INTRODUCTION
The use of differential form in mechanics and its eventual formulation in terms
of symplectic manifolds has been slowly evolving since Cartan
(1922). The first modern exposition of Hamiltonian systems on symplectic
manifolds seems to be due to Reeb (1952).
In this study the Hamiltonian systems formalism given in Abraham
and Marsden (1978) and Arnold (1989) is used to derive
the equations of motion for a particle on a line with a spring force and for
a free particle in nspace from the energy function and the kinetics of the
phase space.
The study of symmetry provides one of the most appealing applications of group theory. Groups were first invented to analyze symmetries of certain algebraic structures called field extensions and because symmetry is a common phenomenon in all sciences, it is still one of the two main ways in which group theory is applied the other way is through group representations. One can study the symmetry of plane figures in terms of groups of rigid motions of the plane. Plane figures provide a rich source of examples and a background for the general concept of group operations. Plane figures have generally bilateral symmetry, rotational symmetry, translational symmetry, glide symmetry and their combination.
HAMILTONIAN SYSTEM
A general Hamiltonian system consists of a manifold X, possibly infinite dimensional together with a (weakly) nondegenerate closed twoform ω on X (i.e., ω is an alternating bilinear form on each tangent space T_{x}X of X, dω = 0 and for xεX, ω_{x} (u, v) = 0 for all uεT_{x}X implies v = 0) and a Hamiltonian function H: X→ú. Then X, H, ω determine in nice cases, a vector field X_{H} called the Hamiltonian vector field determined by the condition:
Flows
Let X be a smooth manifold. A C^{∞}function F:
is called a flow for the vector field v if F_{x}: →X
is an integral solution for v i.e.,
or
and
Hamiltonian Flow
Let (X, H, ω) be a Hamiltonian system. A flow F is called a Hamiltonian
flow if it preserves the symplectic form and the Hamiltonian function (i.e.,
F_{t}^{*} ω = ω and F_{t}^{*}H = H
for tε)
(Abraham and Marsden, 1978).
Group Actions
Let G be a group and let X be a set. An action of G on X is an assignment
of a function S_{g}: X→X to each element gεG such that:
• 
If I is the identity element of the group G, then S_{I}
is the identity map, i.e., for any xεX we have S_{I} (x) =
x 
• 
For any g, hεG we have S_{g}οS_{h}
= S_{gh}, i.e., for every xεXwe have S_{g}(S_{h}
(x)) = S_{gh} (x) 
A Liegroup action should satisfy certain differentiability properties in addition to the algebraic properties given above. The action is called effective if S_{g} = Identity map for only t = 0.
SYMMETRY OF HAMILTONIAN SYSTEMS
The symmetry of Hamiltonian system (X, ω, H) is a function S: X→X that preserves both the symplectic form ω and the Hamiltonian function H.
Motion of a Particle on a Line in the Plane with the Spring Force
The phase space of such a physical system Simmons (1991)
is the simplest nontrivial symplectic manifold, the twodimensional plane X
= R^{2} = {(q, p): qεR, pεR} with the area twoform ω
= dq∧dp.
The Hamiltonian function for such a particle is:
where, second term in the Hamiltonian is the potential energy of the spring.
Using the Eq. 1, we have for qεR and vεT_{q}R
Taking
and
as an arbitrary vector field. We find:
or
or
and
Thus we have:
since, ∂/∂q and ∂/∂p are functions of time t (along a particular trajectory) we can write the vector field:
as time derivative along trajectories on the plane, since ∂/∂q and ∂/∂p are linearly independent we have:
which shows that equation of motion for a particle in a line with spring force is a linear differential equation:
We can draw the useful picture by using the conservation of the Hamiltonian by the Hamiltonian flow because it implies that the orbits of the system must lie inside level sets of H (An orbit is set of all points in phase space that the system must passes through during one particular motion. In other words it is the set of all points on one particular trajectory). The beautiful features of Hamiltonian systems is that we can get information about orbits of the differential equations of motion by solving the algebraic equation H = constant, which is easy to solve. So, here first we determine the Hamiltonian flow of the spring problem.
HAMILTONIAN FLOW OF THE SPRING PROBLEM
For finding the bona fide solutions to our differential equations, i.e., not only the orbit of a trajectory but the trajectory itself (i.e., the position as a function of time). We use the algebraic equation H_{0} = constant to reduce our original system of differential equations:
into one scalar differential equation:
which on integration gives:
The Hamiltonian flow for the linear differential Eq. 4 is given by the function:
or
Geometrically, the flow at time t in phase space is effected by first scaling
the qaxis by a factor of ,
which takes the orbits to circles, second, rotating these circles clockwise
through an angle and
finally rescaling the qaxis back to its original scale. Since, for each t the
function f_{t} is a linear function from R^{2}to R^{2}
and because the determinant of the matrix representing f_{t} is 1, f_{t}
is area preserving. So, the flow preserves the symplectic form.
