TY - JOUR
T1 - Dirac structures in Lagrangian mechanics Part I
T2 - Implicit Lagrangian systems
AU - Yoshimura, Hiroaki
AU - Marsden, Jerrold E.
N1 - Funding Information:
The first author’s research was partially supported by JSPS Grant 16560216.
Funding Information:
The second author’s research was partially supported by NSF-ITR Grant ACI-0204932.
PY - 2006/12/31
Y1 - 2006/12/31
N2 - This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space Q makes use of Dirac structures on T* Q that are induced from a constraint distribution on Q as well as natural symplectomorphisms between the spaces T* T Q, T T* Q, and T* T* Q. Two illustrative examples are presented; the first is a nonholonomic system, namely a vertical disk rolling on a plane, and the second is an L-C circuit, a degenerate Lagrangian system with holonomic constraints.
AB - This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space Q makes use of Dirac structures on T* Q that are induced from a constraint distribution on Q as well as natural symplectomorphisms between the spaces T* T Q, T T* Q, and T* T* Q. Two illustrative examples are presented; the first is a nonholonomic system, namely a vertical disk rolling on a plane, and the second is an L-C circuit, a degenerate Lagrangian system with holonomic constraints.
KW - Dirac structures
KW - Implicit Lagrangian systems
KW - Nonholonomic systems
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U2 - 10.1016/j.geomphys.2006.02.009
DO - 10.1016/j.geomphys.2006.02.009
M3 - Article
AN - SCOPUS:33749141694
SN - 0393-0440
VL - 57
SP - 133
EP - 156
JO - Journal of Geometry and Physics
JF - Journal of Geometry and Physics
IS - 1
ER -