A geometric approach to the generalized Noether theorem
aa r X i v : . [ m a t h - ph ] S e p A geometric approach to the generalized Noether theorem
Alessandro Bravetti ∗ Instituto de Investigaciones en Matem´aticas Aplicadas y en Sistemas,Universidad Nacional Aut´onoma de M´exico, A. P. 70543, M´exico, DF 04510, Mexico
Angel Garcia-Chung † Departamento de F´ısica, Universidad Aut´onoma Metropolitana - Iztapalapa,San Rafael Atlixco 186, Ciudad de M´exico 09340, M´exico andUniversidad Panamericana,Tecoyotitla 366. Col. Ex Hacienda Guadalupe Chimalistac,C.P. 01050 Ciudad de M´exico, M´exico
We provide a geometric extension of the generalized Noether theorem for scaling symmetriesrecently presented in [37]. Our version of the generalized Noether theorem has several positivefeatures: it is constructed in the most natural extension of the phase space, allowing for the sym-metries to be vector fields on such manifold and for the associated invariants to be first integrals ofmotion; it has a direct geometrical proof, paralleling the proof of the standard phase space versionof Noether’s theorem; it automatically yields an inverse Noether theorem; it applies also to a largeclass of dissipative systems; and finally, it allows for a much larger class of symmetries than justscaling transformations which form a Lie algebra, and are thus amenable to algebraic treatments.
I. MOTIVATION AND PREVIOUS WORKS
There are good reasons why the theorems shouldall be easy and the definitions hard. As theevolution of Stokes’ Theorem revealed, a singlesimple principle can masquerade as severaldifficult results; the proofs of many theoremsinvolve merely stripping away the disguise. Thedefinitions, on the other hand, serve a twofoldpurpose: they are rigorous replacements for vaguenotions, and machinery for elegant proofs.M. Spivak [30].
Noether’s theorem is one of the most profound and beautiful results in mathematical physics. This is because itis very easy to state and prove (in fact, it is one of the central topics in elementary courses on mechanics) but itsconsequences are far-reaching, ranging from the standard conservation of energy and angular momentum in classicalmechanics, to the existence of Noether charges in general relativity (one of them being black holes’ entropy [32]), upto the reduction theorems in symplectic, Poisson, and contact geometry [1].However, it is also well-known that there exist some important transformations that act like symmetries but arenot Noether symmetries, for which it has long been believed that it does not exist an associated invariant quantity.The most famous example of such symmetries is that of
Kepler scalings t → λ t , q → λ q , S → λS , λ = const. (1)Indeed, it is known that this is a type of symmetry of the Kepler problem (actually, this is the symmetry underlyingKepler’s third law [36]), but it is not of Noether type, as it is clear because e.g. the action and the dynamics arerescaled by the transformation. This type of symmetries that rescale the dynamics by multiplying it by a constantterm is sometimes referred to in the literature as a scaling symmetry and they are examples of the more general dynamical similarities [28].Let us remark that scaling symmetries have profound physical consequences. For instance, they have been employedto derive generalizations of the virial theorem [12, 13, 37]; moreover, it has been recently argued that one can use such ∗ Electronic address: [email protected] † Electronic address: [email protected] symmetries in order to reduce a Hamiltonian system to a purely relational description in terms of the observables ofthe theory, where the (equivalent) dynamics can be shown to be free of some spurious singularities such as the bigbang singularity of Einstein’s equations [28, 29]. Therefore it is of primary importance to have a general theory ofsuch symmetries that may help recognize and classify them, and even more so if such theory could help identify theassociated conserved quantities (if any) that can then be used to perform the reduction of the system, as it is the casefor the standard Noether theorem.Recently, in [37] a generalized version of Noether’s theorem that applies to scaling symmetries has been proved.They proved that to any one-parameter family of transformations q → q ′ ( t ′ ), t → t ′ that rescales the Lagrangian(and hence the action) as L (cid:18) q ′ , d q ′ dt ′ , t ′ (cid:19) dt ′ dt = Λ L (cid:18) q , d q dt , t (cid:19) (2)up to a boundary term, where Λ is a constant, one can always associate a conserved quantity of the form Q = p · δ q − Hδt − S ( t ) δ Λ , (3)where p = ∂L∂ ˙ q are the conjugate momenta, H is the Hamiltonian, and S ( t ) = R t Ldτ is the on-shell action , namely,the action calculated along the trajectory. For instance, in the case of Kepler scalings (1), the associated invariant (3)is Q K = 2 p · q − tH K − S ( t ) , (4)where H K = p · p m − ǫ √ q · q , (5)is the Kepler Hamiltonian.One of the most intriguing aspects of such generalized Noether theorem is the fact that the invariants (3) are notfunctions on the phase space of the system, since they include also the on-shell action variable. Therefore, a naturalquestion arising about these invariants is the followingIn what sense are the invariants (3) first integrals of motion, that is, well-defined functions from the (possiblyextended) phase space of the system to the real numbers that are constant along trajectories?Or, more precisely,Can we define the generalized Noether theorem geometrically in some appropriate (extension of the) phasespace, so that the infinitesimal generators of scaling symmetries are actual vector fields on such manifold andtheir related invariants (3) are actual functions on the manifold?Moreover, if the above is possible, it would be great if this geometric structure would allow for a simple proof ofthe generalized Noether theorem, possibly paralleling the one of the standard Noether theorem, and also provide adirect geometric relationship between the infinitesimal generators and their corresponding invariants.An interesting approach to solve some of the above puzzles has been presented in [17], where the authors constructa Noether theorem for scaling symmetries from the variational