The Inverse Problem of the Calculus of Variation

In this paper, it is intended to determine the necessary and sufficient conditions for the existence and hence the construction of a Lagrangian ( ) , , L t q qɺ 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 ( ) 1 , , n i i i L H p q L t q q = → = − ∑ ɺ ɺ 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.


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 ( ) , , U Y S with U the input space, Y the output space and S the system is to find a manifold M called the state space with initial conditions ( ) 0 0 x x = and functions realizes the input-output system ( ) , , . U Y S TM is the tangent bundle of M [4].

Formulation of the Inverse Problem
The formulation of the inverse problem is as follows: Consider the Lagrange's equation Consider also a holonomic Newtonian system ( ) or equivalently in fundamental form [3].
The inverse problem then consists of studying the conditions under which there exists Lagrangin ( ) , , L t q q ɺ such that equations (2) coincides with equations (4) i.e.
Expansion of equation (2) Equation (5) then demands the validity of the equations The following definition is necessary for the statement of existence of the Lagrangian. Definition The Lagrangian ( ) , , L t q q ɺ is called regular/degenerate in a region 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 n second order ordinary differential equations Define the variations of admissible one-parameter paths The variational forms of i F are given by holds for all admissible variations [6]. [3] has shown that the possible structures of ( ) By comparing equations (11) and (13) 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 , , t q q ɺ to be self-adjoint in 2 1 n+ ℝ , is that all of the following conditions are satisfied everywhere in 2 1

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 ., , ., w x ε ε is at least 1 C in ε and in t .
Then the variation of ( ) , x w is given by 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. 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 We solve this problem through variation methods. Consider a family of curves in m ℝ given by ( ) ( Also the variational form are defined by If we define The observability distribution of this system will have a constant dimension of 2m 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 2 0 y k y + = ɺɺ .
The solution of the inverse problem is given by ( )

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.