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

VL - 57

SP - 133

EP - 156

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

IS - 1

ER -