principle in the vertical extension of the standardphase space. The advantages of such approach are its general scope and the fact that it effectively yields invariantsassociated with scaling symmetries which do not depend on computing the on-shell action. However, the relationshipbetween the original scaling symmetry and the constructed invariant is not so clear, since it turns out that the scalingsymmetry in the original configuration space is not the Noether symmetry associated with the invariant, and thatanother symmetry can be found to be its associated Noether symmetry. Moreover, the construction is based onperforming first the variational formulation in the vertical extension of the tangent space and then projecting backdown to the original manifold, which makes the derivation a bit obscure and can hamper the analysis of furtherquestions such as e.g. whether the new symmetries obtained from this construction form a Lie algebra.On the other hand, the presence of the term S ( t ) in (3) is remindful of contact Hamiltonian mechanics , wherethe contact phase space directly includes the on-shell action as an additional dynamical variable, thus serving as anatural arena in which the invariants (3) can be described. Indeed, a great deal of work has been devoted recentlyto the study of the so-called contact Hamiltonian systems , which are Hamiltonian systems on contact manifolds,the “odd-dimensional counterpart of symplectic manifolds” (see e.g. [2, 7, 9, 15, 19, 33, 34] for the formal theoryand [5, 8, 10, 14] and the references therein for some applications). In this context, the standard symplectic phasespace is extended to include an additional direction which in local coordinates turns out to be precisely the on-shell action of the system, and there is a natural way in which standard symplectic Hamiltonian systems can beembedded into contact Hamiltonian systems (and viceversa). Moreover contact Hamiltonian systems can be derivedfrom Herglotz’ variational principle [20, 21, 31], and a counterpart of the standard Noether theorem for such case hasbeen proposed both in the variational formulation [20, 21, 23] as well as in the geometric setting [16, 18].Motivated by all the above considerations, in this work we provide a geometric extension of the generalized Noethertheorem for scaling symmetries recently presented in [37]. To do so, we use the geometric formulation for contactHamiltonian systems in their extended phase space (including time). In this way we will be able to prove very easilya generalized Noether theorem and its inverse that include as a particular case the association of scaling symmetrieswith their related invariants found in [37]. Moreover, we can prove directly that all the Noether symmetries definedin this way form a Lie algebra. Further benefits of our approach are that the definition of generalized symmetries inthis context allows for even more general symmetries than just scaling symmetries of the dynamics, and that it alsoapplies directly to a large class of dissipative systems. Finally, in order to concretely show some of the consequencesof our approach, we will present some paradigmatic examples in which the general theory recovers known results andreveals new interesting features. II. A BRIEF INTRODUCTION TO SYMPLECTIC AND CONTACT HAMILTONIAN MECHANICS
In order to make this paper self-contained, we introduce here the very basic tools from symplectic and contactHamiltonian systems that are needed in the following. We refer to [7, 9, 15] for more detailed accounts and to [15, 18]for the Lagrangian counterparts.
A. The symplectic case
We start with some standard definitions, that can be found in any textbook on analytical mechanics, e.g. [1].
Definition 1.
The symplectic phase space of a mechanical system is the cotangent bundle T ∗ Q , where Q is theconfiguration manifold of the system, endowed with the canonical symplectic form Ω = − dα , with α being the Liouville1-form. In local Darboux coordinates ( q a , p a ), we have that α = p a dq a , and thus Ω = dq a ∧ dp a . Definition 2. A symplectic Hamiltonian system is a triple ( T ∗ Q, Ω , H ) , with H : T ∗ Q → R a sufficiently regularfunction called the symplectic Hamiltonian. Definition 3. A symplectic Hamiltonian vector field X H is defined to be the only solution to the condition ι X H Ω = − dH . (6)One can directly check that in Darboux coordinates this leads to the standard Hamilton equations˙ q a = ∂H∂p a ˙ p a = − ∂H∂q a , (7)from which one can recover e.g. the Newtonian dynamics of conservative systems.Now we are ready to define Noether symmetries and conserved quantities. Definition 4. A Noether symmetry of a symplectic Hamiltonian system is a vector field Y ∈ X ( T ∗ Q ) suchthat ι Y Ω = − dF ( Y is Hamiltonian for some Hamiltonian function F ) and L Y H = 0 ( Y preserves H ). Definition 5. A conserved quantity is a function F : T ∗ Q → R such that L X H F = 0 . Furthermore, we note that the condition (6) provides an isomorphism between the Lie algebra of Hamiltonian vectorfields on T ∗ Q with the Lie bracket and the Lie algebra of functions on T ∗ Q with the Poisson bracket { F, G } P := ι X F ι X G Ω . (8)Equipped with this isomorphism, a Noether symmetry can equivalently be expressed as a Hamiltonian vector field X F such that { F, H } P = 0. Thus, Noether’s theorem can be proved in one line using the antisymmetry of the Poissonbracket, to obtain (see [11] for more comments) Theorem 1 (Symplectic Noether) . X F is a Noether symmetry of a symplectic Hamiltonian system if and only if F is a conserved quantity. Now, in order to understand Noether’s theorem from a more general perspective, we need to introduce some generaldefinitions.
Definition 6. A dynamical similarity of a vector field X is a vector field Y ∈ X ( T ∗ Q ) such that [ Y, X ] = Λ X ,for some (in general non-constant) function Λ . As a particular but important case of the above, we have
Definition 7. A dynamical symmetry of a vector field X is a vector field Y ∈ X ( T ∗ Q ) such that [ Y, X ] = 0 . It is easy to verify that any Noether symmetry is a dynamical symmetry of X H ; however the converse is not true,as the following proposition in the particular case λ = λ clearly shows. Proposition 1.
