A Summary on Symmetries and Conserved Quantities of Autonomous Hamiltonian Systems,

A complete geometric classification of symmetries of autonomous Hamiltonian mechanical systems is established; explaining how to obtain their associated conserved quantities in all cases.


Introduction
The interest of the existence of symmetries of mechanical systems is related with the existence of conserved quantities (or constants of motion), which allows us to simplify the integration of dynamical equations, applying suitable reduction methods [1,3,15,22]. In fact, it is well known that, roughly speaking, a symmetry of a dynamical system is a transformation in the phase space of the system that maps solutions to the dynamical equations into solutions, and that every symmetry originates a conservation law or conserved quantity; that is, there is a physical magnitude that is conserved along the dynamical trajectories of the system. Then, these conserved quantities are used to reduce the number of physical degrees of freedom and of equations of the system.
The use of geometrical methods is a powerful tool in the study of these topics. In particular, the most rigorous and complete way to deal with these problems is by means of the theory of actions of Lie groups on (symplectic) manifolds, and the subsequent theory of reduction [23] (see also [11,21] for an extensive list of references that cover many aspects of the problem of reduction by symmetries in a lot of different situations). In this paper we are only interested in the case of regular (i.e., symplectic) Hamiltonian systems. Nevertheless, the problem of reduction will not be addressed in this dissertation.
As it is well known, the standard procedure to obtain conserved quantities consists in introducing the so-called Noether symmetries, and then use the Noether theorem which is stated both for the Lagrangian and the Hamiltonian formalism in mechanics (and field theories). Thus, Noether's theorem gives a procedure to associate conservation laws to Noether symmetries. However, these kinds of symmetries do not exhaust the set of symmetries. As is known, in mechanics there are symmetries which are not of Noether type, but they also generate conserved quantities (see, for instance, [5,6,7,8,9,16,18,17,25,27]), and they are sometimes called hidden symmetries. Different attempts have been made to extend Noether's results or state new theorems in order to include and obtain the conserved quantities corresponding to these symmetries, for mechanical systems (for instance, see [4,8,17,19,30,31]) and also for field theories [12,14,28].
The aim of this paper is to make a broad summary about the geometric study of symmetries of dynamical Hamiltonian systems (autonomous and regular) in the environment of mechanics of symplectic mechanics. In particular, we establish a complete scheme of classification of all the different kinds of symmetries of Hamiltonian systems, explaining how to obtain the associated conserved quantities in each case. We follow the same lines of argument as in the analysis made in [30] for the symmetries of Lagrangian time-dependent systems.
In particular, in Section 2, after stating the main concepts about the geometric (symplectic) description of (autonomous) Hamiltonian systems, we introduce the concept and characterization of symmetries and conserved quantities and we classify the symmetries in two groups: those leaving invariant the geometric structure (the symplectic form) and those leaving invariant the dynamics (the Hamiltonian function).
The first part of Section 3 is devoted to review Noether symmetries; that is, those which are both geometrical and dynamical (i.e., Hamiltonian), and their conserved quantities; stating the Noether theorem and its inverse [1,15,22]. Then, we consider the case of non-Noether symmetries. First the non-Hamiltonian symmetries are also reviewed, explaining how to obtain their associated conserved quantities, depending on whether the symmetry is or not geometric too, and establishing, in this last case, the relation with the so-called bi-Hamiltonian systems [10,27]. The most original part of the paper begins next. Different kinds of non-geometrical symmetries can be defined, depending on how the symplectic form transforms under the symmetry. All of them are studied in detail, showing how to obtain conserved quantities depending, in this case, on whether the symmetry is or not dynamical too. This section ends showing how all the above results can be recovered from an analysis based on some geometrical properties.
Finally, in Section 4 we present some typical examples of dynamical systems that illustrate some of the cases presented.
All manifolds are real, paracompact, connected and C ∞ . All maps are C ∞ . Sum over crossed repeated indices is understood.
2 Symplectic Hamiltonian mechanics: symmetries and conserved quantities 2.1 Hamiltonian systems Definition 1 A (regular) Hamiltonian system is a triad (M, ω, α), where: • (M, ω) is a symplectic manifold; where M represents the phase space of a dynamical system. Usually M = T * Q, where Q is the configuration space of the system.
• α ∈ Z 1 (M ) is the Hamiltonian 1-form, which gives the dynamical information of the system.
-By Poincaré's Lemma we have that, for every p ∈ M , there exists an open set U ⊂ M , with p ∈ U , and h ∈ C ∞ (U ), which is called a local Hamiltonian function, such that α | U = dh. In this case, (M, ω, α) is said to be a local Hamiltonian system.
-If α is an exact form, then there exists h ∈ C ∞ (M ), which is called a (global) Hamiltonian function, such that α = dh, and then (M, Ω, α) is said to be a (global) Hamiltonian system.
If ω is a degenerate form (i.e.; a presymplectic form, in general, then (M, ω, α) is said to be a non-regular (or singular) Hamiltonian system In this paper, only regular Hamiltonian systems are considered.
For regular Hamiltonian systems there exists a unique vector field X h ∈ X(M ), which is called the dynamical Hamiltonian vector field associated with α, such that and the dynamical trajectories of the system are the integral curves σ : R → M of this Hamiltonian vector field X h ∈ X(M ).
In a chart of symplectic (Darboux) coordinates (U ; q i , p i ) in M we have that and the integral curves σ(t) = (q i (t), p i (t)) of X h are the solution to the Hamilton equations: Asσ = X • σ, these last equations can be written also in an intrinsic way as whereσ : R → T M denotes the canonical lifting of σ to the tangent bundle T M . This equation is equivalent to (1).
In this case, the Hamiltonian system is said to be a bi-Hamiltonian system.
From now on we will write α = dh, where the Hamiltonian function h is locally or globally defined. If M = T * Q and Φ = T * ϕ for a diffeomorphism ϕ : Q → Q, then Φ is a natural symmetry.

