INVERSE PROBLEMS IN MECHANICS

Wayne Lawton

Department of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543

Email matwml@nus.edu.sgTel (65) 874-2749

DIRECT and INVERSE PROBLEMS

Direct: given the coefficient vector

compute the roots

,0a),a,a,,a,(a a N011-NN

k1 k1

kkNN ii ii(-1)a,1a

of the equation

.0axaxaxa 01N

1-NN

N1

01 ,,

Inverse: given the roots of the equation, compute the coefficient vector. Answer: up to a scalar multiple

RIGID BODY MODEL

In 1765 the mathematician Leonard Euler usedNewton’s laws and properties of rotations to derive a system of three nonlinear first order differential equations that describe the motion of a rigid body. In matrix notation, these equations are

)A(A is the 3 x 3 inertia operatorA

is the angular velocity in the body

BB)( 1

DIRECT PROBLEM

Attracted the attention of leading mathematicians and physicists for two and a half centuries, including Adler, Appel, Arnold, Audin, Bobylev, Cartan, Chaplygin, Delone, Haine, Hilbert, Jacobi, Jordan, Kirillov, Klein, Kovalevskaya, Lagrange, Lax, Painleve, Picard, Poincare, Poinsot, Poisson, Suslov, Steklov, Weil, and Whittaker.

Motivated much applied and theoretical mathematicsincluding dynamics, elliptic functions, integrablesystems, solitons, and string theories.

INVERSE PROBLEM

Algebraic versus numerical computation

Geometric intuition versus symbolic manipulation

if

A can be computed from

(up to a multiple) only if is not degenerate

))nnA(()nnA( TT

Result 1

Rt,0)t(nT

(lies in a two dimensional subspace), since

INVERSE PROBLEM

B

Angular velocity in space B

momentum in space BAm

E2mT

Energy A5.0E T and angular

are constant

INVERSE PROBLEM

is degenerate if and only if

)A(eig}III{E2

mm321

T

Result 2

is nongegenerate then

If

is bounded below by a positive number, T

is periodic

and

INVERSE PROBLEM

Converse of result 1.Result 3

mMAM

We first observe that

Eu2mM

INVERSE PROBLEM

12 ss

2

1

Ts

sdt)t()t(M

where for

2

1

Ts

sdt)t()t(M

2

1

Tm

s

sdt)t(Bm)t(M

2

1

s

sdt)t(u

INVERSE PROBLEM

Therefore, it suffices to show that V

If it lies in the line }E2mx|x{V T and hence is unbounded since TT

can not

lie in a two dimensional subspace

is bounded below by a positive number. This

contradicts the fact that

TT

is bounded above.

TWO MORE ALGORITHMS

Algorithm 2. Compute a minor eigenvector of

Algorithm 3. Exploit relationships between

and the null vector of the matrix

2

1

ss

dt2||)D(D||)D(Q

jjiiiijjTGij

)A(eig

computed from the moments of

EXTENSIONS

2. Compute A from two measurements of

1. Prove results 1 and 3 for geodesic flows ongeneral Lie groups with invariant metrics, e.g. Euler-Poincare equations for the (polymer) metric

),B( 3. Inverse problems for general classes of differential equations, e.g. integrable, stochastic

volume

2 vuvuv,u

REFERENCES

[2] Arnold, V. I., Mathematical Methods of ClassicalMechanics, Springer, New York, 1978.

[3] Auden, M., Spinning Tops, Cambridge UniversityPress, 1996.

[4] Chang, Y. T., Tabor, M., Weiss, J., Analyticstructure of the Henon-Heiles hamiltonian andintegrable and nonintegrable regimes”, Journal ofMathematical Physics, 23, p. 531, 1982.

[1] Abraham, R. and Marsden, J. E., Foundations ofMechanics, Benjamin, Massachusetts, 1978. 

REFERENCES

[6] V. V. Golubev, Lectures on the Integration of theEquations of Motion of a Rigid Body About a FixedPoint, State Publishing House of Theoretical TechnicalLiterature, Moscow, 1953.

[7] Lawton, W. and Noakes, L., “Computing the inertiaoperator of a rigid body”, Journal of Mathematical Physics,April or May, 2001.

[5] Euler, L., Theoria motus corporum solidorum seurigodorum, 1765.

REFERENCES

[9] Marsden, J. E., Ebin, D. G. and Fischer, A. E.,“Diffeomorphism groups, hydrodynamics and relativity”,Proc. 13th biennial seminar of the Canadian MathematicalCongress (J. R. Vanstone, ed.), Montreal (1972), 135-279.

[10] Weiss, J., Tabor, M. and Carnvale, G., “The Painleve’ property for partial differential equations”, Journal ofMathematical Physics, 24, p. 522, 1983.

[8] Lawton, W. and Lenbury, Y., “Interpolatory solutions of linear ODE’s”, Submitted.