p we specify the generalized forces to be conservative with the time-dependent Newtonian generalized Kepler potential V = − ω(t) r ν where ν is a non-zero real constant and r = x 2 + y 2 + z 2 . This potential for ν = −2, 1 includes respectively the three-dimensional (3d) timedependent oscillator and the time-dependent Kepler potential. For other values of ν it reduces to other important dynamical systems, for example, for ν = 2 one obtains the Newton-Cotes potential (see e.g. [31]). We determine the QFIs of the time-dependent generalized Kepler potential and recover in a systematic way the known results concerning the QFIs of the 3d time-dependent oscillator, the time-dependent Kepler potential and the Newton-Cotes potential. For easier reference we collect all the results in Table 2 of section 14.
Using the well-known result that by a reparameterization the linear damping term φ(t)q a of a dynamical equation is absorbed to a time-dependent force of the form ω(t)Q a (q), we also study the non-linear differential equationẍ = −ω(t)x µ + φ(t)ẋ (µ = −1) and compute the relation between the coefficients ω(t), φ(t) for which QFIs are admitted. It is found that a family of 'frequencies'ω(s) is admitted which for µ = 0, 1, 2 is parameterized with functions whereas for µ = −1, 0, 1, 2 is parameterized with constants. As a further application, we study the integrability of the well-known generalized Lane-Emden equation.
The structure of the paper is as follows. In section 2 we determine the system of PDEs resulting form the condition dI/dt = 0. In section 3, we assume that the KT is proportional to the kinetic metric and derive the point Noether FIs of the time-dependent dynamical system (2). In section 4, we consider the polynomial method and define the general forms of the KT K ab and the vector K a which lead to a new form of the system of PDEs. In section 5, we assume that ω(t) is a general polynomial of t and we find that the resulting time-dependent system admits two independent QFIs as stated in Theorem 1. In section 6, we discuss some special cases of the QFI I n of Theorem 1. In section 7, we consider the basis method. In section 8, we find a basis for the KTs in E 3 in order to apply the basis method to 3d Newtonian systems. In sections 9 -13, we study the time-dependent generalized Kepler potential and find for which functions ω(t) admits QFIs. Particularly, in section 13, we study a special class of time-dependent oscillators with frequency ω 3O (t) as given in equation (123). We collect our results for the several values of ν in Table 2 of section 14. In section 15, we use the independent LFIs I 41i , I 42i given in equations (125), (126) to integrate the equations of the time-dependent oscillators defined in section 13; and the FIs L i , E 2 , A i determined in subsection 11.1 to integrate the time-dependent Kepler potential with ω(t) = k b0+b1t where kb 1 = 0. In section 16, we consider the second order non-linear time-dependent differential equation (154) and show that it is integrable with an associated QFI given in equation (175) iff the functions ω(t), φ(t) are related as shown in equation (174). For the special values µ = 0, 1, 2 we find also that there exist additional relations between ω(t), φ(t) for which the resulting differential equation admits a QFI. For µ = 1 equation (154) admits the general solution (166) provided that condition (165) is satisfied. We apply these results in subsection 16.1 and we study the properties of the well-known generalized Lane-Emden equation. Finally, in section 17, we draw our conclusions and, in the appendix, we give the proof of Theorem 1.

