Integral equations natural constraints

Property 1. In a theory based on the pair of fields (, 0) with action integral equal to (118), submitted to the duality constraint (119), both tensors Fap and Fap obey the Maxwell equations in empty space. As the duality constraint is naturally conserved in time, the same result is obtained if it is imposed just at t = 0. 

By the Lagrange multipliers technique the differential equations are integrated in the natural coordinates and the Lagrange multipliers are used to keep constant the constraints at each step of integration. 

The closed set of product density equations is given by Eqs. (7.4.7) through (7.4.9) in which if is any one of the set if, if2, if3, if4, if 5. The solution of the set of product density equations is, however, most efficiently done by recognizing a set of integral constraints that arise naturally. Taking... 

The steady-state version of the optimization problem is normally termed the linear programming (LP) or nonlinear programming (NLP) problem, depending on the nature of the objective function and constraints. In such problems, the objective is usually a differential function of the optimization variables. In contrast, in dynamic optimization, the objective function is usually a functional and frequently a time integral along the trajectory as illustrated in Equation 18.5. The initial conditions on the state variables have to be specified along with the final desired state ... 

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