Variation Formulations of Physical Laws: The Correct Formulation

In summary, J.N. Reddy states that while all sufficiently smooth fields can be represented by meaningful variational forms, there are physical phenomena that can only be adequately modeled mathematically in a variational setting. These phenomena, such as Maxwell's equations in matter, appear nonsensical when viewed locally due to the presence of discontinuities that prevent the application of the variational principle.
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Source of Question
J.N. Reddy states in Finite Element Method: "Variational forms of the laws of continuum physics may be the only natural and rigorously correct way to think of them. While all sufficiently smooth fields lead to meaningful variational forms, the converse is not true: There exists physical phenomena which can be adequately modeled mathematically only in a variational setting; they are nonsensical when viewed locally"

My Question
What are some examples of physical phenomena that can be adequately modeled mathematically only in a variational setting and why are they nonsensical when viewed locally?
 
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Some approaches are "mean field theories" which result in very smooth fields - Maxwell's equations in matter are an example. But when you look at the microscopic sources of the fields (atoms and molecules, etc) their appear to be discontinuities ... hence you cannot meet the usual requirements for differentiability so that you can carry out the variational principle.
 

What is the concept of variation formulations in physical laws?

Variation formulations in physical laws refer to the mathematical technique of expressing physical laws in terms of variations or changes in a particular quantity. This allows for a more precise and concise representation of the law and makes it easier to solve complex problems.

Why is it important to have a correct formulation of physical laws?

A correct formulation of physical laws is crucial because it ensures that the laws accurately represent the behavior of the physical world. Incorrect formulations can lead to incorrect predictions and interpretations of phenomena, which can hinder scientific progress.

How do variation formulations improve upon traditional formulations of physical laws?

Variation formulations provide a more elegant and efficient way of representing physical laws. They allow for a more direct and intuitive understanding of the laws, as well as easier calculations and derivations. They also provide a unified framework for expressing different physical laws.

What is the role of variational principles in the development of variation formulations?

Variational principles play a key role in the development of variation formulations. They provide a fundamental framework for expressing physical laws in terms of variations, and allow for the derivation of equations of motion and other important relationships.

Can variation formulations be applied to all physical laws?

Yes, variation formulations can be applied to all physical laws as long as they can be expressed in terms of a suitable variational principle. This includes fundamental laws such as Newton's laws of motion, Maxwell's equations, and the laws of thermodynamics.

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