The system of equations
We consider the dynamical systemq where Γ a bc are the Riemannian connection coefficients determined by the kinetic metric γ ab (kinetic energy) of the system and −ω(t)Q a (q) are the time-dependent generalized forces. Einstein summation convention is assumed and the metric γ ab is used for lowering and raising the indices.
We next consider a function I(t, q a ,q a ) of the form where K ab is a symmetric tensor, K a is a vector and K is an invariant. We demand I be a FI of (2) by imposing the condition Using the dynamical equations (2) to replaceq a whenever it appears we find 2 the system of equations where the last two equations 3 (9), (10) express the integrability conditions for the scalar K. Equation (5) implies that K ab is a KT of order 2 (possibly zero) of the kinetic metric γ ab . The solution of the system requires the function ω(t) and the quantities Q a (q) both being quantities which are characteristic of the given dynamical system. There are two ways to proceed. a) Consider a general form for the function ω(t) and let the quantities Q a (q) unspecified. In this case the quantities Q a (q) act as constraints. b) Specify the quantities Q a (q) and determine for which functions ω(t) the resulting dynamical system admits FIs.
However, before continuing with this kind of considerations, we first proceed with the simple geometric choice K ab = N (t)γ ab where N (t) is an arbitrary smooth function. By specifying the KT K ab as above both the function ω(t) and the quantities Q a (q) stay unspecified and can act as constraints.
3 The point Noether FIs of the time-dependent dynamical system (2) We consider the simplest choice K ab = N (t)γ ab (11) where N (t) is an arbitrary smooth function. This choice is purely geometric; therefore, the function ω(t) and the quantities Q a (q) are unspecified and act as constraints, whereas the vector K a is identified with one collineation of the kinetic metric. With this K ab , the system of equations (5) -(10) become (eq. (5) vanishes trivially) We consider the following cases.
3.1 Case K a = K a (q) is the HV of γ ab with homothety factor ψ In this case K a,t = 0 and K (a;b) = ψγ ab where ψ is an arbitrary constant. Equation (12) gives where c is an arbitrary constant. Equation (16) implies that (take ω = 0) is an arbitrary potential. Replacing in (13) we find that where M (t) is an arbitrary function. Substituting the function K(t, q) in (14) we get The remaining condition (15) is just the partial derivative of (17), and hence is satisfied trivially. Moreover, since ω = 0, equation (17) can be written in the form where c 1 , c 2 are arbitrary constants. Therefore equation (18) becomes The QFI is where Q a = V ,a and the quantities ω(t), M (t), V (q), K a (q) satisfy the conditions (19) -(21).

Case
In this case S ;ab = ψγ ab and M (t) = 0 is an arbitrary function. Equation (12) implies N ,t = ψM . From equation (16) we find that there exists a potential function V (q) such that Q a = V ,a . Replacing the above results in (13) we obtain where C(t) is an arbitrary function. Substituting in (14) we get (take ωM = 0) ωM S ,a V ,a + 2ω ,t N V + 2ωψM V + M ,tt S + C ,t = 0 =⇒ S ,a V ,a + 2ψV where d 1 , m, k are arbitrary constants. The remaining condition (15) is satisfied identically. The QFI is where Q a = V ,a , N ,t = ψM and the conditions (23) -(26) must be satisfied.

3.3
Case Q a = V ,a and K a = −M(t)V ,a (q) where V ,a is the gradient HV of γ ab Equation (12) implies N ,t = ψM where ψ is the homothety factor of V ,a . From equation (13) we obtain where C(t) is an arbitrary function. Substituting in (14) we get (take ωM = 0) where d 2 , k are arbitrary constants. The remaining conditions are satisfied identically. The QFI is where Q a = V ,a , N ,t = ψM and the conditions (28) -(30) must be satisfied.
The above results reproduce Theorem 2 of [12] which states that the point Noether symmetries of the time-dependent potentials of the form ω(t)V (q) are generated by the homothetic algebra of the kinetic metric (provided the Lagrangian is regular).
It is interesting to observe that the QFIs (22), (27), (31) produced by point Noether symmetries can be also produced by generalized (gauged) Noether symmetries using the inverse Noether theorem. This proves that a Noether FI is not associated with a unique Noether symmetry!

