Symmetries, conservation and dissipation in time-dependent contact systems

In contact Hamiltonian systems, the so-called dissipated quantities are akin to conserved quantities in classical Hamiltonian systems. In this paper, we prove a Noether's theorem for non-autonomous contact Hamiltonian systems, characterizing a class of symmetries which are in bijection with dissipated quantities. We also study other classes of symmetries which preserve (up to a conformal factor) additional structures, such as the contact form or the Hamiltonian function. Furthermore, making use of the geometric structures of the extended tangent bundle, we introduce additional classes of symmetries for time-dependent contact Lagrangian systems. Our results are illustrated with several examples. In particular, we present the two-body problem with time-dependent friction, which could be interesting in celestial mechanics.

When a classical mechanical system exhibits explicit time dependence, i.e., it is non-autonomous, its underlying geometric structure can be taken either as a contact structure or as a cosymplectic structure [22]. Recently, the so-called cocontact geometry [39,55], a suitable geometric structure describing non-autonomous dissipative systems, combining contact and cosymplectic geometry, has been introduced.
The study of symmetries of mechanical systems is of great interest since it provides a way of finding conserved (or dissipated) quantities. Moreover, reduction procedures can be used in order to simplify the description of a dynamical system whose group of symmetries is known. The relation between symmetries and conserved quantities has been a topic of great interest in mathematical physics since the seminal work by Emmy Noether [50] (see also [36,49]). Since the dawn of geometric mechanics, numerous papers have been devoted to the geometric study of symmetries and conserved quantities for Hamiltonian and Lagrangian systems [7-9, 13, 19-21, 23, 25, 45-47, 51, 52, 57, 58, 60]. However, in the case of contact (or cocontact) systems, it is more natural to consider the so-called dissipated quantities and their associated symmetries [37,53]. Some notions of symmetries for autonomous contact Hamiltonian and Lagrangian systems were independently introduced in [28] and [18]. The study of symmetries and conserved (or dissipated) quantities is also related with Hamilton-Jacobi theory. A first Hamilton-Jacobi equation for autonomous contact systems was obtained in [22], and an alternative one was obtained in [15]. The Hamilton-Jacobi theory for non-autonomous contact systems has been recently done in [14]. Canonical and canonoid transformations [2] and Lie integrability [1] of (co)contact systems have also been studied.
As a matter of fact, when a (co)contact Lagrangian system exhibits a cyclic coordinate, the associated quantity is no longer conserved but dissipated. In [6] the symmetries and dissipated quantities of time-dependent contact systems were studied. Their results are restricted to the socalled extended contact phase space, i.e., the extended cotangent bundle T * Q × R × R endowed with a contact form defined by the canonical contact form of T * Q × R and the Hamiltonian function of the system. Among the advantages of the cocontact formalism it is the fact that one can consider more general manifolds. Moreover, R × T * Q × R is endowed with a canonical cocontact structure, independent of the Hamiltonian function.
In the present paper, the symmetries of time-dependent contact Hamiltonian and Lagrangian systems are studied and classified. A characterization of dissipated quantities and their relation with symmetries is also provided. Firstly, the most general type of symmetries with associated dissipated quantities, the so-called generalized infinitesimal dynamical symmetries, are studied. Secondly, other types of transformations which preserve additional geometric or dynamical structures are discussed, exploring the relations between them. After that, we consider symmetries of time-dependent contact Lagrangian systems which also preserve the geometric structures of the extended tangent bundle. Finally, we study three examples in detail: the free particle with time-dependent mass and linear dissipation, the action-dependent central potential with time-dependent mass, and the two-body problem with time-dependent friction. The latter may have interesting applications in celestial mechanics, allowing to describe the motion of planets with damping provoked by the medium.
In particular, all our results can be applied to time-independent contact Hamiltonian and Lagrangian systems. We review and extend the results from the literature regarding symmetries in autonomous contact systems [18,26,28]. Hence, this paper may also be used as a reference for the reader interested in the symmetries of contact Hamiltonian and Lagrangian systems (even if they do not have an explicit time-dependence).
New results and relation to literature. This paper is, to the best of our knowledge, the first reference studying the symmetries of cocontact Hamiltonian and Lagrangian systems. Cocontact geometry was introduced in [39] in order to provide a geometric framework for action and time dependent systems, combining features of contact and cosymplectic geometry. Furthermore, the present paper may also be used as a reference for the classification of symmetries of autonomous contact Hamiltonian and Lagrangian systems, the relations between them and their associated conserved and dissipated quantities. Several notions of symmetries that we consider had already been studied for the time-independent case in the literature: • Generalized infinitesimal dynamical symmetries were introduced in [18], where they were called "dynamical symmetries".
• Infinitesimal generalized natural symmetries of the Lagrangian L are called generalized infinitesimal symmetries of L in [18].
• Infinitesimal natural symmetries of the Lagrangian L are called infinitesimal symmetries of L in [18]. These symmetries were also studied in [28].
• Infinitesimal action symmetries are called action symmetries in [37]. This kind of transformations are employed in [40] to generate equivalent Lagrangians.
Some relations of these symmetries with dissipated quantities were also studied in the aforementioned papers. Nevertheless, there was a lack in the literature of a systematic classification of symmetries considering the structures they preserve and the relations between them (see Figures 1, 2 and 3).
Structure of the paper. In Section 2, the most important aspects of cocontact geometry are reviewed. Section 3 is devoted to the study of symmetries and dissipated quantities of time-dependent contact Hamiltonian systems. The symmetries and dissipated quantities of time-dependent contact Lagrangian systems are discussed in Section 4. Some examples are studied in Section 5. Finally, Section 6 provides some conclusions and topics for future research.
Notation and conventions. Throughout the paper all the manifolds and mappings are assumed to be smooth, connected and second-countable. Sum over crossed repeated indices is understood. Given a Cartesian product of manifolds M 1 × M 2 , the natural projections will be denoted by pr 1 : M 1 × M 2 → M 1 and pr 2 : M 1 × M 2 → M 2 , and similarly for a product of k manifolds M 1 × M 2 × · · · × M k .

