- Introduction
- Euler Equation
- Functions, Functionals and Neighborhoods
- More Complex Problems
- Constrained Variational Problems
- Closure
- References
- Problems
Generally, there are two procedures used for solving variational problems that have constraints. These are the methods of direct substitution and Lagrange multipliers. In the method of direct substitution, the constraint equation is substituted into the integrand; and the problem is converted into an unconstrained problem as was done in Chapter II. In the method of Lagrange multipliers, the Lagrangian function is formed; and the unconstrained problem is solved using the appropriate forms of the Euler or Euler-Poisson equation. However, in some cases the Lagrange multiplier is a function of the independent variables and is not a constant. This is an added complication that was not encountered in Chapter II.
To illustrate the method of Lagrange multipliers, the simplest case with one algebraic equation will be used. The extension to more complicated cases are the same as that for analytical methods: The Lagrangian function is formed as shown below.
The Lagrange multiplier l is a function of the independent variable, x, and the unconstrained Euler equation is solved as given below. along with the constraint equation G(x, y) = 0. There is a Lagrange multiplier for each constraint equation when the Lagrangian function is formed. A derivation of the Lagrange multiplier method is given by Forray(1), and the following example illustrates the technique.
The classic example to illustrate this procedure is the problem of finding the path of a unit mass particle on a sphere from point (0,0,1) to point (0,0,-1) in time T, which minimizes the integral of the kinetic energy of the particle. The integral to be minimized and the constraint to be satisfied are: The Langrangian function is:
There are three optimal functions to be determined and the corresponding three Euler equations are: Performing the partial differentiation of
L, recognizing that [(x') These can be integrated after some manipulations to give:
which is the equation of a plane through
the center of the sphere. The intersection of this plane and the sphere
is a great circle which is the optimal path. It can be shown that the
minimum kinetic energy is p
Isoperimetric problems(1) are ones where an integral is to be optimized subject to a constraint which is another integral having a specified value. This name came from the famous problem of Dido of finding the closed curve of given perimeter for which the area is a maximum. For the Euler equation the problem can be stated as: where J is a known constant. To solve this problem the Lagrangian function L(x, y, y') is formed as shown below,
and the following unconstrained Euler equation is solved along with the constraint equation. For integral equation constraints, the Lagrange multiplier l is a constant; and each constraint has a Lagrange multiplier when forming the Lagrangian function. The following example illustrates the use of Lagrange multipliers with an integral constraint. It is the classic problem of Dido mentioned previously.
Determine the shape of the curve of length J that encloses the maximum area. The integral to be maximized and the integral constraint are as follows: The Lagrangian function is:
The two Euler equations are: Performing the differentiation and substituting into the Euler equations give: The two equations above can be integrated once to obtain the following results: Squaring both sides and adding the two equations gives the following:
which is the equation of a circle. Thus a circle encloses the maximum area for a given length curve.
To illustrate the method of Lagrange multipliers for differential equation constraints, a simple case will be used. Extensions to more detailed cases are the same as for the two previous types of constraints. The problem is as follows: As was done previously the Lagrangian function is formed as follows:
Then the Lagrange function is used in the Euler equation. In this case the Lagrange multiplier l(x) is a function of the independent variable. This procedure is illustrated in the following example which was given by Beveridge and Schechter(10). Also, they extend these results to obtain Pontryagin's maximum principle for constraints placed on the range of the dependent and independent variables.
The following problem to minimize I[y The Lagrangian function is:
Using equation (8-60) the two Euler equations
for y
The solutions for y where the constants of integration c The previous examples were designed to illustrate the particular extension of the calculus of variations and were essentially simple mathematics problems with no industrial application associated with them. However, the following example was designed to illustrate the application of the calculus of variations to a process, and it employs unsteady material and energy balance equations to determine the optimum way to control the flow rate to an agitated tank. Although the example is relatively simple, it illustrates economic model and process constraints for a dynamic system; and an optimal control function is developed.