Let Y ∈ X ( T ∗ Q ) be such that L Y Ω = λ Ω ( Y is a non-strictly canonical symmetry [12, 13, 28])and L Y H = λ H ( Y rescales H ). Then [ Y, X H ] = ( λ − λ ) X H .Proof. We have ι [ X H ,Y ] Ω = ι X H L Y Ω − L Y ι X H Ω = λ ι X H Ω + d L Y H = ( λ − λ ) dH , (9)and therefore, since Ω is non-degenerate, we conclude that [ Y, X H ] = ( λ − λ ) X H .Clearly, by the isomorphism described above, we know that we cannot associate any function of the phase space to avector field that rescales Ω (these are not Hamiltonian), and therefore we do not have much hope to extend Noether’stheorem in order to include dynamical symmetries and similarities in the standard phase space. As a particularlyrelevant example for our exposition, we first note that Kepler scalings (1) can be shown to be induced by the actionof the following vector field on the symplectic phase space Y KS = 2 q a ∂ q a − p a ∂ p a . (10)Then we have the following result Proposition 2. Y KS is a dynamical similarity of Kepler’s Hamiltonian vector field.Proof. Let H K be the Kepler Hamiltonian (5) and Ω K the canonical symplectic form. Then, as one can directly check, L Y KS H K = − H K and L Y KS Ω K = Ω K , and therefore by Proposition 1, we have [ Y KS , X H K ] = − X H K .We remark that the result [ Y KS , X H K ] = − X H K appearing in the proof of Proposition 2 is expected, as it meansthat the dynamics after the transformation induced by Y KS is rescaled by λ − , with λ being the mock parameter alongthe flow of Y KS . Therefore by scaling the time variable as in (1), we would observe no difference in the trajectories,and therefore we can consider Y KS as a “symmetry” of the Kepler dynamics in some sense (note also that the properscaling of the action in (1) can be recovered by dimensional analysis). However, as a consequence of Proposition 2and the above discussion, we conclude that it would be nonsensical to look for a conserved quantity associated to Y KS in the standard symplectic phase space. B. The contact case
As a further generalization of the above results, we now provide analogue definitions for the contact case andproceed to prove the “standard” Noether theorem in this case.
Definition 8.
The contact phase space is the canonical contactification of the cotangent bundle, that is, themanifold T ∗ Q × R endowed with the contact 1-form η = dS − π ∗ S α , where π S : T ∗ Q × R → T ∗ Q is the standardprojection, and S is the global coordinate on R . Associated to the contact 1-form η , there is a unique vector field, called the Reeb vector field , defined by theconditions ι R dη = 0 and ι R η = 1. In local Darboux coordinates ( q a , p a , S ), we have that η = dS − p a dq a and R = ∂/∂S . Definition 9. A contact Hamiltonian system is a triple ( T ∗ Q × R , η, h ) , with h : T ∗ Q × R → R a sufficientlyregular function called the contact Hamiltonian. Definition 10. A contact Hamiltonian vector field X h is defined to be the only solution to the conditions ι X h dη = dh − R ( h ) η and ι X h η = − h . (11)One can directly check that in Darboux coordinates this leads to the contact Hamiltonian equations˙ q a = ∂h∂p a ˙ p a = − ∂h∂q a − p a ∂h∂S ˙ S = p a ∂h∂p a − h , (12)from which one can recover the standard Hamilton equations (7) for q a and p a whenever h does not depend on S .Note also that from the last equation in (12), the new variable S is the action of the system evaluated along thetrajectories of the dynamics , that is the on-shell action . Moreover, the contact Hamiltonian equations (12) can beused to model mechanical systems with different types of dissipative terms (see e.g. [6, 7, 18]).Now we are ready to define Noether symmetries and the analogue of conserved quantities in the contact case. Firstwe point out that the condition (11) provides a (global) isomorphism between the Lie algebra of contact vector fieldson T ∗ Q × R with the Lie bracket and the Lie algebra of functions on T ∗ Q × R with the Jacobi bracket { F, G } J := ι [ X F ,X G ] η . (13)Then, starting from the symplectic analogue and from (13), we have the following natural definitions. Definition 11. A Noether symmetry of a contact Hamiltonian system is a contact Hamiltonian vector field X F ∈ X ( T ∗ Q × R ) such that { F, h } J = 0 . Moreover, we can generalize the concept of conserved quantities to the case of dissipative systems, which is thegeneral case for contact systems, as follows.
Definition 12. A dissipated quantity of a contact Hamiltonian system is a function F : T ∗ Q × R → R thatis dissipated at the same rate as the contact Hamiltonian, i.e. L X h F = − R ( h ) F (recall that L X h h = − R ( h ) h ). Then, the contact version of Noether’s theorem goes as follows [16, 18]:
Theorem 2 (Contact Noether) . X F is a Noether symmetry of a contact Hamiltonian system if and only if F = − ι X F η is a dissipated quantity.Proof. The proof directly follows from the fact that L X h F = − L X h ( ι X F η ) = − ι X F L X h η − ι L Xh X F η = R ( h ) ι X F η − ι [ X h ,X F ] η = − R ( h ) F − ι [ X h ,X F ] η . Theorem 2 has been proved previously in this form in [16, 18], while similar results for the non-dissipative casewere already elucidated in [3]. Here we discuss a slight difference between our approach and theirs, mostly related tothe definition of symmetries. In [18] a more restrictive definition of symmetry has been considered, that is, Y suchthat [ Y, X h ] = 0. This has been called an infinitesimal dynamical symmetry in [18] and Noether’s theorem (thereincalled dissipation theorem ) has been proved for such symmetries, analogously to the proof presented here. However,the condition for Y to be a Noether symmetry (Definition 11) is more general (it allows for [ X F , X h ] = 0), andtherefore some symmetries may escape the more restrictive definition in [18]. On the other hand, in [16] the mostgeneral definition of symmetry has been considered that can lead to Theorem 2. Indeed, from the proof of Theorem 2,one finds out that the necessary and sufficient condition for the existence of a dissipated quantity associated to avector field Y , is to have ι [ X h ,Y ] η = 0. This is indeed the definition of symmetry considered in [16], where it hasbeen referred to as a dynamical symmetry . However, in such case the definition is too weak, and this hampers the 1:1correspondence between dissipated quantities and symmetries which constitutes part of the beauty of the standardNoether theorem. Note also that the term “dynamical symmetry” employed in such reference may be confusingwith respect to Definition 7. Therefore we will refer to such symmetries as generalized dynamical symmetries in thefollowing discussion. For these reasons we prefer to consider a different concept of symmetry, Noether’s symmetry, asstated in Definition 11. Indeed, for a given dissipated quantity F , our definition selects the associated symmetry inthe class of generalized dynamical symmetries corresponding to F that coincides exactly with the contact Hamiltonianvector field associated with F (cf. [16, Remark 3]).Moreover, the following important Corollary of Theorem 2 has been observed already both in [16] and in [18]. Corollary 1.
