American Journal of Applied Mathematics
Volume 4, Issue 3, June 2016, Pages: 132-136

Symmetries and Conservation Laws for Hamiltonian Systems

Estomih Shedrack Massawe

Department of Mathematics, University of Dar es Salaam, Dar es Salaam, Tanzania

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To cite this article:

Estomih Shedrack Massawe. Symmetries and Conservation Laws for Hamiltonian Systems. American Journal of Applied Mathematics. Vol. 4, No. 3, 2016, pp. 132-136. doi: 10.11648/j.ajam.20160403.13

Received: April 19, 2016; Accepted: May 3, 2016; Published: May 14, 2016


Abstract: In this paper, it is shown that symmetry of a physical system is a transformation which may be applied to the state space without altering the system or its dynamical interaction in any way. The theory is applied to generalize the concept of symmetries and conservation laws with external to Hamiltonian systems with external forces. By this we obtain a generalized Noether’s Theorem which states that for Hamiltonian systems with external forces, a symmetry law generates a conservation law and vice versa.

Keywords: Symmetries, Conservation Laws, Hamiltonian Systems


1. Introduction

Symmetries are among the most important properties of dynamical systems when they exist [1]. The study of symmetries is very important in the sense that they are equivalent to the existence of conservation laws. [2] has shown that in Hamiltonian system, symmetries are very close to the constants of the motion. Noether’s theorem has also advocated this concept. Also [3] applied symmetries and constants of motion and derived the reduced Hamiltonian system. Generally, symmetry of a physical system is a transformation which may be applied to the state space without altering the system or its dynamical interaction in any way. Consider for example motion of a particle in a central force field with potential  where  is the position vector of the particle. This system is not affected by rotations and they are referred to as a symmetry. The existence of such symmetries gives insight into the structure of the system i.e. any solution of the system must reflect these symmetries. Thus it is useful to make use of any symmetry information available in obtaining solutions of the system i.e. constants of the motion (conservation laws) which are defined as mappings  such that . Think for example, the energy of the system. It is usually a mapping on the tangent bundle and it is usually constant of the motion. The connection between the symmetry of a system and its corresponding conservation law is summarized in Noether’s theorem which follows later. It is therefore intended to formulate and analyse Symmetries and Conservation Laws for Hamiltonian Systems which finally summarized by the generalized Noether’s theorem.

2. Formulation of the Concept of Symmetry

Let  be the configuration manifold for a physical system. Let  be the Lagrangian of the system i.e.  is a smooth function on the tangent bundle  of the system. Let  be a smooth map on  and  the corresponding bundle map. A Lagrangian  is said to be invariant under the mapping  if  for any tangent vector  i.e.  [1]. The extension of the symmetry of a physical system to dynamical systems yields the following: [4].

Definition 1

(a)  A symmetry for time-invariant external dynamical system  is a map  which leaves  invariant i.e. if  then also  and if  then there exists  such that . In short

(b)  A symmetry for a dynamical system in state space form  is a mapping  with  and  which leaves  invariant i.e. if  then also  and if  then there exists  such that .

Example

Consider a particle of mass  moving in  subject to a potential  and to which n external force  is applied. The external variables are  and the observation of the position, i.e. . If  is invariant under rotations around the  in  then the external symmetry  is given by simultaneously rotating  and the direction of  around . The internal symmetry  is given by simultaneously rotating the position and the velocity or momentum around the  [7].

We note that if  is a symmetry for , then  is an external symmetry for external behaviour  of

Accordingly, using differential geometry approach we give the following definition.

Definition 2

Let  be a smooth nonlinear system. A symmetry for this system is given by a triplet  such that , , and  are diffeomorphism for which the following diagram commutes. [4].

c

We note that if  is a symmetry for , then  is a symmetry for  and  is a symmetry for  in the sense of definition 1.

Suppose that the symmetry of a physical system is given by a one parameter group of diffeomorphisms  which leaves the Lagrangian invariant i.e.  for all . In the above example of the motion of a particle in a central force field, the parameter  is the rotation angle and the one-parameter group is a group of rotations. Then we have the following definition of infinitesimal symmetries which correspond to 1 and 2.

Definition 3

Let  be a smooth nonlinear system. An infinitesimal symmetry is given by a triple  with  and  vectorfields on  and  respectively such that  is a symmetry for every  and small i.e. if  are the one-parameter groups generated by  and  respectively, then the following diagram commutes. [4].

The consequence of the commutativity of the above diagram is that the one-parameter group  acting on  takes a feasible state/external signal trajectory into a similar pair.

Since the objective of this paper is to relate symmetries when they exist to conservation laws, we shall next define a conservation law.

Definition 4

Let  be a dynamical system in state space form and let  be its external behaviour. Let  be such that for every ,  locally integrible vector-valued functions on , and let . The pair  is called a conservation law if

(1)

Holds for all  and for all .  is called the conserved quantity. [4].

The interpretation of equation (1) is that the change of  along a trajectory  is a function of the external trajectory  only.

We use the differential geometry to equation (1). Let  be a smooth function. Define  by  for [4].

Definition 5

Let  be a nonlinear dynamical system with  such that  and . Let  and  be smooth functions. Then the pair  is called a conservation law if  [4].

If  are fibre respecting coordinates for , then  [5]. Therefore . But  is the time derivative of  in  along a trajectory of the vectorfield . Equation (3) therefore yields  We note that  is the Lie derivative .

If the external influence to a system is absent then   The conservation law amounts to   and .

