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