Given two dissipated quantities F and F , their ratio F /F , whenever it is well-defined, is a conservedquantity (note also that the contact Hamiltonian is, by definition, a dissipated quantity). At this point we are ready to focus on the comparison between Theorem 1 and Theorem 2, and on discussing therole of dynamical similarities (including e.g. Kepler scalings) in the contact case.To begin, a direct comparison between the symplectic and the contact versions of Noether’s theorem leads to thefollowing observation: in the case where h does not depend on S , both h itself and any other dissipated quantity are infact conserved quantities, and therefore we get a generalization of the symplectic version of Noether’s theorem, giventhat now both the symmetries and their associated invariants can depend explicitly on S , similarly to (3). However,it is also clear that the dissipated (or conserved) quantities associated to Noether’s symmetries in the contact caseare functions of positions, momenta and the action only, and thus (3) cannot be recovered, since in general they arefunctions of the time variable too. To further illustrate this point, we proceed to show that, as in the symplectic case,Kepler scalings (1) are not Noether symmetries according to any of the definitions discussed above. To do so, first wenote that Kepler scalings are induced by the action of the following vector field on the contact phase space Y cKS = 2 q a ∂ q a − p a ∂ p a + S∂ S . (14)Then we have the following result, whose proof is exactly analogous to that of the corresponding result in the symplecticcase and can be also obtained by a direct calculation Proposition 3. Y cKS is a non-trivial dynamical similarity of Kepler’s contact Hamiltonian vector field. In particular [ Y cKS , X Kh ] = − X Kh , where X Kh is the contact Hamiltonian vector field associated to Kepler’s Hamiltonian (5) . Finally, we have the following important no-go result.
Proposition 4.
Any non-trivial dynamical similarity of a contact Hamiltonian vector field is not a generalizeddynamical symmetry. In particular they are not Noether symmetries.Proof.
By definition of a non-trivial dynamical similarity, we have [
Y, X h ] = Λ X h , with Λ = 0. Therefore ι [ Y,X h ] η =Λ ι X h η = − Λ h = 0.We conclude that scaling symmetries cannot be Noether symmetries in the contact phase space, as it was the casein the symplectic phase space. In the next section we show how to extend the contact version of Noether’s theoremin order to include such symmetries. III. THE GENERALIZED NOETHER THEOREM
In this section we extend all the previous results to the case of generalized symmetries defined on the extendedcontact phase space and their associated dissipated (conserved) quantities. We will see that this is at the same timean extension of the symplectic and contact versions of Noether’s theorems, and of the generalized Noether theoremfor scaling symmetries proved in [37].As usual, we start with the necessary definitions. In order to consider contact Hamiltonian systems with an explicittime dependence, we follow [7] and define the following.
Definition 13.
We call the extended contact phase space the manifold T ∗ Q × R × R , endowed with the 1-form η E = π ∗ t η + hdt , where π t : T ∗ Q × R × R → T ∗ Q × R is the projection and t is the coordinate on R . Note that this extension is not canonical, in the sense that it depends on the system at hand through h , and that( T ∗ Q × R × R , η E ) in general is a pre-symplectic manifold [27]. Indeed, dη E is non-degenerate (hence symplectic) ifand only if ∂h/∂S = 0.Moreover, we define time-dependent contact Hamiltonian vector fields as follows. Definition 14. A time-dependent contact Hamiltonian vector field X th is the solution to the conditions ι X th dη E = − R ( h ) η E and ι X th η E = 0 , (15)where R ( h ) = ∂h∂S . We point out that this definition is different from that of a pre-symplectic system, as the righthand side of the first condition in (15) is not a closed 1-form [27]. Note that in this case X th is not uniquely fixedby the two conditions in (15), as one has the freedom to choose a global function f : T ∗ Q × R × R → R − { } thatmultiplies X th , that is, X th is uniquely defined up to (nowhere-vanishing) rescalings. For instance, fixing f = 1, in localadapted coordinates one recovers the contact Hamiltonian equations for time-dependent Hamiltonians˙ q a = ∂h∂p a ˙ p a = − ∂h∂q a − p a ∂h∂S ˙ S = p a ∂h∂p a − h , ˙ t = 1 , (16)from which we infer that changing the function f amounts to reparametrizing the dynamics.To avoid clutter of notation and without loss of generality, we will also assume from now on that f = 1, that is, X th = X h + ∂ t , where X h is the contact Hamiltonian vector field associated with h . All the results can be easilygeneralized by including the factor f back into the corresponding equations.Now we are ready to study symmetries and associated dissipated (conserved) quantities in this extended phasespace. Definition 15.
We call Y ∈ X ( T ∗ Q × R × R ) a generalized Noether symmetry of a time-dependent contactHamiltonian system if L Y η E = λ η E for some λ ∈ C ∞ ( T ∗ Q × R × R ) . One can check that this definition includes the previous definitions of Noether symmetries of symplectic and contactHamiltonian systems (cf. Theorem 4 below). However, in this case we allow also for the symmetry to have a non-zerotime component, and therefore such symmetries naturally allow for time rescalings.We have the following important property.
Proposition 5.
Generalized Noether symmetries form a Lie algebra with the Lie bracket.Proof.
