Dynamics and Control Technology Forum → *** Dynamics and Control Theory and Applications *** → Nonlinearity and Modern Mathematics → [Repost] Nonlinear Concept, Nature and Philosophical Significance
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- 释 非线 -
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Some impressive words:
To date, the concept of nonlinearity, nonlinear nature, has not clear, complete understanding, and has not fully explored its philosophical significance.
Linearity: From two angles associated with each other, one: The superposition principle is established; its second: the functional relationship between physical variables is a straight line, and the change rate between the variables is constant.
Nonlinearity: One: φ and ψ are coupled; two, as an equivalent, representation, the primary incremental relationship between physical variables is asymmetrical in the defined domain of variables. It can be said that this symmetrical shortage is the most basic embodiment of nonlinear relationship, and is also the root cause of nonlinear system complexity. For different objects, one of two of its two or two representations have their own convenience.
Contacts on linearity and nonlinearity:
Some non-linear problems can be used to convert them into several linear problems in a linear approximation method, which is widely used in each subject and is equivalent to a method.
It is not only possible from nonlinear perspective, not only possible, but sometimes it is necessary.
About linearity and nonlinear
Essentially difference:
Nonlinear and linear can be transformed with each other by mathematical transformation, in mathematical links, but at the same perspective, the same level, the same reference line, nonlinear and linear is essentially different.
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Mathematics, linear function relationship is a straight line, not linear function relationship, non-straight line, including various curves, folding lines, discontinuous lines, etc .; linear equations satisfy the principle of superposition, nonlinear equations do not satisfy superposition principle; linear equation is easy to find out Solution, and nonlinear equations generally cannot obtain analytical solutions.
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Physically, near linear problems (it is not what we said by nonlinear issues) can be used in linear approximation methods to find a certain accuracy solution, which is based on specific issues to accurately solve certain linear problems, and superimpose them. Get up, you can get a good approximate solution. However, for nonlinear problems, the linear approximation method will fail due to the problem of small parameters divergence and convergence, especially for high-speed motion, strong interaction, long dynamic behavior, etc., linear The method will completely powerless. These limitations of linear approximation methods result in irreplaceable nonlinear methods, and only nonlinear methods can only be used in nonlinear ways that cannot be treated with linear methods. Linear approximation methods are not often effective. This is not just methodological issue. It is also natural concept. Naturally, the existing variations and qualitative changes, [5] In the case, natural circles should experience jumping or turning, which is linear.
It is emphasized that linear or non-linear proposals are relative to physical variables, that is, only the relationship between physical variables is determined whether it is non-linear, rather than physical variables cannot be Become a criterion for nonlinearity.
A determined system, generally have two nature of linear and non-linearity: First, in a given nonlinear system, its nonlinear nature determines whether its balance construct or means that the stabilization mechanism is present, and the existence . Secondly, the linear nature of the system determines the law of small vibration of the system on its balance point (stabilizing structure), that is, the linearity of the system in the vicinity of the stabilizer. It is more important to be more interested in compared to nonlinear mathematical expression.
Nonlinear - one of the most important physical mechanisms, it is to be interacting.
