Abstract
In Chapter 17, Hamilton’s principle was proved by using Euler’s simple approach according to which a one-parameter family of curves is introduced that reduces the search for a minimum of a functional to that of a minimum of a function depending on a single variable. In this chapter, we present a deeper approach to variational calculus that shows how to extend to functionals many concepts of differential calculus relative to functions. Further, we formulate the problems of constrained stationary points of functionals and show how to obtain the momentum equation of continuum mechanics by variational principle.
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Notes
- 1.
When the functions of A verify the boundary conditions (27.5), then \(h(a)=h(b)=0\).
- 2.
To verify (27.35) it is important to notice that \(\partial L/\partial x_j\) denotes the total derivative of L with respect to \(x_j\) whereas \(L,_j\) denotes the partial derivative of L, if L explicitly depends on \(x_j\).
- 3.
We use the formula \(\displaystyle \frac{\partial J}{\partial F_{km}}=J(F^{-1})_{mk}\).
- 4.
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Romano, A., Marasco, A. (2018). Variational Calculus with Applications. In: Classical Mechanics with Mathematica®. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-77595-1_27
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DOI: https://doi.org/10.1007/978-3-319-77595-1_27
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Publisher Name: Birkhäuser, Cham
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