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Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation [ l ] is defined as the linear part of the change in the functional, and the second variation [ m ] is defined as the quadratic part.
Biological constraints are factors which make populations resistant to evolutionary change. One proposed definition of constraint is "A property of a trait that, although possibly adaptive in the environment in which it originally evolved, acts to place limits on the production of new phenotypic variants."
For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be solved using variational calculus, and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational potential energy of the chain.
The result was derived using ideas from the classical calculus of variations. [6] After a slight perturbation of the optimal control, one considers the first-order term of a Taylor expansion with respect to the perturbation; sending the perturbation to zero leads to a variational inequality from which the maximum principle follows. [7]
In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum ( functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf .
In the calculus of variations, and are referred to as the Mayer term and the Lagrangian, respectively. Furthermore, it is noted that the path constraints are in general inequality constraints and thus may not be active (i.e., equal to zero) at the optimal solution. It is also noted that the optimal control problem as stated above may have ...
In mathematics, variational analysis is the combination and extension of methods from convex optimization and the classical calculus of variations to a more general theory. [1] This includes the more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized derivatives.
Developmental constraints are limitations on phenotypic variability (or absence of variation) caused by the inherent structure and dynamics of the developmental system. [1] Constraints are a bias against a certain ontogenetic trajectory, and consequently are thought to limit adaptive evolution. [12] [13]