Abstract
The Bloch-Crouch formulation of LC-circuit dynamics is seen to be an implicitly defined Hamiltonian system on a particular manifold. A particular basis independent Dirac structure is shown to be equivalent to the hybrid input-output representation of the Dirac structure used by Bloch and Crouch thereby allowing circuit dynamics to be written in a basis independent fashion.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Blankenstein, G., van der Schaft, A.J.: Symmetry and reduction in implicit generalized Hamiltonian systems. Rep. Math. Phys. 47, 57–100 (2001)
Bloch, A., Crouch, P.: Representations of Dirac structures on vector spaces and nonlinear L-C circuits. Proc. Sympos. Pure Math. 64, 103–117 (1999)
Chua, L., Desoer, C., Kuh, E.: Linear and Nonlinear Circuits. McGraw-Hill, Boston (1987)
Courant, T.J.: Dirac manifolds. Trans. Amer. Math. Soc. 319, 631–661 (1990)
Dalsmo, M., van der Schaft, A.J.: On representations and integrability of mathematical structures in energy conserving physical systems. SIAM J. Control Optim. 37, 54–91 (1998)
Dorfman, I.: Dirac structures of integrable evolution equations. Phys. Lett. A 125, 240–246 (1987)
Fortney, J.P.: Dirac structures in pseudo-gradient systems with an emphasis on electrical networks. IEEE Trans. Circuits Syst. I. Regul Pap. 57 (2010)
Maschke, B.M., van der Schaft, A.J.: An intrinsic Hamiltonian formulation of the dynamics of L-C circuits. IEEE Trans. Circuits Syst. I. Regul Pap. 42, 73–82 (1995)
Smale, S.: On the mathematical foundations of electrical circuit theory. J. Differ. Geom. 7, 193–210 (1972)
van der Schaft, A.J.: Implicit Hamiltonian systems with symmetry. Rep. Math. Phys. 41, 203–221 (1998)
Yoshimura, H., Marsden, J.E.: Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems. J. Geom. Phys. 57, 133–156 (2006)
Yoshimura, H., Marsden, J.E.: Dirac structures in Lagrangian mechanics Part II: Variational structures. J. Geom. Phys. 57, 209–250 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this paper
Cite this paper
Fortney, J.P. (2017). Basis Independence of Implicitly Defined Hamiltonian Circuit Dynamics. In: Abualrub, T., Jarrah, A., Kallel, S., Sulieman, H. (eds) Mathematics Across Contemporary Sciences. AUS-ICMS 2015. Springer Proceedings in Mathematics & Statistics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-46310-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-46310-0_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-46309-4
Online ISBN: 978-3-319-46310-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)