Let L Y η E = λ η E and L Y η E = λ η E . Then L [ Y ,Y ] η E = L Y L Y η E − L Y L Y η E = ( Y ( λ ) − Y ( λ )) η E . We can also define dissipated quantities in analogy with the standard contact case.
Definition 16. A dissipated quantity in the extended contact phase space is a function F : T ∗ Q × R × R → R that satisfies L X th F = − R ( h ) F . (17)
In the following we will refer to condition (17) as the dissipation equation . Note that the Hamiltonian is not necessarily a dissipated quantity itself. Indeed, as it happens in the symplecticcase where H is not conserved whenever it depends explicitly on t , here h is not a dissipated quantity whenever itdepends explicitly on t .Next, we have the following Lemma. Lemma 1.
Let Y be a generalized Noether symmetry. Then ι [ Y,X th ] η E = 0 .Proof. ι [ Y,X th ] η E = [ L Y , ι X th ] η E = L Y ι X th η E − ι X th L Y η E = − ι X th λη E = 0 . Now we are ready to prove the generalized Noether Theorem.
Theorem 3 (Generalized Noether Theorem) . Let Y be a generalized Noether symmetry of a time-dependent contactHamiltonian system. Then F = − ι Y η E is a dissipated quantity. As usual, the proof is a one-liner:
Proof. L X th F = − L X th (cid:0) ι Y η E (cid:1) = − ι Y L X th η E − ι L Xth Y η E = R ( h ) ι Y η E − ι [ X th ,Y ] η E = − R ( h ) F − ι [ X th ,Y ] η E . We remark that the dissipated quantity associated with a generalized Noether symmetry is obtained exactly in thesame way as in Theorem 2, with η replaced by η E . This replacement is fundamental in order to obtain dissipated(conserved) quantities associated to scaling symmetries of the type obtained in [37] (cf. Equation (3)).Clearly, we could have proved Theorem 3 even under the more general definition of symmetry ι [ Y,X th ] η E = 0.However, in such case we would lose the possibility to associate to any invariant a somewhat “unique” symmetry, aswe now show. Theorem 4 (Inverse generalized Noether Theorem) . Let F be a dissipated quantity. Then Y F = X F + Y t X th (18) is the most general form of its associated generalized Noether symmetry, where X F is the contact Hamiltonian vectorfield associated to F (see Definition 10) and Y t is a free function.Proof. Given F , we wish to find the corresponding Y that satisfies L Y η E = λη E ( Y generalized Noether symmetry)and ι Y η E = − F , (the associated dissipated quantity is F ). Using these two conditions and Cartan’s identity, we getimmediately ι Y dη E = dF + λη E , which, when written in Darboux coordinates is equivalent to the following conditions λ = − Y t ∂h∂S − ∂F∂S (19)and Y a = Y t ∂h∂p a + ∂F∂p a (20) Y a = − Y t (cid:20) ∂h∂q a + p a ∂h∂S (cid:21) − ∂F∂q a − p a ∂F∂S (21) Y a ∂h∂q a + Y a ∂h∂p a + Y S ∂h∂S = − (cid:18) Y t ∂h∂S + ∂F∂S (cid:19) h + ∂F∂t . (22)Now from ι Y η E = − F and the above conditions, we get the additional requirement Y S = Y t (cid:18) p a ∂h∂p a − h (cid:19) + p a ∂F∂p a − F . (23)From (20), (21) and (23) we get that Y = X F + Y t ( X h + ∂ t ) = X F + Y t X th , while equation (22) can be rewrittenafter some algebra as X th ( F ) = − ∂h∂S F , that is, the dissipation equation (17).Theorems 3 and 4 are the main contributions of this work. Let us make some comments about these results:first, we get a generalization of the contact version of Noether’s theorem, as now the symmetries and the associatedinvariants can depend on t explicitly. Additionally, if h does not depend on S and t , then we get a generalization ofthe symplectic version of Noether’s theorem, given that the symmetries and the associated invariants now can dependon S and t explicitly. This will be the case of e.g. Kepler scalings. Secondly, as anticipated, in the most generalcase L X th h = − R ( h ) h + ∂h∂t , and therefore whenever h depends explicitly on t , it is not a dissipated quantity itself.This implies that whenever h depends on t we lose the fact that F/h is a conserved quantity, as it happened in thecontact version of Noether theorem. However, whenever h does not depend on t we have that F/h is still a conservedquantity, even when F depends on t . Moreover, we still have the following analogue of Corollary 1. Corollary 2.
Given two dissipated quantities F and G , the function F/G , whenever defined, is a conserved quantity.
Finally, note that for Y F as in (18), we have ι Y F η E = − F . Moreover observe that, since ι Y t X th η E = 0 and L Y t X th η E = − Y t R ( h ) η E , we can always add a term of the form Y t X th to any given generalized Noether symmetryand obtain another generalized Noether symmetry corresponding to the same invariant. Therefore such “gaugefreedom” in the fixing of the symmetry is unavoidable. We say that the correspondence between symmetries andinvariants is 1:1 up to the mentioned gauge freedom .A fundamental question for our discussion at this point is the relationship between generalized Noether symmetriesand the dynamical similarities of X th . This is the content of the next two propositions. We start with a lemma. Lemma 2.