Symmetries of Hamiltonian systems. Conserved quantities
(Here, T * ϕ denotes the canonical lifting of ϕ to the cotangent bundle).
• An infinitesimal symmetry (or a infinitesimal dynamical symmetry)of the Hamiltonian system is a vector field Y ∈ X(M ) whose local flows are local symmetries.
If M = T * Q and Y = Z C * for Z ∈ X(Q), then Y is a natural infinitesimal symmetry.
(Here, Z C * denotes the canonical lifting of Z to the cotangent bundle).
An immediate consequence of this definition is: As it is pointed out in the Introduction, symmetries can be used to obtain new symmetries: is an infinitesimal symmetry.
Starting from a conserved quantity, more constants of motion could be obtained using symmetries, as follows: Proposition 4 Let (M, ω, α = dh) be a Hamiltonian system.
1. If Φ : M → M is a symmetry and f ∈ C ∞ (M ) is a conserved quantity, then Φ * f is a conserved quantity.
2. If Y ∈ X(M ) is an infinitesimal symmetry and f ∈ C ∞ (M ) is a conserved quantity, then L(Y )f is a conserved quantity.

Geometric and Hamiltonian symmetries
As we have seen in the above section, a symmetry of a dynamical system lets invariant the dynamical vector field. As in Hamiltonian mechanics this vector field is determined by the geometrical structure (the symplectic form ω) and the dynamics (the Hamiltonian function h) through the equation (1); it is expected to have a relationship between the invariance of the dynamical vector field X h and the invariance of these two elements ω and h. Now we explore this relationship.
First we introduce the following terminology: Definition 5 Let (M, ω, α = dh) be a Hamiltonian system.
it is a local Hamiltonian vector field, Y ∈ X lh (M ))).
The first fundamental relation is among dynamical, geometrical and Hamiltonian symmetries is the following: Proposition 5 Every (infinitesimal) geometrical and Hamiltonian symmetry is a (infinitesimal) symmetry.
(Proof ) In fact, we have: The converse of this statement is not true, as we will see in the following sections.