Various laws of conservation are particular cases of Noether’s theorem. Noether’s theorem relates the symmetries of the configuration manifold of a Lagrangian system to conservation Laws. The consequence of the existence of symmetries is the existence of symmetries of a first integral of the equations of motion. This is the content Noether’s theorem and we shall state it. For simplicity only the autonomous case shall be considered.

Theorem 1: (Noether’s theorem)

Let  be a Lagrangian system and let ,  be a one-parameter group of diffeomophism. If the system  admits symmetry under the mapping , thenthe Lagrangian system of equations corresponding to the Lagrangian  has a first integral . In local coordinates of ,  is given by  [4].

Proof

Let  be the coordinate space. Denote the solution of the Lagrange’s equations by  where . It is easy to see that since , it follows that the Lagrangian  is invariant under the mapping . Consequently, the mapping  which is just a translation of the solution of the Lagrange’s equations for any .

Now define the mapping  by . By the hypothesis of invariance of  under the mapping , we have

(2)

The mapping  for fixed  satisfies Lagrange’s equations

(3)

Define  and substitute  for  in (2) equation to get

3. Symmetries and Conservation Laws for Hamiltonian Systems

In this section we specialize the concept of symmetries to Hamiltonian systems. In this case it becomes stronger for the reason that we shall want it to preserve the symplectic structure. Define a symmetry for a Hamiltonian system as follows:

Definition 6

Let  be a full Hamiltonian system. An internal symmetry  is called Hamiltonian if  and  are simplectomorphism i.e.

(i)   

(ii) 

with  and  the pullbacks of  and  by  and  respectively. [7] has pointed out that for minimal systems we don’t have to assume a priori that  is a symplectomorphism.  is implied by the external symmetry  as shown by the following proposition.

Proposition 1

Let  be a full Hamiltonian and minimal system. Let  be an internal symmetry and  a symplectomorphism. Then  is necessarily also a symplectomorphism [6].

Proof

Let . Because  is a symmetry,  is mapped by  and  onto  where  is the derivative map of . Therefore  with  is again a Hamiltonian system. Hence  and  where we have used . This yields  with . [4] has derived that  satisfies the minimality rank condition, then  and .

We shall now consider the case of the infinitesimal symmetries for Hamiltonian systems.

A vectorfield  on a symplectic manifold  is called a symmetry for Hamiltonian vectorfield  on  if [6].

(i)        The Lie derivative ,

(ii)        where  is the Hamiltonian function.

From (i) it follows that  has locally a corresponding Hamiltonian function  and so (ii) implies that  and therefore  is a conserved quantity for . Conversely for  such that  it follows that  satisfies (i) and (ii) and so  is a Hamiltonian symmetry.

The generalization of the above to the Hamiltonian system yields the following definition:

Definition 7

Let  be a full Hamiltonian system. An infinitesimal symmetry  for  is called Hamiltonian if  and  are locally Hamiltonian vectorfields i.e.  and  [9].

A conservation law for a Hamiltonian system can be constructed in the following way:

Consider a Hamiltonian system with an input . For every  we get a Hamiltonian vectorfield on  denoted by . If  is a Hamiltonian symmetry for  then there exists functions  and  with  and  such that  and ,  where  [9] We note that  and  implies that  and  are Hamiltonian functions. The pair  is the conservation law for the Hamiltonian system

The interpretation of the above construction is that the change of  along the trajectories of the system is a function of the external variables. Knowledge of the external variables together with the initial conditions can determine the behaviour of  as a function of time.

We conclude with the generalized Noether’s theorem.

Theorem 2: (Generalized Noether’s theorem)

Let  be an infinitesimal symmetry for a full Hamiltonian system  Then locally there exists a conservation law  Conversely if  is a conservation law, then there exists a Hamiltonian symmetry  such that  and  [4]

The following proposition will be needed for the proof of Noether’s theorem.

Proposition 2

Let  be a nonlinear dynamical system with .Then  is an infinitesimal symmetry iff

    i.      

   ii.       .

 and  are derivative maps of  and  respectively [10].

Proof

We note that  is an infinitesimal symmetry iff diagram (1) commutes for every  with  small. This is equivalent to

(a)     

(b)     

Differenting (a) and (b) with respect to  at  we get (i) and (ii).

Now we proceed with the Generalized Noether’s theorem.

For a Hamiltonian system  we have  (By proof of Proposition 1)

,

, (By proposition 2)

,

.

We have used the fact that  when  [4]. We have thus obtained

.

Therefore  is a conservation law. (c.f. definition 4)

Conversely:

Let  be a conservation law. This is equivalent to  restricted to  equal to zero. For  and  we set  such that .  is therefore a Hamiltonian function. Hence we can define the Hamiltonian vectorfield  on the symplectic manifold . With  the Hamiltonian vectorfield on and  the Hamiltonian vectorfield on  we have

. Because  restricted to  is zero, it follows that  on  for all Hamiltonian vectorfield  tangent to .  is also tangent to  since  is Lagrangian. If we denote  by  and  by  then we say  is tangent to  and for  small we obtain

We construct a Hamiltonian symmetry  by defining a 1-parameter family  such that  for  small and a vectorfield  on  by .

4. Conclusion

The concept of symmetry for Hamiltonian systems has been formulated and analysed. It was shown that symmetry of a physical system is a transformation which may be applied to the state space without altering the system or its dynamical interaction in any way. Symmetries and conservation laws with external to Hamiltonian systems with external forces has been analysed. The conservation law for a Hamiltonian system was constructed and which was concluded by generalized Noether’s theorem.


References

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