The polynomial method for computing the QFIs
In the polynomial approach one assumes a polynomial form in t of the KT K ab (t, q) and the vector K a (t, q) and solves the resulting system for given ω(t), Q a (q). One application of this method can be found in [27] where a general theorem is given which allows the finding of the QFIs of an autonomous conservative dynamical system. In the present work we generalize the considerations made in [27] and assume that the quantity K ab (t, q) has the form where C (N )ab , N = 0, 1, ..., n, is a sequence of arbitrary KTs of order 2 of the kinetic metric γ ab .
This choice of K ab and equation (6) indicate that we set where L (M)a (q), M = 0, 1, ..., m, are arbitrary vectors. We note that both powers n, m in the above polynomial expressions may be infinite. Substituting (32), (33) in the system of equations (5) -(10) (equation (5) is identically zero since C (N )ab are KTs) we obtain the system of equations In this system of PDEs the pairs ω(t), Q a (q) are not specified. As we explained in the introduction we shall fix a general form of ω and find the admitted QFIs in terms of the (unspecified) Q a . In the following section we choose ω(t) to be a general polynomial in t, however any other choice is possible.
where ℓ is the degree of the polynomial. Substituting the function (39) in the system of equations (34) - (38) we find 4 that there are two independent QFIs as given in Theorem 1.
where n = 0, 1, 2, ..., C (0)ab is a KT, the KTs C (N )ab = −L (N −1)(a;b) for N = 1, ..., n, L (n)a is a KV, G(q) is an arbitrary function defined by the condition s is an arbitrary constant defined by the condition and the following conditions are satisfied with k = 2, 3, ...n. Integral 2.
We note that the FI I e exists only when ω(t) = b 0 + b 1 t, that is, for ℓ = 1.

Special cases of the QFI I n
The parameter n in the case Integral 1 of Theorem 1 runs over all positive integers, i.e. n = 0, 1, 2, .... This results in a sequence of QFIs I 0 , I 1 , I 2 , ..., one QFI I n for each value n. A significant characteristic of this sequence is that I k < I k+1 , that is, each QFI I k where k = 0, 1, 2, ... can be derived from the next QFI I k+1 as a subcase.
In the following we consider some special cases of the QFI I n for small values of n.

The QFI I 0
For n = 0 we have This QFI consists of the independent FIs

The QFI I 1
For n = 1 the conditions (41) -(44) become Since b ℓ = 0 the last condition for k = ℓ gives and the remaining equations become The last set of equations exist only for ℓ ≥ 2. From these equations, using mathematical induction, we prove after successive substitutions that The QFI is (I 0 is a subcase of I 1 ) For some values of the degree ℓ of the polynomial ω(t) we have: 1) For ℓ = 1.
We have ω = b 0 + b 1 t + b 2 t 2 and the QFI is

The basis method for computing QFIs
As it has been explained in the introduction, in the basis method instead of considering the KT K ab to be given as a polynomial in t with coefficients arbitrary KTs (see equation (32) ) one defines the KT K ab (t, q) by the requirement where α N (t) are arbitrary smooth functions and the m linearly independent KTs C (N )ab (q) constitute a basis of the space of KTs of the kinetic metric γ ab (q). In this case, one does not assume a form for the vector K a (t, q) which is determined from the resulting system of equations (5) - (10). The basis method has been used previously by Katzin and Levine in [17] in order to determine the QFIs for the time-dependent Kepler potential. As we shall apply the basis method to 3d Newtonian systems we need a basis of KTs (and other collineations) of the Euclidean space E 3 .