.1 Noether symmetries. Noether's theorem
The last proposition leads to stablish the following: If M = T * Q and Φ = T * ϕ for a diffeomorphism ϕ : Q → Q, then Φ is a natural Noether symmetry.
• An infinitesimal Noether symmetry is a vector field Y ∈ X(M ) such that: Thus, a (infinitesimal) Noether symmetry is both a (infinitesimal) geometrical and Hamiltonian symmetry and hence it is a symmetry.
From now on we consider only the case of infinitesimal symmetries.
Noether symmetries generate conserved quantities in the following way: (Proof ) Both statements are immediates, since: Furthermore we have that: This means that no new conserved quantities are generated starting from f Y , under the action of Y (see Prop. 4).
and the result follows.
And finally, the converse statement of Noether's theorem allows to associate a (Noether) symmetry to every conserved quantity:

Non-Hamiltonian symmetries
Now we study all the symmetries which are not of Noether's type (that is, symmetries which are not Hamiltonian and/or geometrical), and how they generate conserved quantities. First we analyze those (infinitesimal) symmetries which are not Hamiltonian; that is, such that , then f is a locally constant function (and hence it is a trivial conserved quantity).
2. If L(Y )ω = 0 (that is, Y is not a geometrical symmetry), then there are two options: (a) If L(Y )ω = c ω, c ∈ R, (that is, Y is not a geometrical symmetry), then f = c h + k, k ∈ R (and it is a conserved quantity).
and we distinguish the following cases: (2) we obtain that df = 0, and then f is locally constant.
2. If L(Y )ω = 0, then: and bearing in mind (2) we conclude that f = c h + k, k ∈ R. (2) says that X h is a bi-Hamiltonian vector field. Therefore: Remarks: • Observe that, in the last case, L(Y )ω is closed, but not necessarily symplectic. If L(Y )ω is degenerate, then X h is not the only solution to the equation (2), but the other solutions are not relevant since they are not solutions to the dynamical equation (1).

Non-geometrical symmetries
Next we start analyzing the (infinitesimal) symmetries which are not geometrical; that is, which verify that L(Y )ω = 0 .
Although in the item 2 of Theorem 3 we have analyzed two particular cases of this situation, there are other possibilities. Next we study all the possible cases that may occur.

Higher-order Noether symmetries
Definition 7 Y ∈ X(M ) is an infinitesimal Noether symmetry of order N if: 1. Y is an infinitesimal symmetry.

Remarks:
• Observe that, if L(Y )h = 0 (Y is not a Hamiltonian symmetry), then we are in the case (2.b) of Theorem 3. Then L(Y )h and, eventually, L N (Y )h, for N > 1, are conserved quantities.
• Notice also that the other cases (1) and (2.a) in Theorem 3 cannot occur in the current situation.
For these kinds of symmetries we have the following generalization of Noether's theorem: Theorem 4 (Noether generalized): Let Y ∈ X(M ) be an infinitesimal Noether symmetry of order N . Then: Then there exists f ∈ C ∞ (U p ), which is unique up to a constant function, such that 2. The function f ∈ C ∞ (U p ) is a conserved quantity; that is, L(Xh)f = 0 (on U P ).
(Proof ) 2. If Y ∈ X(M ) is an infinitesimal Cartan symmetry of order N then it is a symmetry, and then [Y, X h ] = Z ∈ ker ω. Therefore and repeating the reasoning N − 2 times we arrive at the result

Furthermore we have:
Corollary 4 The function f is invariant by Y .
(Proof ) In fact, Thus, in this case no new conserved quantities are generated starting from f , under the action of Y .
(Proof ) In fact, we have that, in U p and the result follows.
Remark: If L(Y )h = 0 (Y is not an infinitesimal Hamiltonian symmetry) we are in the case (2.b) of Theorem 3.
Notice that the other cases (1) and (2.a) in Theorem 3 cannot occur in the current situation.

