By Andre Mercier
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Extra resources for Analytical and canonical formalism in physics
Sample text
6 visualizes this process. The starting point is a real-world system S together with a question Q relating to S. A mathematical model (S, Q, M) then opens up the way into the ‘‘mathematical universe’’, where the problem can be solved using powerful mathematical methods. This leads to a problem solution in mathematical terms (A*), which is then translated into an answer A to the original question Q in the last step. 6 (Mathematical models as door opener) Translating originally nonmathematical problems into the language of mathematics, mathematical models virtually serve as a door opener toward the ‘‘mathematical universe’’ where powerful mathematical methods become applicable to originally nonmathematical problems.
6) is an example of a mathematical model in the form of an AE. 11, black box regression models of this kind are widely used for the modeling for example, of psychological, social, or economic systems (see Chapter 2 for more on regression models). 2 exemplifies the modeling of a biological/chemical system using ODEs (see Chapters 3 and 4 for more examples of DE models). 11a illustrate that mathematical models can be used to solve increasingly challenging problems as the model gradually turns from a Q Linear—nonlinear Analytical—numerical Autonomous—nonautonomous Continous—discrete Difference equations Differential equations Algebraic equations Integral equations (a) Fig.
1, a mathematical model is a triplet (S, Q, M) consisting of a system S, a question Q, and a set of mathematical statements M, and that the details of the system S that are represented in M depend on the question Q that is to be answered by the model. In this case, Q was asking for the time development of the plant biomass, and this can be sufficiently answered based on a model that represents the system S = ‘‘plant’’ as Sr = {r}. Generally one can say that the reduced system of a well-formulated mathematical model will consist of no more than exactly those properties of the original system that are important to answer the question Q that is being investigated.