Review on cocontact mechanics
In this section the main tools of cocontact geometry are presented. This geometric framework is used to develop a geometric formulation of time-dependent contact systems both in the Hamiltonian and the Lagrangian formalisms. See [39] for details.

Contact and Jacobi geometry
First, let us briefly recall the basic notions of contact and Jacobi manifolds that will be employed. For more details see [16,33,42].
where [·, ·] denotes the Schouten-Nijenhuis bracket. The pair (Λ, E) is called a Jacobi structure on M . The Jacobi bracket is the map {·, ·} : This bracket is bilinear and satisfies the Jacobi identity. However, unlike Poisson brackets, in general Jacobi brackets do not satisfy the Leibniz rule. Definition 2.2. A (co-oriented) contact manifold is a pair (M, η) where M is a (2n + 1)manifold, and η is a one-form on M such that η ∧ (dη) n is a volume form on M . The one-form η is called a contact form on M .
Given a contact manifold (M, η), one can define an isomorphism of C ∞ (M )-modules given by Every contact manifold has a unique Reeb vector field R, given by R = ♭ −1 (η). Moreover, to each function f ∈ C ∞ (M ) one can associate a (contact) Hamiltonian vector field X f given by ♭( Additionally, given a contact manifold (M, η), around every point p ∈ M there exist local coordinates (q i , p i , z) such that These coordinates are called canonical or Darboux coordinates. A contact Hamiltonian system is a triple (M, η, H), where (M, η) is a contact manifold and H ∈ C ∞ (M ) is the Hamiltonian function. Its dynamics is given by X H , the Hamiltonian vector field of H. There is also a Lagrangian formalism for time-independent contact systems (see [17]).

Cocontact geometry
where M is a (2n + 2)-manifold, and τ and η are one-forms on M such that dτ = 0 and τ ∧ η ∧ (dη) n is a volume form on M . The pair (τ, η) is called a cocontact structure on M .
Given an n-dimensional smooth manifold Q with coordinates (q i ) and its cotangent bundle T * Q with adapted coordinates (q i , p i ), consider the product manifolds R × T * Q, T * Q × R and R × T * Q × R with adapted coordinates (t, q i , p i ), (q i , p i , z) and (t, q i , p i , z) respectively. The following diagram illustrates this situation and provides some canonical projections: Denote by θ ∈ Ω 1 (R × T * Q × R) the pull-back of the canonical Liouville one-form of the cotangent bundle by the projection π given in the diagram above. Hence, (τ = dt, η = dz − θ) is a cocontact structure on the product manifold R × T * Q × R. This example, also known as canonical cocontact manifold, is just a particular case of the following.
Example 2.4. Let (P, η 0 ) be a contact manifold and consider the product manifold M = R×P . Denoting by dt the pullback to M of the volume form in R and denoting by η the pullback of η 0 to M , we have that (M, dt, η) is a cocontact manifold.

Hamiltonian formalism
where ψ ′ : I ⊂ R → TM is the canonical lift of the curve ψ to the tangent bundle TM . The cocontact Hamiltonian equations for a vector field X ∈ X(M ) are which can also be written as These equations have a unique solution called the cocontact Hamiltonian vector field X ≡ X H .
Given a curve ψ : I ⊂ R → M with local expression ψ(r) = (f (r), q i (r), p i (r), z(r)), the third equation in (2.2) imposes that f (r) = r + c for some constant c, thus we will denote r ≡ t, while the other equations read (2.5) On the other hand, the local expression of the cocontact Hamiltonian vector field in Darboux coordinates is Note that the integral curves of this vector field satisfy the system of differential equations (2.5).