Other kinds of non-geometrical symmetries
If Y ∈ X(M ) is not an infinitesimal geometrical symmetry and it is not a higher-order Noether symmetry, then we have that L m (Y )ω = 0, ∀m ∈ N .
Then, as the module of 2-forms in a finite-dimensional manifold is locally finite generated, after a finite number of Lie derivations we will obtain that the following condition holds (maybe only locally): we have that: i. The form is closed. Then, for every p ∈ M , there exist an open neighbourhood U p ∋ p and a function f ∈ C ∞ (U p ) (unique up to a constant), such that γ = df .
ii. f is a conserved quantity.
2. If (4) holds, we distinguish the following cases: ii. As γ = df and [Y, X h ] = 0 (because Y is an infinitessimal symmetry), we obtain and repeating the procedure N − 2 times we arrive to the result 3. If (5) holds, we distinguish the following cases: (a) The case L(Y )h = 0: as [Y, X h ] = 0, we have: Remarks: • In the case of item 1, it is obvious that the conserved quantities {f 0 , . . . , f N −1 } are not invariant by Y necessarily and hence their Lie derivatives under Y could generate new conserved quantities.
• As we have said in the proof, the case (2.a) is the case (2.b) of Theorem 3 and hence, as it was commented in the first remark after this Theorem, conserved quantities could be generated starting from f , under the action of Y In the case (2.b), we have that and no new conserved quantities are generated starting from f , under the action of Y .
• In the case (3.a), we have that could generate a new conserved quantity.
In the case (3.b), trivially we have that L(Y )f = 0.
Thus X h is a bi-Hamiltonian vector field and hence L j (Y )h is a conserved quantity.
In fact, this is a consequence of the item (2.b) of Theorem 3, which says that L(Y )h is a conserved quantity and hence, as a consequence of the item 2 of Proposition 4, so is L j (Y )h, for j = 1, . . . , N .

Geometric properties
Consider a Hamiltonian system (M, ω, α), and let Y ∈ X(M ) be an infinitesimal symmetry; that is, [Y, X h ] = 0, We define the forms
The other statement follow as a straightforward consequence of this result.
3. Taking into account the above items, we obtain In addition, we have: 1. First observe that, as Y is an infinitesimal symmetry, then [X h , Y ] = 0 and L(Xh) i(Y ) = i(Y ) L(Xh). Then, we have 2. Once again, bearing in mind that [Y, X h ] = 0, we have 3. Taking into account the above items we obtain Consider the case j = 0; then θ (0) = i(Y )ω. We have several options: • θ (0) is a closed form: this is equivalent to demand that L(Y )ω = 0 (item 4 of Lemma 1). Then Y is a geometrical symmetry and: -If L(Y )h = 0, then we have that Y is an infinitesimal Noether symmetry and, from the item 2 of Proposition 7, we recover Noether's theorem (Theorem 1).
-If L(Y )h = 0, then we have trivial conserved quantities as stated in the item 1 of Theorem 3. Otherwise we consider the form θ (1) = L(Y ) i(Y )ω. Then we have the following options: • θ (1) is a closed form: it is equivalent to dθ (1) = L 2 (Y )ω = 0. Then: -If L(Y )h = 0, then Y is an infinitesimal Noether symmetry of order 2 and, from the the item 2 of Proposition 7, we recover the generalized Noether's Theorem 4.
-If L(Y )h = 0, then we are in the case case 2(b) of Theorem 3.
• θ (1) is not a closed form: -If dθ (1) = L 2 (Y )ω = c ω; the condition in the item 3 of Proposition 7 leads us to L 2 (Y )h = c h = 0 (up to a constant), and we are in the case described in Theorem 6.
-If dθ (1) = L 2 (Y )ω = c ω and L(Y )h = 0, then we are in some of the cases described in the items 2 and 3 of Theorem 5.
Otherwise we must go on with a new step, repeating the above analysis as many times as necessary until arriving to one of the cases described in Theorems 3 , 4, 5, and 6.

