A F.E.M and Hamilton's Principle (converting differential equations into integral equations)

AI Thread Summary
The discussion focuses on the conversion of differential equations into integral equations within the context of the Finite Element Method (FEM) and Hamilton's Principle. Both approaches utilize variational methods, with FEM applying the Principle of Virtual Work and Hamilton's Principle reformulating Newtonian Dynamics through the extremization of the Action of the Lagrangian. This conversion simplifies complex problems, making them more amenable to computational solutions. The conversation seeks to explore the philosophical implications of these transformations and their underlying similarities. Overall, the thread emphasizes the significance of understanding these mathematical processes in both theoretical and practical applications.
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Differential to Integral Equations
Hello

May I begin by saying I do not exactly know what I am asking, but here goes...

In the Finite Element Method (as used in Solid Mechanics), we convert the differential equations of continuum mechanics into integral form. Here, I am thinking of the more pragmatic Principle of Virtual Work, rather that exploiting the more mathematically sophisticated strong/weak formulations (but no matter on that detail)

In Hamilton's Principle, we reformulate Newtonian Dynamics into Analytical Dynamics, but extremizing the Action of the Lagrangian.

Now, in both cases, we convert differential equations into integral equations.

So something is happening here... this act of converting differential into integral. Through the haze of my confusion I can sort of see that the result is more easily addressed with computer programming

Could someone elaborate, perhaps a bit more philosophically, on what is happening when we do these things.

In one sense, both processes relate to variational methods, but is something going on here that these two approaches (sort of) resemble each other, in a way)?

Or am I thinking a bit silly?
 
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