Lagrangian formalism
Given a smooth n-dimensional manifold Q, consider the product manifold R × TQ × R equipped with adapted coordinates (t, q i , v i , z). We have the canonical projections which are summarized in the following diagram: The usual geometric structures of the tangent bundle can be naturally extended to the cocontact Lagrangian phase space R × TQ × R. In particular, the vertical endomorphism of T(TQ) yields a vertical endomorphism S : T(R × TQ × R) → T(R × TQ × R). In the same way, the Liouville vector field on the fiber bundle TQ gives a Liouville vector field ∆ ∈ X(R × TQ × R). The local expressions of these objects in Darboux coordinates are The vector fields satisfying the second-order condition can be characterized by means of the canonical structures ∆ and S introduced above, since X is a sode if and only if S(Γ) = ∆.
A Lagrangian function is a function L ∈ C ∞ (R × TQ × R). The Lagrangian energy associated to L is the function E L = ∆(L) − L. The Cartan forms associated to L are where t S denotes the transpose operator of the vertical endomorphism. The contact Lagrangian form is Notice that dη L = ω L . The couple (R × TQ × R, L) is a cocontact Lagrangian system. The local expressions of these objects are Not all cocontact Lagrangian systems (R × TQ × R, L) result in the tuple (R × TQ × R, τ = dt, η L , E L ) being a cocontact Hamiltonian system because the condition τ ∧ η ∧ (dη L ) n = 0 is not always fulfilled. The Legendre map characterizes the Lagrangian functions that will result in cocontact Hamiltonian systems.
Given a Lagrangian function L ∈ C ∞ (R×TQ×R), the Legendre map associated to L is its fiber derivative [34], considered as a function on the vector bundle τ 0 : where FL(t, ·, z) is the usual Legendre map associated to the Lagrangian L(t, ·, z) : TQ → R with the variables t and z fixed.
The Cartan forms can also be defined as θ L = FL * (π * θ 0 ) and ω L = FL * (π * ω 0 ), where θ 0 and ω 0 = −dθ 0 are the canonical one-and two-forms of the cotangent bundle and π is the natural projection π : Proposition 2.6. Given a Lagrangian function L the following statements are equivalent: (1) The Legendre map FL is a local diffeomorphism.
(2) The fiber Hessian nondegenerate (the tensor product is understood to be of vector bundles over R × Q × R).
A Lagrangian function L is regular if the equivalent statements in the previous proposition hold. Otherwise L is singular. Moreover, L is hyperregular if FL is a global diffeomorphism. Thus, every regular cocontact Lagrangian system yields the cocontact Hamiltonian system The local expressions of the Reeb vector fields are If the Lagrangian L is singular, the Reeb vector fields are not uniquely determined, actually, they may not even exist [39].

The Herglotz-Euler-Lagrange equations
Definition 2.7. Given a regular cocontact Lagrangian system (R × TQ × R, L) the Herglotz-Euler-Lagrange equations for a holonomic curve c : The only vector field solution to these equations is the cocontact Lagrangian vector field.
Equations (2.7) and (2.8) are the Lagrangian counterparts of equations (2.2) and (2.3), respectively. The cocontact Lagrangian vector field of a regular cocontact Lagrangian system (R × TQ × R, L) coincides with the cocontact Hamiltonian vector field of the cocontact Hamiltonian system (R × TQ × R, dt, η L , E L ).
Theorem 2.8. If L is a regular Lagrangian, then X L ≡ Γ L is a sode, called the Herglotz-Euler-Lagrange vector field for the Lagrangian L.

The coordinate expression of the Herglotz-Euler-Lagrange vector field is
An integral curve of Γ L fulfills the Herglotz-Euler-Lagrange equations for dissipative systems: d dt These equations can also be obtained variationally from the Herglotz principle [35] (see also [17]). Roughly speaking, the variable z can be interpreted as the action of the Lagrangian system.

Symmetries and dissipated quantities of cocontact Hamiltonian systems
In this section we will study the symmetries of regular time-dependent contact mechanical systems and their associated conserved and dissipated quantities. A summary of the symmetries and their relations can be found in Figure 1. In some cases we will restrict ourselves to the case of cocontact manifolds of the form M = R × N where N is a contact manifold (see Example 2.4). In this case, the natural projection R × N → R defines a global canonical coordinate t on the cocontact manifold R × N .
An infinitesimal conformal (resp. strict) cocontactomorphism is a vector field Y ∈ X(M ) whose flow is a one-parameter group of conformal (resp. strict) cocontactomorphisms.
Proof. Suppose that Φ is a cocontactomorphism. We have Since Φ * η = η and Φ * τ = τ , by the uniqueness of the time Reeb vector field, we get that Φ * R t = R t . Analogously, one can see that the contact Reeb vector field is also preserved.
It is worth noting that the converse is false.
Example 3.4. Consider the cocontact manifold (M, τ, η) where M = R 4 , τ = dt and η = dz − pdq, where (t, q, p, z) are canonical coordinates. Clearly, the vector field Y = ∂/∂p on M preserves the Reeb vector fields R t = ∂/∂t and R z = ∂/∂z. However, it is not an infinitesimal cocontactomorphism. Indeed, Similarly, one can check that the map Φ : M → M, (t, q, p, z) → (t, q, 2p, z) is a diffeomorphism preserving the Reeb vector field, but it is not a cocontactomorphism

