## Classical Analysis

Seminar information archive ～09/23｜Next seminar｜Future seminars 09/24～

### 2018/11/02

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

On the inverse problem of the discrete calculus of variations (ENGLISH)

**Giorgio Gubbiotti**(The University of Sydney)On the inverse problem of the discrete calculus of variations (ENGLISH)

[ Abstract ]

One of the most powerful tools in Mathematical Physics since Euler and Lagrange is the calculus of variations. The variational formulation of mechanics where the equations of motion arise as the minimum of an action functional (the so-called Hamilton's principle), is fundamental in the development of theoretical mechanics and its foundations are present in each textbook on this subject [1, 3, 6]. Beside this, the application of calculus of variations goes beyond mechanics as many important mathematical problems, e.g. the isoperimetrical problem and the catenary, can be formulated in terms of calculus of variations.

An important problem regarding the calculus of variations is to determine which system of differential equations are Euler-Lagrange equations for some variational problem. This problem has a long and interesting history, see e.g. [4]. The general case of this problem remains unsolved, whereas several important results for particular cases were presented during the 20th century.

In this talk we present some conditions on the existence of a Lagrangian in the discrete scalar setting. We will introduce a set of differential operators called annihilation operators. We will use these operators to

reduce the functional equation governing of existence of a Lagrangian for a scalar difference equation of arbitrary even order 2k, with k > 1 to the solution of a system of linear partial differential equations. Solving such equations one can either find the Lagrangian or conclude that it does not exist.

We comment the relationship of our solution of the inverse problem of the discrete calculus of variation with the one given in [2], where a result analogous to the homotopy formula [5] for the continuous case was proven.

References

[1] H. Goldstein, C. Poole, and J. Safko. Classical Mechanics. Pearson Education, 2002.

[2] P. E. Hydon and E. L. Mansfeld. A variational complex for difference equations. Found. Comp. Math., 4:187{217, 2004.

[3] L. D. Landau and E. M. Lifshitz. Mechanics. Course of Theoretical Physics. Elsevier Science, 1982.

[4] P. J. Olver. Applications of Lie Groups to Differential Equations. Springer-Verlag, Berlin, 1986.

[5] M. M. Vainberg. Variational methods for the study of nonlinear operators. Holden-Day, San Francisco, 1964.

[6] E. T. Whittaker. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge, 1999.

One of the most powerful tools in Mathematical Physics since Euler and Lagrange is the calculus of variations. The variational formulation of mechanics where the equations of motion arise as the minimum of an action functional (the so-called Hamilton's principle), is fundamental in the development of theoretical mechanics and its foundations are present in each textbook on this subject [1, 3, 6]. Beside this, the application of calculus of variations goes beyond mechanics as many important mathematical problems, e.g. the isoperimetrical problem and the catenary, can be formulated in terms of calculus of variations.

An important problem regarding the calculus of variations is to determine which system of differential equations are Euler-Lagrange equations for some variational problem. This problem has a long and interesting history, see e.g. [4]. The general case of this problem remains unsolved, whereas several important results for particular cases were presented during the 20th century.

In this talk we present some conditions on the existence of a Lagrangian in the discrete scalar setting. We will introduce a set of differential operators called annihilation operators. We will use these operators to

reduce the functional equation governing of existence of a Lagrangian for a scalar difference equation of arbitrary even order 2k, with k > 1 to the solution of a system of linear partial differential equations. Solving such equations one can either find the Lagrangian or conclude that it does not exist.

We comment the relationship of our solution of the inverse problem of the discrete calculus of variation with the one given in [2], where a result analogous to the homotopy formula [5] for the continuous case was proven.

References

[1] H. Goldstein, C. Poole, and J. Safko. Classical Mechanics. Pearson Education, 2002.

[2] P. E. Hydon and E. L. Mansfeld. A variational complex for difference equations. Found. Comp. Math., 4:187{217, 2004.

[3] L. D. Landau and E. M. Lifshitz. Mechanics. Course of Theoretical Physics. Elsevier Science, 1982.

[4] P. J. Olver. Applications of Lie Groups to Differential Equations. Springer-Verlag, Berlin, 1986.

[5] M. M. Vainberg. Variational methods for the study of nonlinear operators. Holden-Day, San Francisco, 1964.

[6] E. T. Whittaker. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge, 1999.