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I start with the following philosophical view visions: ��(1) Creation is a reflection of the Creater via which He is to be known. �His static eternity beyond Cosmos projects a dynamical existence that necessitates space-time. Thus we got structured constrained variation and relativities, whence Allah is the only absolute. (M. A. Bashir 1996).��(2) Mathematics is the study of variations and relations of Being and ultimately invariants. This indicates our need to investigate equations, space structure and related theories. (M. A. Bashir 2001).
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TOWARDS AN INTEGRATED
MATHEMATICAL KNOWLEDGE
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(3) Being necessitates the creation of cosmos and human. Both are phenomenological and dynamic. We need to acknowledge and understand both the cosmological and human phenomena via their dynamics. This dynamic involves variables for description and invariants for characterization. (M. A. Bashir 1988).
�(4) We have to grasp the invariants, though relative, in order to penetrate into beyond Being which is interpreted and modeled by integrated pure mathematical thought. (M. A. Bashir 1983)��(5) Metaphysics (inspiration and religions) provides us with the most general invariants, and from these, using variations we have to deduce the relative ones. (M. A. Bashir 1986) �
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(2) The variations are not Absolute. They are Relative. So for a specific variation the variables are related by mathematical equations. Thus the variation is described by a Subspace of the Space of variables.
(3) Therefore for any variation, we have the following equivalent descriptions:
High lights
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Both are constraints indicating the relativity of variations:
The interplay between Algebra , Analysis and Geometry is quite obvious in contemporary mathematical thought ( look for instance at Atiyah- Singer index theorem).
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Difficulties In Solving Non-linear D.E:
Difficulties in solving non-linear DE lead to the following approaches in order to solve the mathematical problems
(A) Numerical Analysis and Numerical Methods.
(B) Non-linear Analysis:
(i) Non- linear Operator Theory.
(ii) Geometrical Formulation of non-linear problems using the concept of symmetry.
Notice that quite often researchers treat only local mathematical problems expressed interms of coordinates, while global treatments, though very important, are handelled by few mathematicians who are geometers or topologists.
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Symmetry is the characteristic of nature and inherited in the human thought. Thus most variations are symmetrical. This symmetry has been used to apprehend the variations and even utilized as a technique to describe and solve these variations. The idea was developed into Galois's theory for algebraic equations and Lie theory for differential equations.
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Invariant (algebraic or differential) may be used to construct a solution for an algebraic or differential equation. If symmetries are known or calculated, then invariants can be evaluated. Physicists , engineers, and scientists, having their problems at hands, know the symmetries very well. That is the reason why most solutions of applied problems are constructed by them.
And let : be a symmetric transformation. This means that the above form of differential equation is preseved under G; i.e. :
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In this case the group action is generated by 1- parameter and the infinitesimal generator of the group is given by the vector field
The transformed coordinates under the symmetry group satisfy:
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Equation (4) implies that y is invariant. Equation (2) implies that in the new coordinates . Solution of (4) and (5) yields the required change of variables that separates the differential equation.
Example: consider the homogeneous differential equation :
Which has obvious symmetry transformation
(scaling transformation)
the infinitesimal generator of the above transformation is given by:
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The required transformation is therefore :
And the transformed differential equation in the new variables can be calculated as:
Thus we get:
so:
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suppose the differential equation :
M dx + N dy = 0
Has a one – parameter symmetry group with infinitesimal generator:
Then the integrating factor is given by:
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Example: Consider the rotation group in the plane:
The infinitesimal generator is given by:
The most general ordinary differential equation admitting the rotation group SO(2) as a symmetry is given by:
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The new coordinates satisfying (4) and (5) should be the polar coordinates since is equal to in these coordinates. So :
Substituting into (12) and solving for , we get:
Which provides the general solution:
Now if we utilize the integrating factor method provided by (9) for equation (12) we get:
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For instance if H(r) = 1 , our equation (12) becomes:
And multiplying by the integrating factor we get the exact equation :
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(5) Theorem:
Let be an s-parameter normal subgroup of a Lie group acting on a manifold M such that H has s-dimensional orbits. Let be an n-th order ordinary differential equations admitting H as a symmetry group with invariants:
Then can be reduced to order on .
Also there is an induced action of the quotient group on and
Admits, all of G as a symmetry group if and only if the H-reduced equation
Admits the quotient group as a symmetry group.
Example: Consider the D.E
The above equation admits the two-parameter symmetry group:
APPLICATIONS OF LIE GROUPS
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Infinitesimal generators are:
So we need first to determine the Invariants of v:
Where in our example:
And therefore: .
Then we can see that:
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So equation (13) reduces to the first order equation
Which is a separable equation with implicit solution
We also see that equation (14) is invariant under the reduced group
Determined by . Illustrating the reduction method, we proceed further to reconstruct the general solution to equation (13) by solving the linear equation
Using the integrating factor determined from the symmetry group generated by v. Doing this we get
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Construction of Group Invariant solutions to system of partial differential equations:
(6) Theorem:
Let G be a symmetry group of transformations for a system of partial differential equations Δ defined over Ϻ. Then there is a reduced system of differential equations over , the space of independent invariants, with the property that any G-invariant function on Ϻ is a solution to Δ if and only if its representative on is a solution to .