The geometric quantities of E 3
In E 3 the general KT of order 2 has independent components where a I with I = 1, 2, ..., 20 are arbitrary real constants. The vector L a generating the KT x + a 13 y + 2a 7 z + a 14 −a 2 x 2 − a 12 y 2 − 2a 16 xy + a 11 xz + a 8 yz + 2(a 19 − a 1 )x + 2(a 20 − a 7 )y + a 9 z + a 10   (50) and the generated KT is   (51) which is a subcase of the general KT (49) for a 1 = a 4 = a 6 = a 7 = a 10 = a 14 = 0.
We note that the covariant expression of the most general KT M ij of order 2 of E 3 is (see [32], [33]) where A mn , B l i , D ij are constant tensors all being symmetric and B l i also being traceless; λ k is a constant vector; and ε ijk is the 3d Levi-Civita symbol. This result is obtained from the solution of the Killing tensor equation in Euclidean space.
Observe that A mn , D ij have each 6 independent components; B l i has 5 independent components; and λ k has 3 independent components. Therefore M ij depends on 6 + 6 + 5 + 3 = 20 arbitrary real constants, a result which is in accordance with the one given above in equation (49).
The Lagrangian of the system is and the corresponding Euler-Lagrange equations arë For this system the Q a = νq a r ν+2 where q a = (x, y, z) whereas the ω(t) is unspecified. We shall determine those ω(t) for which the resulting FIs are not combinations of the angular momentum.
The LFIs and the QFIs of the autonomous generalized Kepler potential, that is, ω(t) = k = const, have been determined in [27] using the direct method and are listed in Table 1. Table 1: The LFIs/QFIs of the autonomous generalized Kepler potential for ω(t) = k = const.
In Table 1 H ν is the Hamiltonian of the system, L i are the components of the angular momentum, R i are the components of the Runge-Lenz vector and B ij are the components of the Jauch-Hill-Fradkin tensor.
Using Q a = νq a r ν+2 conditions (5) -(10) become (see [17]) From the Lagrangian (53) we infer that the kinetic metric is δ ij = diag(1, 1, 1). According to the basis approach, the KT K ab (t, q) of (55) is the KT given by (49) but the 20 arbitrary constants a I are assumed to be time-dependent functions a I (t).
Condition (56) gives From the first three conditions (61) -(63) we find where A, B, C are arbitrary functions. Substituting these results in (64) -(66) we obtain By taking the second partial derivatives of (67) with respect to (wrt) x, y, of (68) wrt x, z and of (69) wrt y, z we find that are arbitrary constants. Then equations (67) -(69) become By suitable differentiations of the above equations we obtain Then A =ȧ 14 zy 2 +ȧ 10 yz 2 +ȧ 15 y 2 +ȧ 11 where σ k (t), τ k (t), η k (t) for k = 1, 2, 3, 4 are arbitrary functions. Substituting in (70) -(72) we find from which we have finally where c 4 , c 5 , c 6 are arbitrary constants. Therefore the KT K ab is x + a 8 y + a 9 and the vector K a is Replacing the above results in the constraint (60) we find the following set of equations: Therefore the KT K ab becomes and the vector Since the ten parameters a 3 (t) and c A where A = 1, 2, ..., 9 are independent (i.e. they generate different FIs) we consider the following two cases.

a 3 (t) = 0
In this case the conditions (79) are satisfied identically leaving the function ω(t) free to be any function.
Therefore the KT (80) becomes and the vector (81) becomes the general non-gradient KV Then the constraint (58) implies that (since K a q a = 0) K = G(x, y, z) which when replaced in (57) gives (since K ab q b = 0) G ,a = 0. Hence K = const ≡ 0.
The QFI I = K abq aqb + K aq a leads only to the three components L i of the angular momentum. We note that I contains nine independent parameters each of them defining a FI: a) c 7 , c 8 , c 9 lead to the components L 1 = yż − zẏ, L 2 = zẋ − xż, L 3 = xẏ − yẋ of the angular momentum (LFIs); and b) c 1 , c 2 , c 3 , c 4 , c 5 , c 6 lead to the products (QFIs depending on L i ) L 2 1 , L 2 2 , L 2 3 , L 1 L 2 , L 1 L 3 and L 2 L 3 . We have the following result. In this case the conditions (79) imply that a 3 (t) = b 0 + b 1 t + b 2 t 2 and where k, b 0 , b 1 , b 2 are arbitrary constants and the index (ν) denotes the dependence of ω(t) on the value of ν.
Since c A = 0 the quantities (80) and (81) become Substituting in the remaining constraints (57) and (58) we find The QFI is We note that the resulting time-dependent generalized Kepler potential is a subcase of the Case III potential of [18] if we set the function Then the associated QFI (3.13) of [18] (for K 1 = K 2 = 0) reduces to the QFI J ν .
For some values of ν we have the following results: -ν = 1 (time-dependent Kepler potential).
The ω (2) = k = const and the QFI is

