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Copy file name to clipboardExpand all lines: lectures/cont_indir_calculus_of_variations.qmd
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As a consequence, $L_{y'}y' - L$ is constant along the optimal curve.
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The two new functions whose values are *preserved along the extremal* are so special that they deserve their own symbols and names:
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The two new functions whose values are *preserved along the extremal* (under the respective conditions) are so special that they deserve their own symbols and names:
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$$\boxed{
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p(x) \coloneqq L_{y'},}
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$$
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and you will see that the choice of the symbol $p$ is intentional becase this variable will be seen to play the role of *momentum*as you know it from physics, and
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and the choice of the symbol $p$ is intentional as this variable plays the role of *momentum*in physics (when the independent variable $x$ is time and $L$ is the difference between the kinetic and potential energies), and
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$$\boxed{
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H(x,y,y',p) \coloneqq py'-L}
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$$
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and you will see that the choice of the symbol $H$ is intentional becase this variable will be seen to play the role of *Hamiltonian* as you know it from physics.
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::: {.callout-warning}
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Unfortunately, we have to give a warning here, that while the above definitions are well accepted in the physics-related fields of science, most control theory books adopt a slightly different convention, which may be confusing. We will have more on that in a while (or see directly 3.4.4 in Liberzon's book).
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:::
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and the choice of the symbol $H$ is intentional becase this variable plays the role of *Hamiltonian* in physics.
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Let us now see how $y$ and $p$ develop as functions of $x$, and we will use Hamiltonian for that purpose. First, it is immediate from the definition of Hamiltonian that
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