Dissipated and conserved quantities of cocontact systems
Notice that, unlike in the time-independent contact case, the Hamiltonian function is not a dissipated quantity. Taking into account that  Proof. The Jacobi bracket of f and H is given by equation (2.1): so, taking into account equations (2.3), In particular, the right-hand side vanishes if and only if f is a dissipated quantity.
The symmetries that we shall present yield dissipated quantities. However, we are also interested in finding conserved quantities.
Taking into account that every dissipated quantity changes with the same rate R z (H), we have the following result, whose proof is straightforward.  (2) if f is a dissipated quantity and g is a conserved quantity, then f g is a dissipated quantity, (3) if f 1 and f 2 are dissipated quantities, a 1 f 1 + a 2 f 2 is also a dissipated quantity for any a 1 , a 2 ∈ R, (4) if g 1 and g 2 are conserved quantities, a 1 g 1 + a 2 g 2 + a 3 is also a conserved quantity for any a 1 , a 2 , a 3 ∈ R.

Generalized infinitesimal dynamical symmetries
The following result motivates the definition of the most general type of symmetries with associated dissipated quantities.
On the other hand, given a dissipated quantity where we have used equations (2.4).
This result motivates the following definition. In particular, if H is a time-independent Hamiltonian function, then H is a dissipated quantity and its associated generalized infinitesimal dynamical symmetry is the Hamiltonian vector field X H .
Theorems 3 and 4 of [6] are the analogous of Theorem 3.9 in the extended contact phase space (instead of the cocontact) formalism.
Remark 3.11. Despite the condition τ (Y ) = 0, the dissipated quantity associated to a generalized infinitesimal dynamical symmetry Y may be time-dependent. Indeed,

