Hamilton function

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A function introduced by W. Hamilton (1834) to describe the motion of mechanical systems. It is used, beginning with the work of C.G.J. Jacobi (1837), in the classical calculus of variations to represent the Euler equation in canonical form. Let be the Lagrange function of a mechanical system or the integrand in the problem of minimization of the functional

of the classical calculus of variations, where , . The Hamilton function is the Legendre transform of with respect to the variables or, in other words,

where is expressed in terms of by the relation and is the scalar product of the vectors and . If the Hamilton function is used, the Euler equation

(also known as the Lagrange equation (cf. Lagrange equations (in mechanics)) in problems of classical mechanics) is written in the form of a system of first-order equations:

These equations are called the Hamilton equations, the Hamiltonian system and also the canonical system. The Hamilton–Jacobi equations for the action function (cf. Hamilton–Jacobi theory) can be written in terms of a Hamilton function.

In problems of optimal control a Hamilton function is determined as follows. One has to find a minimum of the functional

under the differential constraints

for given boundary conditions and with constraints on the control . Here is an -dimensional vector of phase coordinates, is an -dimensional control vector and is a closed set of admissible values of . The Hamilton function in this problem has the form

where , and are conjugate variables (Lagrange multipliers, momenta) analogous to the canonical variables mentioned above. If is a minimum in the above problem and ( may then be considered as equal to ), then


The expression obtained for the Hamilton function has the same structure as in the classical calculus of variations. According to the Pontryagin maximum principle, the Euler equations for the optimal control problem may be written using a Hamilton function as follows:

The optimal control for each should constitute a maximum of the Hamiltonian:


[1] G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947)
[2] L.S. Pontryagin, V.G. Boltayanskii, R.V. Gamkrelidze, E.F. Mishchenko, "The mathematical theory of optimal processes" , Wiley (1962) (Translated from Russian)



[a1] H. Goldstein, "Classical mechanics" , Addison-Wesley (1950)
[a2] P.E. Crouch, A.J. van der Schaft, "Variational Hamiltonian control systems" , Lect. notes in control and inform. science , 101 , Springer (1987)
[a3] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
How to Cite This Entry:
Hamilton function. I.B. Vapnyarskii (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098