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Research Article
 

Symmetry of Hamiltonian Systems



V.G. Gupta and P. Sharma
 
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ABSTRACT

In the present study we use the formalism of Hamiltonian system on symplectic manifold due to Reeb, given in Abraham and Marsden and Arnold to derive the equation of motion for a particle on a line in a plane with a spring force and for a free particle in n-space. The time flows for both the problems mentioned above are also determined and proved that the determined flow is a Hamiltonian flow i.e., the symmetry of a Hamiltonian system. A non-Hamiltonian flow is also considered and it is shown that by changing the symplectic form and the phase space of the system we can convert it into a Hamiltonian flow. The translation and rotational symmetry related to linear and angular momentum respectively for the motion of a free particle in n-space is also considered, which is useful in reducing the phase space of a mechanical system.

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  How to cite this article:

V.G. Gupta and P. Sharma, 2009. Symmetry of Hamiltonian Systems. Asian Journal of Earth Sciences, 2: 39-48.

DOI: 10.3923/ajes.2009.39.48

URL: https://scialert.net/abstract/?doi=ajes.2009.39.48
 

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 n-space 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) non-degenerate closed two-form ω on X (i.e., ω is an alternating bilinear form on each tangent space TxX of X, dω = 0 and for xεX, ωx (u, v) = 0 for all uεTxX implies v = 0) and a Hamiltonian function H: X→ú. Then X, H, ω determine in nice cases, a vector field XH called the Hamiltonian vector field determined by the condition:

Image for - Symmetry of Hamiltonian Systems
(1)

Flows
Let X be a smooth manifold. A C-function F:Image for - Symmetry of Hamiltonian Systems is called a flow for the vector field v if Fx: Image for - Symmetry of Hamiltonian Systems→X is an integral solution for v i.e.,

Image for - Symmetry of Hamiltonian Systems

or

Image for - Symmetry of Hamiltonian Systems

and

Image for - Symmetry of Hamiltonian Systems

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., Ft* ω = ω and Ft*H = H for tεImage for - Symmetry of Hamiltonian Systems) (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 Sg: X→X to each element gεG such that:

If I is the identity element of the group G, then SI is the identity map, i.e., for any xεX we have SI (x) = x
For any g, hεG we have SgοSh = Sgh, i.e., for every xεXwe have Sg(Sh (x)) = Sgh (x)

A Lie-group action should satisfy certain differentiability properties in addition to the algebraic properties given above. The action is called effective if Sg = 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 non-trivial symplectic manifold, the two-dimensional plane X = R2 = {(q, p): qεR, pεR} with the area two-form ω = dq∧dp.

The Hamiltonian function for such a particle is:

Image for - Symmetry of Hamiltonian Systems
(2)

where, second term in the Hamiltonian is the potential energy of the spring.

Using the Eq. 1, we have for qεR and vεTqR

Image for - Symmetry of Hamiltonian Systems

Taking

Image for - Symmetry of Hamiltonian Systems

and

Image for - Symmetry of Hamiltonian Systems

as an arbitrary vector field. We find:

Image for - Symmetry of Hamiltonian Systems

or

Image for - Symmetry of Hamiltonian Systems

or

Image for - Symmetry of Hamiltonian Systems

and

Image for - Symmetry of Hamiltonian Systems

Thus we have:

Image for - Symmetry of Hamiltonian Systems

since, ∂/∂q and ∂/∂p are functions of time t (along a particular trajectory) we can write the vector field:

Image for - Symmetry of Hamiltonian Systems

as time derivative along trajectories on the plane, since ∂/∂q and ∂/∂p are linearly independent we have:

Image for - Symmetry of Hamiltonian Systems
(3)

which shows that equation of motion for a particle in a line with spring force is a linear differential equation:

Image for - Symmetry of Hamiltonian Systems
(4)

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 H0 = constant to reduce our original system of differential equations:

Image for - Symmetry of Hamiltonian Systems

into one scalar differential equation:

Image for - Symmetry of Hamiltonian Systems
(5)

which on integration gives:

Image for - Symmetry of Hamiltonian Systems
(6)

The Hamiltonian flow for the linear differential Eq. 4 is given by the function:

Image for - Symmetry of Hamiltonian Systems
(7)

or

Image for - Symmetry of Hamiltonian Systems
(8)

Geometrically, the flow at time t in phase space is effected by first scaling the q-axis by a factor of Image for - Symmetry of Hamiltonian Systems, which takes the orbits to circles, second, rotating these circles clockwise through an angle Image for - Symmetry of Hamiltonian Systemsand finally rescaling the q-axis back to its original scale. Since, for each t the function ft is a linear function from R2to R2 and because the determinant of the matrix representing ft is 1, ft is area preserving. So, the flow preserves the symplectic form.

Also the Hamiltonian flow preserves the Hamiltonian for ft* H = H, i.e., Hοft, we have, for gεSO(2), (special orthogonal group):

Image for - Symmetry of Hamiltonian Systems

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 Image for - Symmetry of Hamiltonian Systems. Since the motion is continuous, it follows that each orbit is contained in the curveImage for - Symmetry of Hamiltonian Systems (Fig. 1):

The spring Hamiltonian given in Eq. 2 is an action of the group (R, +) on R2, for:

Image for - Symmetry of Hamiltonian Systems
Image for - Symmetry of Hamiltonian Systems

Image for - Symmetry of Hamiltonian Systems
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 Image for - Symmetry of Hamiltonian Systemsthen:

Image for - Symmetry of Hamiltonian Systems

Which also shows that the flow is periodic with period Image for - Symmetry of Hamiltonian Systems.

MOTION OF A FREE PARTICLE IN n-SPACE

Consider the motion of a free particle in n space. Let q = (q1, ..., qn) be the position vector of the particle and p = (p1, ..., pn) be the corresponding momentum vector of the particle. Then the phase space of the particle is the manifold Image for - Symmetry of Hamiltonian Systemswith the symplectic form:

Image for - Symmetry of Hamiltonian Systems

and the Hamiltonian function

Image for - Symmetry of Hamiltonian Systems

Then ω, H determine the vector field XH by the condition (1)

Let Image for - Symmetry of Hamiltonian Systemsbe arbitrary vector fields, then using (1), we have:

Image for - Symmetry of Hamiltonian Systems

or

Image for - Symmetry of Hamiltonian Systems

This gives;

ai = pi/m and bi = 0, (i=1,…,n).

Thus the vector field is given by:

Image for - Symmetry of Hamiltonian Systems
(9)

Taking the vector field:

Image for - Symmetry of Hamiltonian Systems
(10)

as time derivative along trajectories, we have:

Image for - Symmetry of Hamiltonian Systems
(11)

This gives:

Image for - Symmetry of Hamiltonian Systems
(12)

the required equation of motion of the free particle in n- space

HAMILTONIAN FLOW OF THE PARTICLE IN n-SPACE

The Hamiltonian flow of the Particle in n-space is determined by taking the algebraic equation:

Image for - Symmetry of Hamiltonian Systems
(13)

with the system of differential equations (1.9.3) and initial condition p(t) = p(0), t = 0, we have:

Image for - Symmetry of Hamiltonian Systems
(14)

Thus for any fixed time t, the map:

Image for - Symmetry of Hamiltonian Systems

defined by:

Image for - Symmetry of Hamiltonian Systems
(15)

is a Hamiltonian flow, for:

Image for - Symmetry of Hamiltonian Systems

and ft* H = H.

