Symmetries and Conservation Laws for Hamiltonian Systems

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.


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 ( ) U x where x 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 : I TM → ℝ such that 0 dI dt = . 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.

Formulation of the Concept of Symmetry
Let M be the configuration manifold for a physical system. Let ( ) , , L q q t ɺ be the Lagrangian of the system i.
Example Consider a particle of mass m moving in 3 ℝ subject to a potential V and to which n external force F is applied.   Suppose that the symmetry of a physical system is given by a one parameter group of diffeomorphisms s h which leaves the Lagrangian invariant i.e. * s L h L = for all s . In the above example of the motion of a particle in a central force field, the parameter s is the rotation angle and the oneparameter group is a group of rotations. S T R are the one-parameter groups generated by , S T and R respectively, then the following diagram commutes. [4].
The consequence of the commutativity of the above diagram is that the one-parameter group ( ) , t t S T acting on X W × 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.
Holds for all ( ) , i x w ∈ ∑ and for all 2 1 t t ≥ . F is called the conserved quantity. [4]. The interpretation of equation (1) is that the change of F along a trajectory x is a function of the external trajectory w only.
We use the differential geometry to equation (1). Let : F X → ℝ be a smooth function. Define : [4].
is the time derivative of F in x along a trajectory of the vectorfield ( ) If the external influence to a system is absent then e w ∀ ∈ ∑ The conservation law amounts to 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.
The mapping constant :

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  . [4] has derived that ∑ satisfies the minimality rank condition, then 0 ω = and * ϕ ω ω = .
We shall now consider the case of the infinitesimal symmetries for Hamiltonian systems. The following proposition will be needed for the proof of Noether's theorem.

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.