This expression contains the independent QFIs
where H 2 is the Hamiltonian of the system. These are the FIs found in [27] (see also -ν = −2 (time-dependent oscillator).
We infer the following new general result which includes the time-dependent Kepler potential and the timedependent oscillator as subcases.
Proposition 3 (3d time-dependent generalized Kepler potentials which admit FIs) For all functions ω(t) the time-dependent generalized Kepler potential V (t, q) = − ω(t) r ν admits the LFIs of the angular momentum and QFIs which are products of the components of the angular momentum. However for the function the resulting time-dependent generalized Kepler potential admits the additional QFI J ν given by (83). where c 10 , c 11 , c 12 , c 13 are arbitrary constants. Finally, we have From the last conditions follow that in order QFIs to be admitted the function ω(t) can have only three possible forms: -ω(t) a general function; -ω(t) = ω 2K (t) = c11 b0+b1t where c 11 b 1 = 0; and -ω(t) = ω 3K (t) = k (b0+b1t+b2t 2 ) 1/2 where k = 0 and b 2 1 − 4b 2 b 0 = 0. This result confirms the results found previously in [12], [17], [18]. We note that the time-dependent Kepler potential V = − ω2K (t) r is a subcase of the Case II potential of [18] for µ 0 = c 11 and φ = b 0 + b 1 t, whereas the potential V = − ω3K (t) r is a subcase of the Case III potential of [18] (see subsection 10.2). In the following we discuss the cases for the special functions ω 2K (t) and ω 3K (t) because the case for a general function ω(t) reproduces the results of the subsection 10.1.

ω(t)
In that case conditions (86) give Substituting the resulting vector K a and the KT K ab in (58) we find the solution K(q, t) = − 2c 10 b 1 t c 11 r + G(q).
The QFI I contains the already found LFIs L i of the angular momentum; the QFI E 2 which for b 1 = 0 reduces to the Hamiltonian of the Kepler potential V = − c11 b0r ; and the QFIs A i which may be considered as a generalization of the Runge-Lenz vector b0 . The expressions (88) -(90) are written compactly as follows where ω 2K (t) = c11 b0+b1t . We remark that only five of the seven FIs E 2 , L i , A i are functionally independent because they are related as follows
Then the constraint (59) implies thaẗ ... Therefore and Before we proceed with considering various subcases it is important that we discuss the ordinary differential equations (ODEs) (100) and (101).

The Lewis invariant
where a = a(t) can be written as follows By putting a = −ρ 2 where ρ = ρ(t) equation (105) becomes For 2ω(t) = −ψ 2 (t) equation (106) is writtenρ Equation (107) is the auxiliary equation (see [8], [34], [35]) that should be introduced in order to derive the Lewis invariant for the one-dimensional (1d) time-dependent oscillator By eliminating the ψ 2 using (108) and multiplying with the factor xρ − ρẋ equation (107) gives which is the well-known Lewis invariant for the 1d time-dependent harmonic oscillator or, equivalently, a FI for the two-dimensional (2d) time-dependent system with equations of motion (107) and (108).

