Reduction theory and the Lagrange–Routh equations

Marsden, Jerrold E.; Raţiu, Tudor S.; Scheurle, Jürgen

doi:10.1063/1.533317

Cited by 120 publications

(179 citation statements)

References 110 publications

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“…͑IV.6͒ of Ref. 10 differs from our reconstruction equation ͑expressed in terms of I ͒ by having an additional term on the right-hand side involving the mechanical connection for the G -action . This arises because the authors start with a more general class of curves on M than we do.…”

Section: Simple Mechanical Systemsmentioning

confidence: 99%

“…10 and called there the mechanical connection for the G -action. This is a connection on the principal fiber bundle M → M / G , i.e., a G -invariant splitting of the short exact sequence…”

Section: Simple Mechanical Systemsmentioning

confidence: 99%

“…10, the theorem that states the reduced equations obtained by following a variational approach to Routh's procedure. We leave it to the reader to verify that the second equation, in its form ͑III.37͒, is, in fact,…”

Section: Simple Mechanical Systemsmentioning

confidence: 99%

“…10, we explain how solutions of the Euler-Lagrange equations with a fixed momentum can be reconstructed from solutions of the reduced equations. The method relies on the availability of a principal connection on an appropriate principal fiber bundle.…”

Section: Introductionmentioning

confidence: 99%

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Routh’s procedure for non-Abelian symmetry groups

Crampin

Mestdag

2008

Journal of Mathematical Physics

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We extend Routh's reduction procedure to an arbitrary Lagrangian system ͑that is, one whose Lagrangian is not necessarily the difference of kinetic and potential energies͒ with a symmetry group which is not necessarily Abelian.To do so, we analyze the restriction of the Euler-Lagrange field to a level set of momentum in velocity phase space. We present a new method of analysis based on the use of quasivelocities. We discuss the reconstruction of solutions of the full EulerLagrange equations from those of the reduced equations.

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Section: Simple Mechanical Systemsmentioning

confidence: 99%

Section: Simple Mechanical Systemsmentioning

confidence: 99%

Section: Simple Mechanical Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Routh’s procedure for non-Abelian symmetry groups

Crampin

Mestdag

2008

Journal of Mathematical Physics

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“…Our approach to systematic reduction of dimension is based on the framework of geometric mechanics [1,2] and hyperspherical coordinates [3]. In the hyperspherical coordinates, one can express the internal motions of an n-atom system in terms of the three gyration radii and the (3n − 9) hyperangular variables.…”

Section: Collective Coordinates and Dynamical Reaction Barriermentioning

confidence: 99%

Collective coordinates and the mechanism for conformational transitions of complex molecules

Yanao

Koon

Marsden

et al. 2007

Proc Appl Math and Mech

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Reduction of dimensionality is crucial for the deeper understanding of the mechanism for large-amplitude conformational transitions of complex molecules. By taking up a six-atom cluster as an illustrative example, we present a general methodology to understand conformational transitions of molecules in terms of the low-dimensional dynamics of molecular gyration radii. The dynamics of gyration radii is generally governed by the interplay between the ordinary potential force and a dynamical force called the internal centrifugal force. We show that the internal centrifugal force can be more important than the original potential barrier and gives rise to a new dynamical barrier that truly dominates the conformational transitions of the system. This kind of dynamical effect should be crucially important in a wide class of molecular reaction dynamics.

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A connection‐theoretic approach to reduction of second‐order dynamical systems with symmetry

Mestdag

Crampin

2007

Proc Appl Math and Mech

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We deal with reduction of Lagrangian systems that are invariant under the action of the symmetry group. Unlike the bulk of the literature we do not rely on methods coming from the calculus of variations. Our method is based on the geometrical analysis of regular Lagrangian systems, where solutions of the Euler-Lagrange equations are interpreted as integral curves of the associated second-order differential equation field. In particular, we explain so-called Lagrange-Poincaré reduction [1] and Routh reduction [3] from the viewpoint of that vector field. The Euler-Lagrange equations in an adapted frameThis contribution is based on references [2] and [4]. Let (x α ) be coordinates on a manifold M and (x α , u α ) coordinates on its tangent manifold T M . We will assume that the Lagrangian L(x, u) is regular, i.e. that the matrix of functions (∂ 2 L/∂u α ∂u β ) is everywhere non-singular. Then, the Euler-Lagrange equations may be written explicitly in the form of a set of second-order differential equationsẍ α = f α (x,ẋ) and its solutions can be interpreted as integral curves of the second-order differentialThe Euler-Lagrange equations may then be expressed in the form Γ (∂L/∂u α ) − ∂L/∂x α = 0. These equations, together with the assumption that it is a second-order differential equation field, determine the vector field Γ.There are two canonical lifts of a vector field Z = Z α ∂/∂x α on M to a vector field on T M . The flow of the so-called complete or tangent lift Z C = Z α ∂/∂x α + u β ∂Z α /∂x β ∂/∂u α consists of the tangent maps of the flow of Z. The vertical lift Z V = Z α ∂/∂u α is tangent to the fibres of τ : T M → M and on T m M it coincides with Z(m). We use these two concepts to cast the Euler-Lagrange field in terms of a non-coordinate basis. If {Z α } is a basis of vector fields on M , then it can easily be verified that an equivalent expression for the Euler-Lagrange equations is Γ(Z V α (L)) − Z C α (L) = 0. We will assume throughout that the system is invariant under a proper, free (left) action ψ M : G × M → M . Let X i be the G-invariant horizontal lifts of a coordinate basis of vector fields on M/G (horizontal w.r.t. a principal connection on π M : M → M/G). We will also need two sets {Ẽ a } or {Ê a } of vector fields on M , associated to a basis {E a } of the Lie algebra G. The vector fieldsẼ a of the 'moving' basis are the fundamental vector fields corresponding to the action. If we set locally π M : U × G → U (and ψ M g (x, h) = (x, gh)), the 'body-fixed' basis consists of the vector fields defined bŷ E a : (x, g) → T ψ M g Ẽ a (x, e) . Note that the basis {X i ,Ê a } is invariant, but the basis {X i ,Ẽ a } is not. We can now express the Euler-Lagrange equations in either one of these two adapted frames. By doing so, we derive in the next sections both the Lagrange-Poincaré equations [1] and the Lagrange-Routh equations [3] in a relatively straightforward fashion. g(t)v H (t) is a reconstructed integral curve of Γ.

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Reduction theory and the Lagrange–Routh equations

Cited by 120 publications

References 110 publications

Routh’s procedure for non-Abelian symmetry groups

Routh’s procedure for non-Abelian symmetry groups

Collective coordinates and the mechanism for conformational transitions of complex molecules

A connection‐theoretic approach to reduction of second‐order dynamical systems with symmetry

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