Other symmetries
We are now interested in other types of symmetries which preserve more properties of the system, such as the dynamical vector field or the Hamiltonian function. (1) If M = R × N with N a contact manifold, a dynamical symmetry is a diffeomorphism Φ : M → M such that Φ * X H = X H and Φ * t = t.
(2) An infinitesimal dynamical symmetry is a vector field Y ∈ X(M ) such that L Y X H = [Y, X H ] = 0 and ι Y τ = 0. In particular, if M = R × N , the flow of Y is made of dynamical symmetries.
Generalized infinitesimal dynamical symmetries receive that name since they satisfy weaker conditions than infinitesimal dynamical symmetries. It is clear that every infinitesimal dynamical symmetry is a generalized infinitesimal dynamical symmetry. We also define a generalization of dynamical symmetries as follows: Definition 3.13. Let (M, τ, η, H) be a cocontact Hamiltonian system, where M = R × N with N a contact manifold, and let X H be its cocontact Hamiltonian vector field. A generalized dynamical symmetry is a diffeomorphism Φ : M → M such that η(Φ * X H ) = η(X H ) and Unlike other symmetries with infinitesimal counterparts, the flow of a generalized infinitesimal dynamical symmetry is not necessarily made of generalized dynamical symmetries.
Example 3.14. Consider the cocontact Hamiltonian system (R 4 \{0}, τ, η, H), with τ = dt, η = dz − pdx and where (t, x, p, z) are the canonical coordinates in R 4 . The family of diffeomorphisms for r ∈ R, is generated by the vector field Y = ∂ ∂p . One can check that Y is a generalized infinitesimal dynamical symmetry, but Φ r is not a generalized dynamical symmetry for r = 0. Indeed, for and η(Φ r * X H ) = η(X H ). The (infinitesimal) dynamical symmetries defined above are the counterparts of (infinitesimal) dynamical symmetries in symplectic Hamiltonian systems (see [21,56] and references therein). They are of interest since they map trajectories of the system onto other trajectories. As a matter of fact, if σ : R → M is an integral curve of X H and Φ is a dynamical symmetry, then Φ • σ is also an integral curve of X H . In addition, we have the following result. In other words, given two infinitesimal dynamical symmetries Y 1 , Y 2 ∈ X(M ), its Lie bracket [Y 1 , Y 2 ] is also an infinitesimal dynamical symmetry.
Moreover, dynamical symmetries form a Lie subgroup of Diff(M ), that is, for any pair of dynamical symmetries Φ 1 and Φ 2 , the composition Φ 1 • Φ 2 is also a dynamical symmetry.
Proof. Using the Jacobi identity, In addition, On the other hand, if Φ 1 and Φ 2 are dynamical symmetries, then Obviously, Φ ≡ id is a dynamical symmetry. Finally, if Φ is a dynamical symmetry, then and similarly (Φ −1 ) * t = t. This proves that dynamical symmetries form a group under composition.
Generalized infinitesimal dynamical symmetries do not close a Lie algebra, as the counterexample below shows.
Example 3.16. Consider the cocontact Hamiltonian system from Example 3.14. Given the vector fields one can check that Y is a generalized infinitesimal dynamical symmetry and Z is an infinitesimal dynamical symmetry. Nevertheless, is not a generalized infinitesimal symmetry.
A natural type of objects that conserve the geometry of the system are the (infinitesimal) f -conformal cocontactomorphisms (see Definition 3.1). Since the function H is independent of the cocontact structure (τ, η), in general f -conformal cocontactomorphisms are not generalized dynamical symmetries. The necessary and sufficient condition is shown in the next result.   Proof. If X H is the solution of the cocontact Hamiltonian system (M, τ, η, H), we have that If Φ is a generalized dynamical symmetry, then ι Φ * X H η = ι X H η, and therefore Φ * H = f H.
Since f = 0 everywhere, we conclude that ι Φ * X H η = ι X H η. The infinitesimal case is proved with a similar argument using the relation This result justifies the following definition. (1) A f -conformal Hamiltonian symmetry is a diffeomorphism Φ : M → M such that where f ∈ C ∞ (M ) does not vanish anywhere, M = R × N with (N, η) a contact manifold, and t is the canonical coordinate of R. If Φ is a cocontactomorphism (i.e., if f ≡ 1), we say that Φ is a strict Hamiltonian symmetry.
(2) An infinitesimal ρ-conformal Hamiltonian symmetry is a vector field Y ∈ X(M ) such that where ρ ∈ C ∞ (M ). In particular, if M = R × N , the flow of Y is made of conformal Hamiltonian symmetries. If Y is an infinitesimal cocontactomorphism (i.e., if ρ ≡ 0), Y is said to be an infinitesimal strict Hamiltonian symmetry.
If a conserved quantity is known, (infinitesimal) dynamical symmetries can be used to compute additional conserved quantities. Similarly, if a dissipated quantity is known, (infinitesimal) strict Hamiltonian symmetries can be used to compute new dissipated quantities. Proposition 3.19. Suppose that g ∈ C ∞ (M ) is a conserved quantity and f ∈ C ∞ (M ) is a dissipated quantity.
(1) If Φ : M → M is a strict Hamiltonian symmetry and a dynamical symmetry, then f = f • Φ = Φ * f is also a dissipated quantity.
(2) If Y ∈ X(M ) is an infinitesimal strict Hamiltonian symmetry and an infinitesimal dynamical symmetry, then f = L Y f is also a dissipated quantity.
(3) If Φ : M → M is a dynamical symmetry, then g = g • Φ = Φ * g is also a conserved quantity.
(4) If Y ∈ X(M ) is an infinitesimal dynamical symmetry, then g = L Y g is also a conserved quantity.
Proof. Let f and g be a dissipated and a conserved quantity, respectively. Suppose that Φ : M → M is an strict Hamiltonian symmetry and a dynamical symmetry. Then, Similarly, if Φ is a dynamical symmetry, then If Y ∈ X(M ) is an infinitesimal dynamical symmetry, then Finally, if Y ∈ X(M ) is an infinitesimal strict Hamiltonian symmetry and an infinitesimal dynamical symmetry, we have that The results from Proposition 3.19 cannot be extended to generalized infinitesimal dynamical symmetries. As a matter of fact, we have the following counterexample.
Example 3.20. Consider the same system as in Example 3.14. Let Y ∈ X(R 4 \ {0}) be the vector field Y = ∂ ∂p . We have that [Y, X H ] = 0, but η([Y, X H ]) = 0 therefore, it is a generalized infinitesimal symmetry but it is not a dynamical symmetry.
The function f (t, x, p, z) = p is a dissipated quantity, but L Y f = 1 is not a dissipated quantity. Likewise, L Y H = p is not a dissipated quantity either. Finally, is not a conserved quantity.
It is also worth mentioning that preserving the Hamiltonian is not a sufficient condition for a diffeomorphism (vector field) to be a (infinitesimal) dynamical symmetry. It is not a sufficient condition for being a generalized (infinitesimal) dynamical symmetry either. where (t, x, p, z) are the canonical coordinates in R 4 . Its Hamiltonian vector field is given by x, p, 2z) is a diffeomorphism preserving the Hamiltonian function H but not the vector field X H .
Furthermore, we can consider the following generalization of infinitesimal ρ-conformal Hamiltonian symmetries.
Proof. Suppose that Y is a (ρ, g)-Cartan symmetry. Then, by Theorem 3.23, the function f = g − ι Y η is a dissipated quantity, so, by Theorem 3.9, Z = X f − R t is a generalized infinitesimal dynamical symmetry. The Hamiltonian vector field of f is given by and, similarly, L Rt ι Y η = L Rt g. In addition, Thus, On the other hand, Remark 3.25. If Y is a (ρ, g)-Cartan symmetry and Z = Y − gR z is its associated generalized infinitesimal dynamical symmetry, then the dissipated quantities associated to Y and to Z via Theorems 3.9 and 3.23 coincide.
(2) Infinitesimal strict Hamiltonian symmetries close a Lie subalgebra from the Lie algebra of infinitesimal conformal Hamiltonian symmetries.
In general, Cartan symmetries do not close a Lie subalgebra. where (t, q, p, z) are the canonical coordinates in R 4 . The vector field is a (0, q)-Cartan symmetry and There is no function f ∈ C ∞ (R 4 ) such that f η + e q−z dq is exact, so it is not possible to write L [Y 1 ,Y 2 ] η = ρη + dg for any functions ρ, g ∈ C ∞ (R 4 ), and hence [Y 1 , Y 2 ] is not a Cartan symmetry.
The types of symmetries and the relations between them are summarized in Figure 1.