The system of equations (98) -(101)
The conditions (99) are not involved into the conditions (98), (100) and (101). This means that the parameters σ 4 , τ 4 , η 4 give different independent FIs from the remaining parameters a 3 , a 9 , a 13 , a 17 , a 19 , a 20 . Therefore without loss of generality they can be treated separately. This leads to the following two cases.
Since the remaining ODEs (99) are all independent (i.e. each one generates an independent FI) and of the same form without loss of generality we assume where k 1 , k 2 are arbitrary constants.
From (99) for σ 4 = 0 we get The parameters c A where A = 1, 2, ..., 9 produce the FIs of the angular momentum and we fix them to zero. Therefore Substituting in the remaining constraints (57) and (58) we find The QFI is which contains the irreducible LFIs (see eq. (6.25) in [8]) where f (t) is an arbitrary non-zero function satisfying (114). We note that the LFIs (115) can be derived directly from the equations of motion for ω(t) =f 2f .
From the above two cases we arrive at the following conclusion.

A special class of time-dependent oscillators
In proposition 5 it has been shown that the time-dependent oscillator (ν = −2) for the frequency where f (t) is an arbitrary non-zero function admits the six QFIs and for the frequency where g(t) is an arbitrary non-zero function admits the three LFIs We consider the class of the 3d time-dependent oscillators for which ω 1O (t) = ω 2O (t). These oscillators admit both the six QFIs Λ ij and the three LFIs I 4i .
The condition ω 1O (t) = ω 2O (t) relates the functions f (t), g(t) as follows It can be easily proved that and satisfy the requirement (120) for any non-zero function f (t). In other words all the time-dependent oscillators with frequency admit the six QFIs and the six LFIs These are the LFIs J k 3 , J k 4 derived in eqs. (44), (45) in [10] using Noether point symmetries and Noether's theorem.
We note that and Next we consider the LFIs of the angular momentum L i = q i+1qi+2 − q i+2qi+1 which can be expressed equivalently as components of the totally antisymmetric tensor where ε ijk is the 3d Levi-Civita symbol and L i = L i since the kinetic metric γ ij = δ ij . Then (see eq. (51) in [10]) Proposition 6 For the class of 3d time-dependent oscillators with potential V (t, q) = −ω(t)r 2 where ω(t) is defined in terms of an arbitrary non-zero (smooth) function f (t) as in (123), the only independent FIs are the LFIs I 41i , I 42i .
In order to recover the results of [10], we assume a time-dependent oscillator with ω 3O (t) given by (123) and we write the non-zero function f (t) in the form f (t) = ρ 2 (t). Then equation (123) becomes The relations (121), (122) become and the LFIs (125), (126) take the form These latter expressions for c 0 = 2 coincide with the independent LFIs (44) and (45) found in [10].
Finally we note that if we consider in this special class of oscillators the simple case f = 1, we find ω 3O (t) = const = − c0 4 ≡ k which is the 3d autonomous oscillator (for k < 0). Then it can be shown that the exponential LFIs I 3i± (see Table 1) found in [27] can be written in terms of I 41i , I 42j . Indeed we have I 3i± (k > 0) = I 41i ∓iI 42i and I 3i± (k < 0) = I 41i ± iI 42i .

Collection of results
We collect the results concerning the time-dependent generalized Kepler potential for all values of ν in Table  2. We note that for ν = −2, 1, 2 the dynamical system is the time-dependent 3d oscillator, the time-dependent Kepler potential and the Newton-Cotes potential respectively. Concerning notation we have q i = (x, y, z), ν ω(t) LFIs and QFIs Table 2: The LFIs/QFIs of the time-dependent generalized Kepler potential V = − ω(t) r ν .

Integrating the equations
In this section we use the independent LFIs I 41i , I 42i to integrate the equations of the special class of 3d timedependent oscillators (ν = −2) defined in section 13 with ω(t) given by (123). We also use the FIs L i , E 2 , A i to integrate the time-dependent Kepler potential (ν = 1) with ω(t) = k b0+b1t where kb 1 = 0 (see subsection 11.1).

