Science Journal of Applied Mathematics and Statistics
Volume 4, Issue 2, April 2016, Pages: 48-51

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

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

Contents

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

1. A. J. Van der Schaft, "Controllability and observability or affine nonlinear nonlinear Hamiltonian systems", IEEE Trans, Automomatic Control, Vol AC-27, pp. 490-492, 1992.
2. W. M. Tulczyjew, "The Legendre transformation, Annales de I’Institut Henri Poincare–Section A – Vol XXII no. 1, pp 102-114, 1977.
3. S. Bernet and R. G. Cameroon, "Introduction to mathematical control systems", Oxford University Press, New York, 1985.
4. J. C. Willems and J.C Van der Shaft, "Modelling of dynamical systems using external and internal variables with applications to Hamiltonian systems, Dynamical systems and Microphysics, pp 233-263, Academic Press, New York, 1982.
5. R. A. Abraham, J. E., Marsden, T. Ratiu, "Manifolds, Tensor analys and Applications, 2nd edition, Springer-Verlag, New York, 1988.
6. S. P. Banks, "Mathematical theories of Nonlinear systems", Prencice Hall, New York, 1988.
7. R. M. Santili, Foundations of Theoretical Mechanics I, Springer-Verlag, New York Inc., 1978.
8. A. J. Van der Schaft, System theoretic description of physical systems, Doctoral Thesis, Mathematical Centrum, Amsterdam, 1984.
9. W. M. Tulczyjew, Lagrangian submanifolds, statics and dynamics of mechanical systems, Dynamical systems and Microphysics, pp 3-25, Academic Press, New York, 1982.
10. A. J. Van der Schaft, Controllability and observability for affine nonlinear Hamiltonian systems, IEEE Trans, Automatic Control, Vol AC-27, pp 490-492, 1982.

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