Abstract
The calculus of variations provides a well-defined methodology for formulating differential equation models from problems which seek solutions that optimise some property of interest. The derivation of Euler–Lagrange equations from variational principles will be given in detail. Applications to problems in mechanics will be discussed. The elementary analytical framework will be extended to more advanced problems with free boundaries and constrained optimisation problems, including isoperimetric, holonomic and optimal control problems.
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Notes
- 1.
The behaviour at a local maximum of J(y) follows similarly.
- 2.
The \(O(\varepsilon ^n)\) order symbol will be defined precisely in Chap. 6, but for the current context we use this to refer to the \(\varepsilon \rightarrow 0\) limit of the remainder of the terms in the series with coefficients \(\varepsilon ^N\) for \(N\ge n\).
- 3.
Showing that a solution given by a critical point is a local maximum or minimum involves evaluating the second variation at \(y_*(x)\).
- 4.
In this simplified statement of this result, we are assuming that g(x) is smooth and hence has no discontinuities.
- 5.
The du Bois Reymond lemma (2.11) is closely related to this lemma, choosing \(h(x)=1\) on arbitrary sub-intervals, and otherwise \(h=0\).
- 6.
For the area and arclength examples, the functionals become unbounded for large amplitude solutions and hence there are no local maxima.
- 7.
Given the lack of elementary approaches if the fundamental lemma can not be used, it is tempting to call them necessary boundary conditions.
- 8.
Note that in a different context, there is another definition of the Hamiltonian having \(H=-\mathscr {H}\). Despite the difference in sign conventions, both are called Hamiltonians, see Exercise 3.7.
- 9.
Sometimes called the Pontryagin minimum principle, depending on the choice of sign convention used for H versus \(\mathscr {H}\), recall page 71.
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Witelski, T., Bowen, M. (2015). Variational Principles. In: Methods of Mathematical Modelling. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-23042-9_3
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DOI: https://doi.org/10.1007/978-3-319-23042-9_3
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