15.2
The solution of the time-dependent Kepler potential with ω 2K (t) = In subsection 11.1 it is shown that this system admits the following FIs: The components of the generalized Runge-Lenz vector are written Since the angular momentum is a FI the motion is on a plane. We choose without loss of generality the plane z = 0 and on that the polar coordinates x = r cos θ, y = r sin θ. Then Using the relationθ = L3 r 2 to replaceθ, the above relations are written By multiplying equation (143) with cos θ and (144) with sin θ we find that where k 1 ≡ A1 k and k 2 ≡ A2 k . Applying the transformation k 1 = α cos β and k 2 = α sin β, equation (145) is written (see also section 5 in [17] which for ω 2K (t) = const (standard Kepler problem) reduces to the analytical equation of a conic section in polar coordinates. In that case α is the eccentricity. It is also worthwhile to mention that the relation (94) becomes Finally, in the polar plane the equations of motion (54) for ν = 1 becomë (149) Substituting (149) in (145) we obtain which coincides with eq. (5.17) in [17].

A class of 1d non-linear time-dependent equations
In this section we use the well-known result [12] that the non-linear dynamical system is equivalent to the linear dynamical system (without damping term) where φ(t) is an arbitrary function such that We apply this result to the following problem: Consider the second order differential equation where the constant µ = −1 and determine the relation between the functions ω(t), φ(t) for which the equation admits a QFI, therefore it is integrable. This problem has been considered previously in [36], [37] (see eq. (28a) in [36] and eq. (17) in [37]) and has been answered partially using different methods. In [36] the author used the Hamiltonian formalism where one looks for a canonical transformation to bring the Hamiltonian in a time-separable form. In [37] the author used a direct method for constructing FIs by multiplying the equation with an integrating factor. In [37] it is shown that both methods are equivalent and that the results of [37] generalize those of [36]. In the following we shall generalize the results of [37]; in addition we discuss a number of applications.
Equation (154) is equivalent to the equation where the functionω(s) is given by (153). Replacing with Q 1 = x µ in the system of equations (5) -(10) we find that 7 K 11 = K 11 (s) and the following conditions where b 1 (s), b 2 (s) are arbitrary functions. Then the general QFI (3) becomes We consider the solution of the system (156) -(158) for various values of µ.
3) Case µ = 2. We find the functionω = K −5/2 11 and the QFI where c 4 , c 5 are arbitrary constants and the function K 11 (s) is given by Using the transformation (153) the above results become and ...
The QFI (159) becomes and the functionω It can be checked that (172), (173) for µ = 0, 1, 2 give results compatible with the ones we found for these values of µ.

The generalized Lane-Emden equation
Consider the 1d generalized Lane-Emden equation (see eq. (6) in [44]) where k is an arbitrary constant. This equation is well-known in the literature because of its many applications in astrophysical problems (see Refs. in [44]). In general, to find explicit analytic solutions of equation (176) is a major task. For example, such solutions have been found only for the special values µ = 0, 1, 5, in the case that the function ω(t) = 1 and the constant k = 2. New exact solutions, or at least the Liouville integrability, of equation (176) are guaranteed, if we find a way to determine its FIs. We see that equation (176) is a subcase of the original equation (154) for φ(t) = − k t , therefore we can apply the results found earlier in section 16. In what follows we discuss only the fourth case where µ = −1 in order to compare our results with those found in Table 1 of [44]. In particular, for φ(t) = − k t the function (174) and the associated QFI (175) become and where the function M (t) = t −k dt.
Concerning the form of the function M (t) there are two cases to be considered: a) k = 1; and b) k = 1.
We note also that for k = µ+3 µ−1 where µ = 1 the function ω(t) = A = const. This reproduces the first subcase of Case 1 in Table 1 of [44] which is the Case 5.1 of [45].
We note also that for k = µ+3 µ+1 the function ω(t) = A = const. This recovers the second subcase of Case 1 in Table 1 of [44] which is the Case 5.2 of [45].
We conclude that the seven cases 1-7 found in Table 1 of [44] are just subcases of the above two general cases a) and b). To compare with these results one may adopt the notation ω = f , k = n and µ = p.