Let A ∈ X ( T ∗ Q × R × R ) be such thati) A ∈ ker η E ii) ι A dη E = f A η E , for some function f A .Then A = A t X th , with f A = − ( ∂h∂S ) A t .Proof. We have A ∈ ker η E = ⇒ A S = p a A a − hA t . Then we can directly compute that ι A dη E = f A η E = ⇒ A a = A t ∂h∂p a (24) A a = − A t ∂h∂q a + f A p a (25) f A = − ∂h∂S A t (26) f A h = A a ∂h∂q a + A a ∂h∂p a + A S ∂h∂S . (27)Using (26) we can rewrite the other equations as A a = A t ∂h∂p a (28) A a = − A t (cid:18) ∂h∂q a + p a ∂h∂S (cid:19) (29) A S = A t (cid:18) p a ∂h∂p a − h (cid:19) , (30)meaning that A = A t X h + A t ∂ t = A t X th , as claimed.Now we can prove the following important implication, which states that generalized Noether symmetries are asubset of the dynamical similarities of X th . Proposition 6. Y generalized Noether symmetry = ⇒ [ Y, X th ] = ˜ λX th .Proof. To prove the result, we will prove that [
Y, X th ] ∈ ker η E and that ι [ Y,X th ] dη E = f η E for some function f , andthen we use Lemma 2. We start with L Y η E = λη E = ⇒ ι Y dη E − dF = λη E (31)On the other hand, ι [ X th ,Y ] η E = L X th ι Y η E − L Y ✟✟✟ ι X th η E − ι X th ι Y dη E = − ι X th dF + ι Y ι X th dη E = (31) λι X th η E = 0 , (32)and therefore [ X th , Y ] ∈ ker η E . Finally, a direct computation shows that ι [ Y,X th ] dη E = L [ Y,X th ] η E = L Y L X th η E − L X th L Y η E = − h L Y R ( h ) + L X th λ i η E (33)and therefore by Lemma 2 we conclude.The following example illustrates that in general the inclusion defined by Proposition 6 is strict, i.e., generalizedNoether symmetries are a proper subset of dynamical similarities. Example 1.
Let X tK be the contact Hamiltonian vector field associated with the Kepler Hamiltonian (5) in theextended contact phase space and consider the vector field Y = H K ∂ S . One can directly check that [ Y, X tK ] = 0 , andthus Y is a (trivial) dynamical similarity. However, it is not a generalized Noether symmetry, as L Y η E = λη E forany function λ . Despite the above remark, we have the next proposition, which is an immediate corollary of Theorem 4 and providesin some sense the inverse of Proposition 6, as it guarantees that for any dynamical similarity there exists an associateddissipated (conserved) quantity, and thus a related generalized Noether symmetry.
Proposition 7.
Let Y be such that [ Y, X th ] = λX th . Then F := − ι Y η E is a dissipated quantity and there exists ageneralized Noether symmetry associated with F . Proof.
Since [
Y, X th ] = λX th , then ι [ Y,X th ] η E = 0 and thus from the proof of Theorem 3 we know that F is a dissipatedquantity. Hence, from Theorem 4 it follows that there is a generalized Noether symmetry associated to F .At this point we have reached our aim, that is, we have found the appropriate extension of the phase space andthe appropriate definition of (generalized) Noether symmetries in such space such that we can guarantee that to anydynamical similarity of X th we can associate a dissipated (conserved) quantity via the generalized Noether theorem.This will be the case of e.g. Kepler scalings, as we will prove shortly. IV. SCALINGS AS GENERALIZED NOETHER SYMMETRIES
To keep the discussion as general as possible, let us consider a contact Hamiltonian of the form h = 12 m p · p + f ( t ) V ( q ) + g ( S ) , (34)where V ( ξ q ) = ξ k V ( q ) , g ( ξS ) = ξ κ g ( S ) , (35)i.e., V ( q ) and g ( S ) are homogeneous functions of degree k and κ respectively. The functions f ( t ) and g ( S ) are so fararbitrary functions of time t and the variable S .Consider now a general scaling transformation q ′ = ζ α q , p ′ = ζ β p , S ′ = ζ γ S, t ′ = ζ σ t, (36)where ζ ∈ R > is the scaling parameter. The corresponding infinitesimal transformation is of the form δ q = (cid:18) ∂∂ζ q ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ζ =1 = α q , δ p = (cid:18) ∂∂ζ p ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ζ =1 = β p , δS = (cid:18) ∂∂ζ S ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ζ =1 = γS, δt = (cid:18) ∂∂ζ t ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ζ =1 = σt, (37)and thus the associated generator Y = Y a ∂ q a + Y a ∂ p a + Y S ∂ S + Y t ∂ t has components Y a = αq a , Y a = βp a , Y S = γS, Y t = σt. (38)We now insert these components in (18) and obtain the following equations αq a = ∂F∂p a + Y t p a m , (39) βp a = − ∂F∂q a − p a ∂F∂S − Y t (cid:18) f ( t ) ∂V∂q a + p a dgdS (cid:19) , (40) γS = p a ∂F∂p a − F + Y t (cid:18) m p · p − f ( t ) V ( ~q ) − g ( S ) (cid:19) , (41) σt = Y t . (42)In order to find the solution, we proceed as follows. First, we replace equation (42) into (39), (40) and (41). Thisyields the new set of equations αq a = ∂F∂p a + σt p a m , (43) βp a = − ∂F∂q a − p a ∂F∂S − σt (cid:18) f ( t ) ∂V∂q a + p a dgdS (cid:19) , (44) γS = p a ∂F∂p a − F + σt (cid:18) m p · p − f ( t ) V ( ~q ) − g ( S ) (cid:19) . (45)1Then, we solve for ∂F∂p a in equation (43) and obtain ∂F∂p a = αq a − σt p a m , (46)which can now be inserted in (45) to obtain the associated dissipated quantity F = α q · p − σt h ( q , p , S, t ) − γS , (47)which should now be compared to (3). The third step is to insert this result into equation (44), which leads to thefollowing condition ( β − γ + α ) + σt dgdS = 0 . (48)As g ( S ) is a time independent function, there are only two possibilities to find a solution of (48): (i) dgdS = 0, whichimplies g ( S ) = g = constant, or (ii) σ = 0. Let us analyze each case separately. A. Case (i): Homogeneous non-dissipative time-dependent systems
In this case g ( S ) = constant and this constant can be taken to be zero without loss of generality. Therefore, theHamiltonian function (34) takes the following expression h = 12 m p · p + f ( t ) V ( q ) . (49)Recall that since in this case h does not depend on S , then any dissipated quantity is in fact a conserved one.We now return to (48) and notice that it can be used to fix β in terms of α and γβ = γ − α. (50)Now, for F as in (47) and h as above, we can rewrite the dissipation equation (17) as (cid:16) α − γ − σ (cid:17) p · p m + ( γ − σ ) f ( t ) V ( q ) − αf ( t ) q · ∇ V − σt ˙ f ( t ) V = 0 . (51)Clearly, the coefficient of the p · p term must be zero, hence α = γ σ . (52)On the other hand, the homogeneity of the potential V ( q ) yields the following relation q · ∇ V ( q ) = kV ( q ) , (53)which is used to replace q · ∇ V ( q ) in (51). As a result we obtain the condition f ( t ) (cid:20) γ (cid:18) − k (cid:19) − σ (cid:18) k (cid:19)(cid:21) = σt ˙ f ( t ) . (54)We can now consider three cases: Case (1). If σ = 0, then (54) gives γ (cid:18) − k (cid:19) = 0 . (55)which admits two solutions: γ = 0 and k = 2. The first option yields a trivial solution, F = 0, and the second solutiongives a conserved quantity F of the form F ( q , p , S ) = q · p − S. (56) Case (2).
