Abstract
In this contribution we show, mainly based on an example, how Hamiltonian counterparts for partial differential equations that allow for a variational principle can be derived in a systematic manner. The main tool will be the appropriate use of the Lagrange multiplier technique, which allows us to obtain several well-known Hamiltonian formulations by using a common principle. The Mindlin plate will be used to visualize the presented approach
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Notes
- 1.
In the integrand \(\mathcal{L}\circ \phi _{\epsilon }(\sigma )\) it is implicitly assumed that one plugs in the derivative of ϕ ε (σ) as the derivative variables x i α are present in \(\mathcal{L}\). Using a more precise notation this could be written as \(\mathcal{L}\circ j^{1}(\phi _{\epsilon }(\sigma ))\), but to enhance the readability we will not indicate the jet-prolongations (i.e. j 1) when it follows from the context.
- 2.
\(\rfloor\) denotes the natural contraction, i.e., \(\partial _{X^{i}}\rfloor \varOmega =\mathrm{ (-1)^{i-1}d}X^{1} \wedge \ldots \wedge \hat{\mathrm{d}X}^{i} \wedge \ldots \wedge \mathrm{d}X^{p}\) where \(\hat{\mathrm{d}X}^{i}\) is omitted.
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Acknowledgements
This work has been partially supported by the Linz Center of Mechatronics (LCM) in the framework of the Austrian COMET-K2 programme.
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Schöberl, M., Schlacher, K. (2017). Variational Principles for Different Representations of Lagrangian and Hamiltonian Systems. In: Irschik, H., Belyaev, A., Krommer, M. (eds) Dynamics and Control of Advanced Structures and Machines. Springer, Cham. https://doi.org/10.1007/978-3-319-43080-5_7
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DOI: https://doi.org/10.1007/978-3-319-43080-5_7
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