Symmetries and dissipated quantities of cocontact Lagrangian systems
Consider a regular cocontact Lagrangian system (R × TQ × R, L), with cocontact structure (dt, η L ). Since (R × TQ × R, dt, η L , E L ) is a cocontact Hamiltonian system, every result from Section 3 can be applied to this case. Moreover, making use of the geometric structures of the tangent bundle [21,61] (and their natural extensions to R × TQ × R) we can consider additional types of symmetries. A summary of these symmetries and their relations can be found in Figure 2. The relation between (extended) natural symmetries of the Lagrangian and Hamiltonian symmetries is depicted in Figure 3. Consider a diffeomorphism ϕ = (ϕ Q , ϕ z ) : Q×R → Q×R, where ϕ Q : Q → Q and ϕ z : R → R are diffeomorphisms (in an abuse of notation we omit the projections). Then, the actiondependent lift of ϕ is the diffeomorphism ϕ = (id R , Tϕ Q , ϕ z ) : is split if it is projectable by pr Q : Q × R → Q and by pr R : Q × R → R. Given a split vector field Y ∈ X(Q × R), its action-dependent lift is the vector fieldȲ C ∈ X(R × TQ × R) whose local flow is the action-dependent lift of the local flow of Y . In other words, if Y is locally of the form its action-dependent complete lift is the vector field given locally bȳ Given a function f ∈ C ∞ (Q), its vertical lift is the function f V = f • τ Q • τ 2 ∈ C ∞ (R × TQ×R), where τ Q •τ 2 : R×TQ×R → Q is the projection (see Section 2.4). A 1-form ω ∈ Ω 1 (Q) can be regarded as a function ω ∈ C ∞ (TQ). Locally, if ω = ω i (q)dq i , then ω = ω i (q)v i . The vertical lift of a vector field X ∈ X(Q) to TQ is the unique vector field X V ∈ X(TQ) such that X V ( ω) = (ω(X)) V for any ω ∈ Ω 1 (Q). The vertical lift of an split Y ∈ X(Q×R) to R×TQ×R is the vector fieldȲ V ∈ X(R × TQ × R) given by the vertical lift of T pr Q Y ∈ X(Q) to T Q. Locally, if Y has the local expression (4.1), its vertical lift reads The following properties hold for any X, Y ∈ X(Q × R): where S and ∆ denote the vertical endomorphism and the Liouville vector field, with local expressions (2.6).

Lagrangian symmetries
We will denote φ ′ ≡ dφ dz . Henceforth, all the Lagrangian systems are assumed to be regular.
is called an extended symmetry of the Lagrangian if Φ * L = Φ ′ z L. In addition, if Φ is the action-dependent lift of some ϕ ∈ Diff(Q×R), then it is called an extended natural symmetry of the Lagrangian.
A vector field Y ∈ X(R × TQ × R) of the form is called an infinitesimal extended symmetry of the Lagrangian if L Y L = ζ ′ L. In addition, if Y is the action-dependent complete lift of some X ∈ X(Q × R), then it is called an infinitesimal extended natural symmetry of the Lagrangian.
Proof. Clearly, ιȲ C τ = 0. Moreover, where we have used that the action-dependent complete lift of a vector field commutes with the Liouville vector field (see properties (4.2)), and Therefore,Ȳ C is a ζ ′ -conformal Hamiltonian symmetry. The case for extended natural symmetries of the Lagrangian is similar. Proof. We have that where we have used the second of the properties (4.2), so If Γ L is the Herglotz-Euler-Lagrange vector field (given by equations (2.8)), A particular case of extended natural symmetries are those with ζ = 0. That is, symmetries which are lifted from Q.
is called a symmetry of the Lagrangian if Φ * L = L and Φ * t = t. In addition, if Φ is the canonical lift of some ϕ ∈ Diff(Q), then it is called a natural symmetry of the Lagrangian.
A vector field Y ∈ X(R×TQ×R) is called an infinitesimal symmetry of the Lagrangian if L Y L = 0 and ι Y τ = 0. In addition, if Y is the complete lift of some X ∈ X(Q), then it is called an infinitesimal natural symmetry of the Lagrangian.  It is worth noting that a symmetry of the Lagrangian which is not natural is not, in general, a Hamiltonian symmetry. Moreover, in general, it is not an extended symmetry of the Lagrangian either.
Example 4.6. Consider the Lagrangian L(t, x, v, z) = 1 2 v 2 − V (t, x, z) on R × TR × R. Clearly, the vector field ∂v is an infinitesimal symmetry of the Lagrangian (but it is not natural). However, Y (E L ) = 0. Moreover, we have η L = dz − vdx, so for any ρ ∈ C ∞ (M ) From Proposition 4.3 we have that: Corollary 4.7. Let Y be a vector field on Q and assume that L is regular. Then Y C is an infinitesimal natural symmetry of L if, and only if, Y V (L) is a dissipated quantity.  Proof. If Y 1 , Y 2 ∈ X(R × TQ × R) are symmetries of the Lagrangian L, then