An agitated tank contains W pounds of water
at 32°F.
It is desired to raise the temperature of the water in the tank to 104°F
in (2.0) An unsteady-state energy balance on the
water in the tank at time t gives the following equation relating the
temperatures T For water the heat capacity, C The calculus of variations problem can now be formulated as: with T Two optimal functions are determined, and the solution of two Euler equations are required using equation (8-60). The Lagrangian function is:
and the Euler equations are: The results of performing the differentiation are: Substituting into the Euler equations gives the following set of equations. The third equation is the constraint, and
these equations are solved for T
With the boundary conditions of T where 9.91 = 72/(e The solutions for the optimal functions,
T
Some of the important results from the chapter
are summarized in an abbreviated form in Table
8-1. First, a set of Euler equations is shown in the table to be solved
when the integrand contains several optimal functions and their first
derivatives. Corresponding boundary conditions are required on each of
these Euler equations which are second order ordinary differential equations.
Next in the table is the integral that has higher order derivatives in
the integrand. For this case the Euler-Poisson equation has to be solved,
and it is of order 2m, where m is the order of the highest derivative
in the integrand. Also appropriate boundary conditions on y and its derivatives
at x When the optimal function involves more than one independent variable a partial differential equation has to be solved, and the table shows the case for two independent variables, a second order partial differential equation. Equation (8-57) gives the comparable equation for n independent variables. However, the results given in the table and the chapter are only for an optimal function with first partial derivatives in the integrand. Results comparable to the Euler-Poisson equation with higher order derivatives are available in Weinstock(3). When constraints are involved the Lagrange function is formed as shown in the table. This gives an unconstrained problem that can be solved by the Euler and/or Euler Poisson equation, along with the constraint equations. The purpose of the chapter was to give some of the key results of the calculus of variations, and to emphasize the similarities and differences between finding an optimal function and an optimal point. Consequently, it was necessary to select the methods given here from some equally important methods that were omitted. Two of these are the concept of a variation and the use of the second variation for the sufficient conditions to determine if the function was actually a maximum or minimum. These are discussed by Courant and Hilbert(5) along with the problem of the existence of a solution. Also, most texts discuss the moving (or natural) boundary problem where one or both of the limits on the integral to be optimized can be a function of the independent variable. This leads to extensions of the Brachistochrone problem and Forray's discussion(1) is recommended. With the background of this chapter, extension to Hamilton's principle follows, and is typically the next topic presented on the subject. Also, this material leads to extensions that include Pontryagin's maximum principle, Sturm-Liouville problems, and application in optics, dynamics of particles, vibrations, elasticity, and quantum mechanics. The calculus of variations can be used to solve transport phenomena problems, i.e., obtain solutions to the partial differential equations representing the conservation of mass, momentum and energy of a system. In this approach the partial differential equations are converted to the corresponding integral to be optimized from the calculus of variations. Then approximate methods of integration are used to find the minimum of the integral, and this yields the concentration, temperature and/or velocity profiles required for the solution of the original differential equations. This approach is described by Schechter(4) in some detail. Again the purpose of the chapter was to introduce the topic of finding the optimal function. The references at the end of the chapter are recommended for further information; they include the texts by Fan(12) and Fan and Wang(13) on the maximum principle and Kirk(14) among others (7,15,16) on optimal control.
1. Forray, M. J., 2. Ewing, G. M. 3. Weinstock, Robt. 4. Schechter, R. S. 5. Courant, R. and D. Hilbert, 6. Sagan, Hans 7. M.M. Denn, 8. Wylie, C. R. and L. C. Barrett, 9. Burley, D. M., 10. Beveridge, G. S. G. and R. S. Schechter,
11. Fan, L.T., E.S. Lee, and L.E. Erickson,
12. L.T. Fan, 13. L.T. Fan and C.S. Wang, 14. Kirk, D.E. 15. Miele, A. , 16. Connors, M. M. and D. Teichroew, _{
top
} |