If ˙ f ( t ) = 0, then (54) results in γ (cid:18) − k (cid:19) − σ (cid:18) k (cid:19) = 0 , (57)which admits three possible solutions:22.1) If k = 2, then σ = 0, and this recovers the second solution of Case (1) , where F was defined.2.2) If k = −
2, then γ = 0, and the conserved quantity takes the form F ( q , p , t ) = q · p − t h ( q , p , t ) . (58)Note that in this particular case the dependence on S in the conserved quantity disappears. This is because inthis case scaling transformations are canonical symmetries [22].2.3) If k = ±
2, then γ = (2+ k )(2 − k ) σ . In this case, the conserved quantity is F ( q , p , S, t ) = 22 − k q · p − t h ( q , p , t ) − (2 + k )(2 − k ) S . (59)Note that this case contains the Kepler system with Hamiltonian (5). Given that in such case the potential ishomogeneous of degree k = −
1, the conserved quantity takes the form F ( K )2 = 23 q · p − tH K − S , (60)which coincides with (4) up to multiplication by an irrelevant factor of 3. Observe also that using (4) in (18) andchoosing Y t = 3 t one obtains Y Q K = 3 t ∂ t + 2 q a ∂ q a − p a ∂ p a + S∂ S , which is precisely the generator of Kepler scalings in the extended contact phase space, thus confirming that theseare generalized Noether symmetries. Case (3). If σ, ˙ f ( t ) = 0, then (54) admits a solution for a very specific function f ( t ), depending on the parameterΛ := γ/σ as follows f ( t ; Λ) = t (2 − k )Λ2 − (2+ k )2 , (61)and F is of the form F ( q , p , S, t ) = Λ F ( q , p , S ) + F ( q , p , t ; Λ) . (62)As a particular example of this case, we can consider a time-dependent Kepler problem with the following Hamiltonianfunction H tK ( t ) = p · p m − f ( K ) ( t ; Λ) 4 ǫ √ q · q , (63)where f ( K ) ( t ; Λ) can be derived using (61) and recalling that k = −
1, that is, f ( K ) ( t ; Λ) = t − . (64)From (62) we obtain the following expression for the conserved quantity for this system F ( q , p , S, t ) = (Λ + 1) q · p − H tK t − S . (65)
B. Homogeneous dissipative time-dependent systems If dgdS = 0, then σ = 0 in (48) and this yields a condition for β as given in (50). Moreover, the Hamiltonian in thiscase is still of the form (34) but the expression for the function F is now given as F = α q · p − γS. (66)Inserting this expression into the dissipation equation (17), we obtain (cid:16) α − γ (cid:17) p · p m − αf ( t ) kV ( q ) + γf ( t ) V ( q ) + γg ( S ) − γS dgdS = 0 , (67)3where the homogeneity condition of the potential (53) was used. As before, the coefficient of p · p has to be zero,which means α = γ , (68)and now (67) takes the form γf ( t ) kV ( q ) (cid:18) − k (cid:19) + γg ( S ) (1 − κ ) = 0 , (69)where κ is the homogeneity degree of g ( S ).Again, there are three cases to be considered to obtain solutions of this algebraic equation. The case in which γ = 0gives the trivial solution, F = 0. The other two cases, (i) k = 0 and κ = 1 and (ii) k = 2 and κ = 1 yield the samedissipated function F which turns out to be F (cf. equation (56)). Remarkably, in both cases the dissipative term g ( S ) is forced to be of the form g ( S ) = g S .In summary, a Hamiltonian of the type (34) admits a scaling symmetry (36) only in cases where g ( S ) is a linearfunction and the homogeneity degree of the potential is k = 0 or k = 2. We conclude by remarking that for thecase of the harmonic oscillator, where k = 2, the dissipated function (56) has been already found in [7] by a directcalculation. V. HARMONIC TYPE POTENTIALS WITH LINEAR DISSIPATION
Clearly, Theorems 3 and 4 do not apply only to scaling symmetries (36). To illustrate this point, let us concludeby considering one-dimensional systems with Hamiltonians of the type h = 12 m p + m f ( t ) q + g S, (70)where f ( t ) is an arbitrary function of time t and g is an arbitrary real parameter. These systems can model e.g. aone dimensional harmonic oscillator with a linear dissipation and a time-dependent factor in the potential. We recallthat in this case, due to the explicit time dependence, the Hamiltonian h is not a dissipated quantity.To derive the symmetries of this system let us write the dissipation equation (17) explicitly pm ∂F∂q − ( mf ( t ) q + λp ) ∂F∂p + (cid:18) m p − m f ( t ) q − g S (cid:19) ∂F∂S + ∂F∂t + g F = 0 . (71)Let us look for generalized Noether symmetries of the Hamiltonian (70) for two particular cases: (i) the non-dissipative Hamiltonians, where g = 0, and (ii) linear dissipative term, where g = 0. Each of these cases will betreated using the following ansatz for F : F ( q, p, S, t ) = A ( q, S, t ) p + B ( q, S, t ) p + C ( q, S, t ) . (72)In the first case, when g = 0, the ansatz (72) yields a solution of the form F ( q, p, S, t ) = C F ( q, p, S ) + C F LR ( q, p, t ) , (73)where F was defined in (56) and F LR is given by F LR ( q, p, t ) = ρ ( t ) m p − ρ ( t ) ˙ ρ ( t ) qp + m (cid:18) ˙ ρ ( t ) + ρ ρ ( t ) (cid:19) q . (74)Note that, since the dissipation equation is linear, we have obtained a linear combination of two independent solutions, F and F LR . Moreover, F LR is the well-known Lewis-Riesenfeld invariant [24, 25], and ρ ( t ) has to satisfy the auxiliaryErmakov equation ¨ ρ ( t ) + f ( t ) ρ ( t ) = ρ ρ ( t ) , (75)where ρ is an arbitrary real constant. It is not surprising that we have obtained the Lewis-Riesenfeld invariant as ageneralized Noether invariant, as it can be shown to be a standard Noether invariant [26]. Moreover, we emphasizethat both F and F LR are conserved quantities in this case.4To generalize the above discussion, let us consider now the case where g = 0 and use the same ansatz (72). Adirect calculation shows that in this case the solution of the dissipation equation (71) takes the form F ( q, p, S, t ) = C F ( q, p, S ) + C F GLR ( q, p, t ) + C F EM ( q, p, t ) (76)where F GLR and F EM are defined as F GLR ( q, p, t ) = a ( t ) p m + 12 ( g a ( t ) − a ( t ) ˙ a ( t )) qp + (cid:20) ˙ a ( t ) − g a ( t ) ˙ a ( t ) + g a ( t )4 + a a ( t ) (cid:18) a g (cid:19)(cid:21) mq , (77) F EM ( q, p, t ) = b ( t ) p + m ˙ b ( t ) q , (78)and the auxiliary functions a ( t ) and b ( t ) satisfy the equations¨ a ( t ) + f ( t ) a ( t ) − g a ( t ) = a a ( t ) (cid:18) a g (cid:19) , (79)¨ b ( t ) + g ˙ b ( t ) + f ( t ) b ( t ) = 0 , (80)where a is an arbitrary real constant.The function F GLR can be considered as a generalization of the Lewis-Riesenfeld invariant F LR to the case wherethe system has a further linear dissipative term. This can be checked by taking g = 0 in (77) and (79) and observingthat they reduce to (74) and (75) respectively. Contrary to the previous case where g = 0, these dissipated quantitiesare now not conserved. However, according to Corollary 2, their quotient is a conserved quantity.To conclude this section, let us write the generalized Noether symmetry associated with the generalized Lewis-Riesenfeld dissipated quantity F GLR , which reads X F GLR = (cid:20) a ( t ) pm + 12 ( g a ( t ) − ˙ a ( t )) q (cid:21) ∂∂q + (cid:20)
12 ( g a ( t ) − ˙ a ( t )) p + m (cid:18) f ( t ) a ( t ) − g a ( t ) + ¨ a ( t )2 (cid:19) q (cid:21) ∂∂p ++ (cid:20) a ( t ) p m − m (cid:18) f ( t ) a ( t ) − g a ( t ) + ¨ a ( t )2 (cid:19) q (cid:21) ∂∂S , (81)where we are considering Y t = 0 in (18). VI. CONCLUSIONS AND FUTURE WORK
We have proved a geometric extension of the generalized Noether theorem for scaling symmetries recently putforward in [37]. Our construction stems from the observation that the invariants associated with scaling symmetries,cf. (3), in general include an explicit dependence on the on-shell action of the system and on the time variable, andtherefore we argued that the extended contact phase space is the appropriate minimal geometric setting to includesuch invariants and their related symmetries. Indeed, by carefully constructing the extended contact phase space andthe related Hamiltonian dynamics, we have shown that a sensible definition of Noether symmetries exists such that anextension of the generalized Noether theorem for scaling symmetries that applies to all dynamical similarities and alsoto some dissipative systems can be immediately found (Theorem 3), together with its inverse statement (Theorem 4).As we have argued, these theorems have several positive features: in the first place, their proofs are very simpleand are natural generalizations of their counterparts in the standard symplectic and contact cases; moreover, theyapply equivalently to conservative systems and to those dissipative systems that a admit a description in terms oftime-dependent contact Hamiltonian systems; by construction, in this space the thus-obtained conserved or dissipatedquantities are actual functions on the manifold and do not contain spurious terms involving integrals over the dynamics;finally, one can directly show that the generalized Noether symmetries thus derived form a Lie algebra, and thereforethey are amenable to be treated in algebraic terms, in the lines of e.g. [27].We hope that the analysis initiated in this work will be helpful to clarify the origin and structure of scalingsymmetries and their related invariants, by putting them on the same footing as other standard Noether symmetries.As further developments, we will address the comparison of our results with the “Eisenhart-Duval lift” of mechanicalsystems employed in [36], with the “vertical extension” of the tangent space presented in [17], with the reduction in thepre-symplectic setting described in [27], and also with the “unit-free approach” to Hamiltonian mechanics introduced5in [35], and we will provide a deeper analysis of the Lie-algebraic structure of the generalized Noether symmetries forvarious systems of interest in physics, e.g. in cosmology [28, 29]. 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