Infinitesimal symmetries of the Lagrangian
is a symmetry of the Lagrangian. In particular, if Y 1 = X C 1 and Y 2 = X C 2 (for some X 1 , X 2 ∈ X(Q)) are natural symmetries of the Lagrangian, then [Y 1 , Y 2 ] = [X 1 , X 2 ] C . Therefore, [Y 1 , Y 2 ] is also a natural symmetry of the Lagrangian.
Similarly, suppose thatȲ C 1 andȲ C 2 are extended natural symmetries of the Lagrangian L, Then, is an extended natural symmetry of L.
Infinitesimal extended natural symmetries of the Lagrangian Infinitesimal natural symmetries of the Lagrangian

Symmetries of the action
Another relevant class of symmetry are transformations on the "z" variable, or changes of action, which preserve the dynamics. This kind of transformations are used in [40] to generate equivalent Lagrangians. A vector field Z ∈ X(R × TQ × R) an infinitesimal change of action if T pr R×TQ •Z = 0.
If a change of action has the form Φ : (t, q, v, z) → (t, q, v, Φ z (t, q, v, z)) , then, in particular ∂Φ z ∂z = 0 everywhere. Clearly, the flow of an infinitesimal change of action is made up of changes of action. Moreover, if Y ∈ X(R × Q × R) is a sode and Φ is a change of action, then Φ * Y is also a sode.
is a generalized dynamical symmetry if, and only if, An infinitesimal change of action Z ∈ X(R × TQ × R) with local expression is a generalized infinitesimal dynamical symmetry if, and only if, ζ is a dissipated quantity, i.e., Γ L (ζ) = ζ∂L/∂z.
Proof. Given two sode Y and X, we have that θ L (Y ) = θ L (X) = ∆(L) and t S(Y ) = ∆. Let Γ L be the Herglotz-Euler-Lagrange vector field of the system, given by equations (2.8). If Φ is a change of action, then In addition, On the other hand, ι Γ L η L = −E L = L−∆(L). Therefore, Φ is a generalized dynamical symmetry Furthermore, if Z is an infinitesimal change of action we have that This result motivates the following definition.
where q 0 = q(0), p 0 = p(0), z 0 = z(0) are the initial conditions. The term of H linear in the variable z permits to model a damping phenomena. As a matter of fact, in the particular case where m(t) is constant the linear momenta (and hence the velocity) of the system decreases exponentially.
ds is a dissipated quantity. Hence, by Theorem 3.9, the vector field is a generalized infinitesimal dynamical symmetry. In addition, one can verify that Y f is an infinitesimal dynamical symmetry, namely Y f commutes with X H . Now, and ds .
Moreover, f 2 (t, q, p, z) = p is also a dissipated quantity, whose associated generalized infinitesimal dynamical symmetry is It is clear that Y f 2 is an infinitesimal dynamical symmetry, i.e., Y f 2 commutes with X H . Moreover, L Y f 2 η = 0 and Y f 2 (H) = 0, so Y f 2 is an infinitesimal strict Hamiltonian symmetry. The Lagrangian counterpart of this system is characterized by the Lagrangian function L : R × TR × R → R given by The vector field Z ∈ X(R × TR × R) with local expression is an infinitesimal action symmetry, since it is an infinitesimal change of action and we know that ζ is a dissipated quantity.

An action-dependent central potential with time-dependent mass
Consider a Lagrangian function L : R × TR 2 × R → R of the form where m(t) is a positive-valued function. Let Y ∈ X(R 2 ) be infinitesimal generator of rotations on the plane, namely, Its complete lift is given byȲ Clearly,Ȳ C is an infinitesimal natural symmetry of the Lagrangian, i.e.,Ȳ C (L) = 0. Hence, by is a dissipated quantity. This quantity is the angular momentum for a particle with timedependent mass.