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(7) Solvable Groups:
In order to apply the reduction procedure, using normal subgroups of G, so as to find the general solution by quadratures, our reduction should be full. This means that we should have sufficient normal subgroups. A group with the property that it has sufficient normal subgroups to solve the D.E by quadratures is called a solvable group defined as follows:
Definition: Let G be a lie group with Lie algebra Ğ. Then G is solvable if there exists a chain of Lie subgroups:
Such that for each is a k-dimensional subgroup of G and is a normal subgroup of . Equivalently, there exists a chain of subalgebras such that for each k, and is a normal subalgebra of :
This requirement for solvability is equivalent to the existence of a basis
of Ğ such that:
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The Geometrical Set up (Structured Space)
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A good example is the symplectic structure (𝓜, ) describing a general dynamical system, where is the symplectic form that interprets geometrically the dynamics of Hamilton, i.e.
X is the Hamiltonian vector field generating the motion:
with
we get:
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Proving the equivalence of Hamilton equation on the face of coordinates free global form, namely using the Cartan formula:
and the global Hamilton equation:
One get: ( is closed ) .Thus is invariant under X . If is given and we are solving for , ,then is in fact given by:
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provided the infinitesimal generator X of the symmetry is calculated ( from ).
In fact the original equation and the symmetry condition
are equivalent.
(8) We tackle with the existence and classification of solutions of the above most important problem . We need the representation theory of groups. Since we have symmetry our solutions are generated by group action which is regarded as a geometrical description of variations instead of equations. The solutions are thus orbits of the group which are in fact homogenous spaces and such that:
, H is the isotropy subgroup of a point s in S .
The most appropriate representation is really the linear representation of G in ( The dual of the Lie algebra g of G).
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But before that since we have then G acts on , which is itself a representation of the action of G on M. The problem is then transformed to algebraic one where classification is possible (in order to classify the variation described by M we need only classify (possibly using the corresponding Lie algebra)
(8) The Co-adjoint representation:
Definition: let G be a Hamiltonian group of transformations acting on the Poisson manifold M, generated by the real-valued functions
the momentum map for G is the smooth map given by
In which are the dual basis to for the basis of G relative to which the structure constants were computed.
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Now any Hamiltonian system with G as a Hamiltonian symmetry group naturally restricts to a system of ordinary differential equations on the common level set of the given first integrals. Note that these common level sets are just the level sets of the momentum map, denoted where . Moreover, the reduced system will automatically remain invariant under the residual symmetry group
Of group elements leaving the chosen level set invariant. There is an easy characterization of this residual group:
Proposition: Let be the momentum map associated with a Hamiltonian group action. Then the residual symmetry group of a level set is the isotropy subgroup of the element :
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Moreover, if has the property that it takes one point to a point , then , and has this property for all .
Let M be a Poisson manifold and G a regular Hamiltonian group of transformations. Let . Assume that the momentum map is of maximal rank everywhere on the level set , and that the residual symmetry group acts regularly on the submanifold . Then there is a natural immersion making into a Poisson submanifold of in such a way that the diagram Commutes:
i
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(Here and are the natural projections and i the immersion realizing as a submanifold of M). Moreover, any Hamiltonian system on M which admits G as a Hamiltonian symmetry group naturally restricts to systems on the other spaces in the above diagram, which are Hamiltonian on and , and which are related by the appropriate maps. In particular, we obtain a Hamiltonian system on by first restricting to and then projecting using .
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In addition to dynamical system (M,W) we had so many problems that may be formulated in a geometrical set up.
We have the following as examples:
(1) Einstein field equation:
Given a metric g satisfying Einstein equation, we need to find a manifold M such that (M, g) is a pseudo-Riemannian. The underlying symmetry in this case is the group of all isometries satisfying
Where X is an infinitesimal generator (Killing vector field). If H is isotropy subgroup then the solution of the problem is given by: where G is the group of symmetries. We need beforehand a coordinate-free expression of Einstein equation (considered as a Gauge theory).
(9) Geometrically structured problems
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The candidate for space is the so called Fiber bundle, where the dynamic is described by a cross-section of this bundle (Group action formulates the dynamic) with symmetry group called the Gauge group that is essentially involved in the construction of the bundle (solution of Yang-Mills equations) with the help of the transition functions.
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(10) Existence and Classification of Solutions:
A bout existence of solutions algebraic topology is of particular interest in both the existence and classification of solutions of variation. Cohomology groups have solved most of contemporary mathematical problems.
Examples:
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Casmir operators: for SU(n) there are (n-1) Casmir operators , which commutes with Lie algebra of symmetry group and with themselves. Casmir operators ( such as the Hamiltonian) are constructed from the Lie algebra of the symmetry group. For instance if H is a Hamiltonian , then the general group action can be written as :
(1) , (Separation of variables)
(11) A research Problem:
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The Lie algebra should be closed in order that their distribution gives an integral manifold representing the solution (Frobenius theorem). This solution may be found via the action.
Equation (1) and (2) provide a methodology for solving an operator equation. However we can proceed for a classification scheme as follows: let H be any Casmir operator. Then we have the Eigenvalue problem :