The Inverse Problem of the Calculus of Variation
Estomih Shedrack Massawe
Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam, Dar es Salaam, Tanzania
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To cite this article:
Estomih Shedrack Massawe. The Inverse Problem of the Calculus of Variation. Science Journal of Applied Mathematics and Statistics. Vol. 4, No. 2, 2016, pp. 48-51. doi: 10.11648/j.sjams.20160402.15
Received: February 8, 2016; Accepted: March 9, 2016; Published: March 28, 2016
Abstract: In this paper, it is intended to determine the necessary and sufficient conditions for the existence and hence the construction of a Lagrangian of a dynamical system from its equations of motion. The existence of a Lagrangian is vital importance for the Hamiltonian description of a dynamical system since via the Legendre transformation we get the Hamiltonian of the system [1, 2]. It is also intended to show that the solution of the realization problem for the Hamiltonian system reduces to solving an inverse problem.
Keywords: Inverse Problems, Calculus of Variation, Realization Problem, Hamiltonian Systems
1. Introduction
In practice it often happens that the mathematical description of a system in terms of state variables is very complex or even not known but the external variables can be determined from experimental measurements or other considerations [3]. This difficulty is one reason for the frequent use of a "black-box" description. However it is very useful to find a system in state space form to which the set of the external variables correspond. In "Realization problem" we start with the external behaviour of a system and attempt to obtain the state space description. This idea is very necessary in control theory.
The Hamiltonian realization problem is described with a view of the inverse problem in classical being its special case; that is if the inverse problem can be solved, then there is a special case of the Hamiltonian realization problem that is solved. In realization problem for nonlinear input-output system with the input space, the output space and the system is to find a manifold called the state space with initial conditions and functions , such that
(1)
realizes the input-output system is the tangent bundle of [4].
2. Formulation of the Inverse Problem
The formulation of the inverse problem is as follows:
Consider the Lagrange’s equation
(2)
Consider also a holonomic Newtonian system
(3)
or equivalently in fundamental form [3].
(4)
The inverse problem then consists of studying the conditions under which there exists Lagrangin such that equations (2) coincides with equations (4) i.e.
(5)
Expansion of equation (2) yields
(6)
Equation (5) then demands the validity of the equations
(7)
(8)
The following definition is necessary for the statement of existence of the Lagrangian.
Definition
The Lagrangian is called regular/degenerate in a region of points when the Hessian determinant , is non-null/null in it with the possible exception of a (finite) number of isolated zeros, [5]
The solution of the inverse problem needs the following ingredients:
Consider a system of second order ordinary differential equations
(9)
Define the variations of admissible one-parameter paths , , , which are at least in and in but not necessarily solutions of by
(10)
The variational forms of are given by
(11)
where
, , ,
Definition
A system of variational forms is called the adjoint system of defined by equations (11) if there exists a function such that the Lagrange identity
(12)
holds for all admissible variations [6].
[3] has shown that the possible structures of and are
(13)
If coincides with its adjoint i.e. if , , , then is called self-adjoint.
By comparing equations (11) and (13) we get the conditions of self-adjointness for the variational forms:
(14)
A system of ordinary differential equations is self-adjoint when its variational forms are self-adjoint.
[7] has further shown that
(A) A necessary and sufficient condition for a holonomic one-dimensional Newtonian system in the fundamental form ; , , to be self-adjoint in a region of points is that the condition
(15)
Holds everywhere in .
(B) A necessary and sufficient condition for a holomic Newtonian system , satisfying the continuity and regularity conditions , , , in a region of points to be self-adjoint in , is that all of the following conditions are satisfied everywhere in :
(16)
3. Realization of Hamiltonian Systems
In practice, it often happens that the mathematical description of system in terms of state variables is very complex or even not known but the external variables can be determined from experimental measurements or other considerations. This difficulty is one reason for the frequent use of "black-box" description. However it is very useful to find s system in state space form to which the set of the external variables correspond. In "Realization problem", we start with the external behaviour of a system of a system and attempt to obtain the state space description. The idea is very necessary in control theory.
We shall now establish the necessary conditions on the external behaviour of a Hamiltonian system such that we can construct a Hamiltonian system which generates this external behaviour. Let and be a trajectory in . The variational principle assumes the existence of a parameter such that a family of functions exists such that
i. .
ii. is at least in and in .
iii. .
Then the variation of is given by
(17)
Then
(18)
It can be shown [8].
Consequently,
and of
If has compact support in then
(19)
This is the general variational principle and it involves only the external behaviour of the system. This formulation has a useful consequence for the Hamiltonian realization problem which can be seen in the following way.
Let denote a symplectic manifold and denote a space of function .We define a weak symplectic form on by
(20)
where and , are variations of with respect to with compact support. If is an external system on such that is a submanifold of , then can be realized by a Hamiltonian system iff is a Lagrangian submanifold of [9].
One procedure for realizing a Hamiltonian system is through solving an inverse problem of the calculus of variations as described below.
Consider a system of second order differential equations , . If we assume that the square matrix , has , then the inverse problem is to find the conditions under which there exists the Lagrangian such that
(21)
We solve this problem through variation methods. Consider a family of curves in given by . The variations , are defined by
(22)
Also the variational form are defined by
(23)
where
.
Assume that there exists other variational forms such that we can define a function where
(24)
and of defined by equations (22).
If , then there exists locally a function such that
(25)
Instead of taking now we take , . If the inverse problem of calculus of variation for has a solution for the Lagrangian , then the realization of the system , is
(26)
implies that .
If we define and , , then the above system is equivalent to the Hamiltonian system
(27)
The observability distribution of this system will have a constant dimension of so the Hamiltonian system (27) above is locally minimal i.e. it is controllable and observable [10]
Example
Consider the one-dimensional harmonic oscillator. The equation of motion is given by
.
The solution of the inverse problem is given by
.
The Hamiltonian is given by
.
The Hamiltonian system is given by
4. Conclusions
In this paper it was shown how the Hamiltonian realization problem is described with a view of the inverse problem in classical mechanics being its special case; that is if the inverse problem can be solved, then there is a Hamiltonian realization problem that is solved.
References