Conclusions
The purpose of the present work was to compute the QFIs of time-dependent dynamical systems of the form q a = −Γ a bcq bqc − ω(t)Q a (q), where the connection coefficients are computed from the kinetic metric, using the direct method instead of the Noether symmetries as it is usually done. In the direct method one assumes that the QFI is of the form I = K abq aqb + K aq a + K and demands that dI/dt = 0. This leads to a system of PDEs whose solution provides the QFIs. One key result is that the tensor K ab is a KT of the kinetic metric.
We have discussed the solution of the system of equations at two levels. The first level is purely geometric and concerns the KT K ab ; and the second level is the physical one which concerns the quantities ω(t), Q a (q) defining the dynamical system.
Concerning the first level we have applied two different methods: a. The polynomial method in which one assumes a general polynomial form in the variable t both for the KT K ab and for the vector K a . b. The basis method where one computes first a basis of the KTs of order 2 of the kinetic metric and then expresses K ab in this basis assuming that the 'components' are functions of t. In both methods the key point is to compute the scalar K.
Concerning the dynamical quantities ω(t), Q a (q) we have chosen to work in two ways: a. First we considered the polynomial method and assumed the function ω(t) to be a polynomial leaving the quantities Q a unspecified. It is found that in this case the resulting dynamical system admits two independent QFIs whose explicit expression together with conditions involving the quantities Q a and the collineations of the kinetic metric are given in Theorem 1. b. In the basis method we worked the other way. That is, we assumed the quantities Q a (q) to be given by the time-dependent generalized Kepler potential V = − ω(t) r ν and determined the functions ω(t) for which QFIs exist.
The results of this detailed study are displayed in Table 2 for all values of ν. For the values ν = −2, 1, 2 we recovered the known results concerning the time-dependent 3d oscillator, the time-dependent Kepler potential and the Newton-Cotes potential respectively. We note that these latter results have appeared over the years in many works whereas in the present discussion occur as particular cases of a single geometric approach. The last part of our considerations concerns the well-known proposition that under a reparameterization the linear damping φ(t)q a can be absorbed to a time-dependent generalized force. We used this proposition in the case of a 1d non-linear second order time-dependent differential equation, we determined the condition that the time-dependent coefficients of the equation must satisfy in order a QFI to exist and we computed this QFI. As an application we studied the properties of the well-known generalized Lane-Emden equation.
We note that one is possible to consider other dynamical quantities and/or kinetic metric and compute the QFIs. What is the same in all cases is the method of work which we hope we have presented adequately in the present work.

Appendix
Substituting the polynomial function ω(t) given by (39) in the system of equations (34) - (38) we have the following cases.
We continue with the remaining constraints (35) and (36) in order to determine the scalar coefficient K(t, q).
We consider the following subcases. a. For λ = µ: From (188) we have that C ab = 0 and L a is a KV. From (191) we find that L a = 0. Therefore, the QFI I e (λ = µ) = const which is trivial.
b. For λ = µ: From (188) we have that C ab = − 1 λ L (a;b) . Therefore L (a;b) is a KT. We consider two cases according to the degree ℓ of the polynomial ω(t).
-Case ℓ = 1. From (191) we find that Replacing with C ab = − 1 λ L (a;b) and by substituting (192) in (193) we obtain The solution of (190) is which when replaced in (189) gives G ,a = 0, that is G = const ≡ 0. The QFI is I e (ℓ = 1) = −e λt L (a;b)q aqb + λe λt L aq a + b 0 − b 1 λ e λt L a Q a + b 1 te λt L a Q a where L (a;b) is a KT, L b Q b ,a = λ 3 b1 L a and λ 3 L a = −2b 1 L (a;b) Q b . -Case ℓ > 1. From (191) we find that L b Q b ,a = 2λC ab Q b , C ab Q b = 0 and λ 2 L a = 2b 1 C ab Q b . Therefore L a = 0 and hence C ab = − 1 λ L (a;b) = 0. We end up with a trivial FI I e = const.