Also the Hamiltonian flow preserves the Hamiltonian for f_{t}^{*}
H = H, i.e., Hοf_{t}, we have, for gεSO(2), (special orthogonal
group):
the level sets of H in phase space are ellipses (Fig. 1),
Here if k is large the ellipses are tall and skinny, while if k is close to
0 then the ellipses are short and wide. If k = 1/m the ellipses degenerate to
circles. As the flow preserves the Hamiltonian, each solution of the system
must lie entirely with in one ellipse in phase space. The conservation of the
Hamiltonian by the Hamiltonian flow tells us that orbits must lie inside sets
of the form .
Since the motion is continuous, it follows that each orbit is contained in the
curve
(Fig. 1):
The spring Hamiltonian given in Eq. 2 is an action of the group (R, +) on R^{2}, for:
• 


Fig. 1: 
Phase space of the particle on the line with level sets of
the spring Hamiltonian 
This action is not effective because if t is an integer multiple of
then:
Which also shows that the flow is periodic with period .
MOTION OF A FREE PARTICLE IN nSPACE
Consider the motion of a free particle in n space. Let q = (q_{1},
..., q_{n}) be the position vector of the particle and p = (p_{1},
..., p_{n}) be the corresponding momentum vector of the particle. Then
the phase space of the particle is the manifold with
the symplectic form:
and the Hamiltonian function
Then ω, H determine the vector field X_{H} by the condition (1)
Let be
arbitrary vector fields, then using (1), we have:
or
This gives;
a_{i} = p_{i}/m and b_{i} = 0, (i=1,…,n).
Thus the vector field is given by:
Taking the vector field:
as time derivative along trajectories, we have:
This gives:
the required equation of motion of the free particle in n space
HAMILTONIAN FLOW OF THE PARTICLE IN nSPACE
The Hamiltonian flow of the Particle in nspace is determined by taking the algebraic equation:
with the system of differential equations (1.9.3) and initial condition p(t) = p(0), t = 0, we have:
Thus for any fixed time t, the map:
defined by:
is a Hamiltonian flow, for:
and f_{t}^{*} H = H.
To show that every flow is not a Hamiltonian flow. If we take the flow of the problem particle in nspace as:
defined by:
for any hus
g_{t} is not a Hamiltonian flow of a Hamiltonian system with the canonical
symplectic form on
Taking as
the symplectic form on then
the flow g_{t} defined Eq. 16 preserves ω, for:
The Hamiltonian function for this system can be determined by taking
and
as arbitrary vector fields, then Eq. 1, we have:
or
which gives:
which on integration yields:
Now:
Hence, g_{t} preserves H. Thus g_{t} defined Eq. 16 is a Hamiltonian flow for the Hamiltonian system (M, ω, H), where,
M = ^{2n}{0}
and H is given Eq. 17.
The Hamiltonian flow of the Particle in nspace given Eq. 15 can be written as:
and satisfying the condition of group action, for:
• 
0εR, f_{0} is indeed an identity matrix 
• 

The action is also effective. Thus the time flows of the Spring problem and Particle in nspace problem are symmetry of Hamiltonian system. But a Hamiltonian system may have other type of symmetries in addition to the time flow. For the problem the phase space of such a particle motion is:
with symplectic form and
the Hamiltonian function .
Consider the translation action of the group (R^{n}, +) on X, for each
g = (g_{1}, ..., g_{n}) in R^{n}, define the function:
by
Then S_{g} is the symmetry of the Hamiltonian system for any gεR^{n}, for:
and S_{g}^{*} H =H.
Since, S_{g} gives a onetoone correspondence from X to X, shows that S_{g} preserves the symplectic manifold.
Next, consider the rotational symmetry of a free particle in nspace.
The Hamiltonian system for such a particle is:
The action of SO(n) on X is defined by S_{g}(p, q) = (gq, pg^{T}) and is called the rotation action. Here, S_{g} preserves manifold X and symplectic form, since, g is constant and an orthogonal matrix g g^{T}, also we have:
and S_{g}^{*} H = H, i.e., HοS_{g} = H, for:
Hence, the action of the Lie group (R^{n}, +) on :
preserves both the symplectic form and the Hamiltonian function called the translation and rotation symmetry of the mechanical system. These symmetries are linear symmetries so they can express in matrix form.
CONCLUSION
In this study we have shown that every flow is not Hamiltonian but by changing
the symplectic form and with some restriction on the phase space one can successfully
change the non Hamiltonian flow into the Hamiltonian flow. Also, we have discussed
the symmetry group properties of the mechanical system. For two body problem
only these symmetries are sufficient for consideration but for other system
nonlinear symmetries also arise (for further discussion about symmetry of differential
equations and Hamiltonian systems see Artin (1991) and
Marsden and Ratiu (1999)). The above symmetries of a mechanical
system are useful in reducing the phase space of the system using the MarsdenWeinstein
theorem.