The two-body problem with time-dependent friction
The 2-body problem describes the dynamics of two particles under the effects of a force that depends on the distance between the particles, usually the gravitational force. To model timedependent friction, we will add a linear term on the action in the Lagrangian, with a timedependent coefficient. The two-body problem is one of the most important problems in celestial mechanics. The addition of a friction term may allow to describe the motion of celestial bodies in a dissipative medium. The phase space is R×TR 6 ×R, with coordinates (t, q 1 , q 2 , v 1 , v 2 , z). The superindex denotes each particle, and the bold notation is a shorthand for the three spatial components, namely q 1 = (q 1 1 , q 1 2 , q 1 3 ) and q 2 = (q 2 1 , q 2 2 , q 2 3 ). The relative distance between the particles is r = q 2 − q 1 , whose (Euclidean) length will be denoted r = |r|.
The Lagrangian function is where m 1 , m 2 ∈ R are the masses of the particles which we assume to be constant, U (r) is the central potential and γ is a time-dependent function. The Lagrangian energy is and the cocontact structure is given by the one-forms η = dz − m 1 v 1 · dq 1 − m 2 v 2 · dq 2 , τ = dt .
The evolution of the system is given by the Herglotz-Euler-Lagrange vector field Γ L , defined by equations (2.8) and with local expression (2.9). Its solutions satisfy the Herglotz-Euler-Lagrange equations: The dot notation indicates time derivative and F = − dU dr r r is the force of the potential U . Proceeding as in the classical 2-body problem, we study the evolution of the center of masses R = m 1 q 1 + m 2 q 2 m 1 + m 2 .
That is, every component ofṘ is a dissipated quantity. Along a solution, it evolves aṡ In particular, if γ is a positive constant, as the time increases the center of mass tends to move on a line with constant speedṘ 0 . By Noether's Theorem 3.9, the corresponding generalized infinitesimal dynamical symmetries are YṘ = XṘ − R L t , where R L t is subtracted to every component. A short computation shows that Each component of YṘ is an action dependent complete lift and L YṘ L = 0 therefore, they are infinitesimal natural symmetries of the Lagrangian.
The fact that the center of mass is moving in a very concrete way, may indicate that one could express the system using only the relative position. Indeed, from equations (5.1) and (5. The angular momentum along a solution is L(t) = L 0 e − γ(t)dt .
Since the direction of L remains constant, the movement takes place on a plane perpendicular to L 0 . If γ is a positive constant, the angular momentum tends to 0. The associated generalized infinitesimal dynamical symmetries are Each component of Y L is an action dependent complete lift and L Y L L = 0, therefore they are infinitesimal natural symmetries of the Lagrangian. Finally, the Lagrangian energy E L evolves as Γ L (E L ) = −R L z (E L )E L + R L t (E L ) = −γE L +γz , and it is not a dissipated quantity due to the time-dependence of γ.
The evolution of the mechanical energy, namely the sum of the kinetic and the potential energies, is given by We could proceed by rewriting the reduced system in polar coordinates and describe the possible orbits. Unfortunately, in this case it is not evident how to express the relation between the radial and angular coordinates.

Conclusions and further research
In this paper, we have characterized the symmetries and dissipated quantities of time-dependent contact Hamiltonian and Lagrangian systems. Firstly, we have studied generalized infinitesimal dynamical symmetries, a type of symmetries which are in bijection with dissipated quantities. After that, we have considered other types of symmetries which preserve (up to a conformal factor) additional objects, such as the cocontact structure or the Hamiltonian function. Moreover, making use of the canonical structures of the tangent bundle, we have discussed Lagrangian symmetries and symmetries of the action. We have concluded with three illustrative examples: the free particle with time-dependent mass and linear dissipation, the action-dependent central potential with time-dependent mass, and the two-body problem with time-dependent friction.
In particular, the two-body problem could be interesting in celestial mechanics, where the friction could be used to model the damping caused by the medium. The formalism presented in this paper may also be applied to more complex systems in celestial mechanics. In a future work, we plan to extend this study to the restricted three-body problem with friction. It would be particularly interesting to study how the friction affects the stability of the system.
The study of symmetries and dissipated quantities made in this work is the first step towards investigating the symmetries and dissipation laws in non-conservative field theories using the k-(co)contact [27,29,54] and multicontact [38] settings. Furthermore, the classification of symmetries could provide a new insight towards a reduction method for time-(in)dependent contact systems. received financial support from the Spanish Ministry of Science and Innovation (MCIN/AEI/ 10.13039/501100011033), under the grants PID2019-106715GB-C21 and CEX2019-000904-S and the predoctoral contract PRE2020-093814. X. Rivas also acknowledges financial support of the Novee Idee 2B-POB II project PSP: 501-D111-20-2004310 funded by the "Inicjatywa Doskona lości -Uczelnia Badawcza" (IDUB) program.