Some examples
In addition to the ones presented here, other interesting examples of non-Noether symmetries and their associated conserved quantities can be found, for example, in [8,13,20,26,32] (see also [2,24], and the references quoted therein, for another collection of (quantum-mechanical) systems having nontrivial integrals of motion).
The Hamiltonian function of the system is h = 1 2 (p 2 θ + p 2 ϕ (1 + tan 2 θ)) + Ω 2 (1 + sin θ)) , where Ω 2 = g/l; and the Hamiltonian vector field is For this system, the vector field Y = ∂ ∂ϕ is an infinitesimal Noether symmetry, since and the corresponding Noether conserved quantity is given by that is, the angular momentum with respect the vertical axis. In this case L(Y )f = 0 and no new constant of motion arise from f . This is a typical example of a integrable system: we have two independent conserved quantities h and p ϕ (the same number of degrees of freedom of the system).
Now, the Hamiltonian function is where Ω 1 , Ω 2 are constants. The Hamiltonian vector field is This system has two infinitesimal non-Noether symmetries (geometric but non-Hamiltonian) which are in fact, we have that and the corresponding constants of motion are which, in this case, are constant functions; that is, they are trivial conserved quantities. (See also [29] for an analysis of the algebra of symmetries of this model in the case of commensurable frequencies).
Now we have that L(Y )f = L 2 (Y )h = 4h, and no new conserved quantities arise from f . Nevertheless, it is well known that this dynamical system is an example of a superintegrable system [6,16]. In fact, the Hamiltonian function can be split as h = h 1 + h 2 , where h i = 1 2 (p i ) 2 + Ω 2 (q i ) 2 (i = 1, 2), and h 2 and h 2 are also constants of motion, in addition to h, since L(Xh)hi = 0, for i = 1, 2. Thus, we have 3 = 2n − 1 independent conserved quantities (notice that h 2 , h 2 and h are not independent, but h 2 , h 2 and f are).
As stated in Theorem 2, there are infinitesimal Noether symmetries which originate these new conserved quantities: their Hamiltonian vector fields, which are and, obviously, X h = X h 1 + X h 2 . Nevertheless they can be also associated with other kinds of infinitesimal symmetries. In fact, observe that the infinitesimal symmetry Y can be split into and each one of these vector fields is a non-Hamiltonian and non-geometrical infinitesimal symmetry. In fact, [Y 2 , X h ] = 0 , L(Y1)ω = dq 2 ∧ dp 1 + dq 1 ∧ dp 2 , L(Y2)ω = dq 2 ∧ dp 1 + dq 1 ∧ dp 2 , L 2 (Y 1 )ω = 2 dq 2 ∧ dp 2 , L 2 (Y 2 )ω = 2 dq 1 ∧ dp 1 , Therefore, as it is stated in Theorem 6, we have that h 2 , h 2 and f are three independent conserved quantities.

Conclusions and outlook
In this paper a complete classification of the symmetries for (autonomous and regular) Hamiltonian systems has been done, and the associated conserved quantities are obtained in each case. We have followed a systematic procedure manly based on the techniques used in [30] for the study of non-Noether symmetries of Lagrangian time-dependent systems. In this way, we have reviewed and completed previous analysis about this problem (for instance, in [4,5,8,16,17,19,27,30,31]).
First, we have reviewed the case of the Noether symmetries (which are both geometrical and dynamical) and the Hamiltonian version of the classical Noether theorem (and its converse) which gives a procedure to get the corresponding conserved quantities.
Next, we have considered the non-Noether symmetries. We have analyzed the non-dynamical symmetries. In this case the way to obtain conserved quantities depend on whether the symmetry is also geometrical or not and, in the last case, it is related with the fact that the system is bi-Hamiltonian.
The main contribution of the paper is the analysis of the non-geometrical symmetries. Following the guidelines established in [30], we have seen that there are several types of them, according to the behaviour of the symplectic structure under the action of the symmetry. As for the non-Hamiltonian case, the procedure for obtaining the conserved quantities depends on whether the symmetry is also dynamic or not. In particular, for some special cases, it consist in applying suitable generalizations of the Noether theorem.
A similar study to what we have done here could be done for autonomous Lagrangian systems, although this case is more difficult since the symmetries of the Lagrangian must be also considered. Finally all these results could also be extended to classical field theories in order to do a classification of their symmetries and the corresponding conservation laws; completing, in this way, the partial results already obtained in [12,14,28] for non-Noether symmetries.