To show that every flow is not a Hamiltonian flow. If we take the flow of the problem particle in n-space as:

Image for - Symmetry of Hamiltonian Systems

defined by:

Image for - Symmetry of Hamiltonian Systems
(16)

for any Image for - Symmetry of Hamiltonian Systemshus gt is not a Hamiltonian flow of a Hamiltonian system with the canonical symplectic form on Image for - Symmetry of Hamiltonian Systems

Taking Image for - Symmetry of Hamiltonian Systemsas the symplectic form on Image for - Symmetry of Hamiltonian Systemsthen the flow gt defined Eq. 16 preserves ω, for:

Image for - Symmetry of Hamiltonian Systems

The Hamiltonian function for this system can be determined by taking

Image for - Symmetry of Hamiltonian Systems

and

Image for - Symmetry of Hamiltonian Systems

as arbitrary vector fields, then Eq. 1, we have:

Image for - Symmetry of Hamiltonian Systems

or

Image for - Symmetry of Hamiltonian Systems

which gives:

Image for - Symmetry of Hamiltonian Systems

which on integration yields:

Image for - Symmetry of Hamiltonian Systems
(17)

Now:

Image for - Symmetry of Hamiltonian Systems

Hence, gt preserves H. Thus gt defined Eq. 16 is a Hamiltonian flow for the Hamiltonian system (M, ω, H), where,

M = Image for - Symmetry of Hamiltonian Systems2n-{0}

Image for - Symmetry of Hamiltonian Systems

and H is given Eq. 17.

The Hamiltonian flow of the Particle in n-space given Eq. 15 can be written as:

Image for - Symmetry of Hamiltonian Systems

and satisfying the condition of group action, for:

0εR, f0 is indeed an identity matrix
Image for - Symmetry of Hamiltonian Systems

The action is also effective. Thus the time flows of the Spring problem and Particle in n-space 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:

Image for - Symmetry of Hamiltonian Systems

with symplectic form Image for - Symmetry of Hamiltonian Systemsand the Hamiltonian function Image for - Symmetry of Hamiltonian Systems. Consider the translation action of the group (Rn, +) on X, for each g = (g1, ..., gn) in Rn, define the function:

Image for - Symmetry of Hamiltonian Systems

by

Image for - Symmetry of Hamiltonian Systems
(18)

Then Sg is the symmetry of the Hamiltonian system for any gεRn, for:

Image for - Symmetry of Hamiltonian Systems

and Sg* H =H.

Since, Sg gives a one-to-one correspondence from X to X, shows that Sg preserves the symplectic manifold.

Next, consider the rotational symmetry of a free particle in n-space.

The Hamiltonian system for such a particle is:

Image for - Symmetry of Hamiltonian Systems

The action of SO(n) on X is defined by Sg(p, q) = (gq, pgT) and is called the rotation action. Here, Sg preserves manifold X and symplectic form, since, g is constant and an orthogonal matrix g gT, also we have:

Image for - Symmetry of Hamiltonian Systems

and Sg* H = H, i.e., HοSg = H, for:

Image for - Symmetry of Hamiltonian Systems

Hence, the action of the Lie group (Rn, +) on :

Image for - Symmetry of Hamiltonian Systems

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 Marsden-Weinstein theorem.

REFERENCES

1:  Abraham, R. and J. Marsden, 1978. Foundations of Mechanics. 2nd Edn., Addison-Wesley, Reading, MA., USA

2:  Arnold, V.I., 1989. Mathematical Methods of Classical Mechanics. 2nd Edn., Springer-Verlag, New York

3:  Artin, M., 1991. Algebra. Prantice Hall, Upper Saddle River, New Jersey

4:  Cartan, E., 1922. Lecons sur les Invariants Integraux. Hermann, Paris

5:  Marsden, J.E. and T.S. Ratiu, 1999. Introduction to Mechanics and Symmetry. 2nd Edn., Springer-Verlag, New York

6:  Reeb, G., 1952. Varietes symplectiques varietes presque-complexes et systemes dynamiques. C.R. Acad. Sci. Paris, 235: 776-778.

7:  Simmons, G.F., 1991. Differential Equations with Applications and Historical Notes. 2nd Edn., McGraw Hill, Inc., New York

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