Lagrangian Lie Subalgebroids Generating Dynamics for Second-Order Mechanical Systems on Lie Algebroids

Abrunheiro, Lígia; Colombo, Leonardo

doi:10.1007/s00009-018-1108-x

Cited by 5 publications

(21 citation statements)

References 30 publications

(65 reference statements)

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“…(i) r is an algebroidal relation, (ii) (a) For any section s 1 ∈ Sec M 1 (E 1 ) and I = L, R, the vector fields ρ 2I (r(s 1 )) and ρ 1I (s 1 ) are r-related and (b) for any sections s 1 , s 1…”

Section: Description Of (Lie) Algebroids In Terms Of Vbcsmentioning

confidence: 99%

“…Our construction was motivated by the procedure of reduction of a higher tangent bundle of a Lie groupoid. In fact, as we shall see shortly, these objects provide natural examples of (Lie) higher algebroids in the sense of Definition 4. which can be defined directly as κ [1] [27,Proposition 4.6]):…”

Section: Prolongations Of An Al Algebroidmentioning

confidence: 99%

“…Moreover, prolongations of Lie algebroids are Lie higher algebroids. On T 2 E consider induced coordinates (x a , y i ,ẋ b ,ẏ j ,ẍ c ,ÿ k ), and coordinates x a , y i , y j, (1) ,ẋ b ,ẏ k ,ẏ l,(1) on TE [2] . The inductive formula (4.10) allows to calculate the expression for κ [2] : T 2 E → TE [2] (recall a general local form of a second-order algebroid given in Remark 4.2) which reads as κ [2] :…”

Section: Prolongations Of An Al Algebroidmentioning

confidence: 99%

“…Observe that the equation forẍ i can be obtained from the formulaẋ a = Q a i (x)y i by a formal derivation, assuming that (y i )˙= y i, (1) . In a similar manner, formulaẏ i, (1) can be derived from the equationẏ i =ẏ i + Q i jk (x)y j y k . Thus it is instructive to think about κ [2] as of first-order differential consequences of κ = κ [1] .…”

Section: Prolongations Of An Al Algebroidmentioning

confidence: 99%

“…In a similar manner, formulaẏ i, (1) can be derived from the equationẏ i =ẏ i + Q i jk (x)y j y k . Thus it is instructive to think about κ [2] as of first-order differential consequences of κ = κ [1] . We will use this property later in Section 5.2.…”

Section: Prolongations Of An Al Algebroidmentioning

confidence: 99%

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Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms

Rotkiewicz

2018

SIGMA

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We introduce the concept of a higher algebroid, generalizing the notions of an algebroid and a higher tangent bundle. Our ideas are based on a description of (Lie) algebroids as vector bundle comorphisms -differential relations of a special kind. In our approach higher algebroids are vector bundle comorphism between graded-linear bundles satisfying natural axioms. We provide natural examples and discuss applications in geometric mechanics.

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Section: Description Of (Lie) Algebroids In Terms Of Vbcsmentioning

confidence: 99%

Section: Prolongations Of An Al Algebroidmentioning

confidence: 99%

Section: Prolongations Of An Al Algebroidmentioning

confidence: 99%

Section: Prolongations Of An Al Algebroidmentioning

confidence: 99%

Section: Prolongations Of An Al Algebroidmentioning

confidence: 99%

See 3 more Smart Citations

Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms

Rotkiewicz

2018

SIGMA

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Euler–Lagrange–Herglotz equations on Lie algebroids

Anahory Simoes,

Colombo,

de León

et al. 2023

Anal.Math.Phys.

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We introduce Euler–Lagrange–Herglotz equations on Lie algebroids. The methodology is to extend the Jacobi structure from $$TQ\times \mathbb {R}$$ T Q × R and $$T^{*}Q \times \mathbb {R}$$ T ∗ Q × R to $$A\times \mathbb {R}$$ A × R and $$A^{*}\times \mathbb {R}$$ A ∗ × R , respectively, where A is a Lie algebroid and $$A^{*}$$ A ∗ carries the associated Poisson structure. We see that $$A^*\times \mathbb {R}$$ A ∗ × R possesses a natural Jacobi structure from where we are able to model dissipative mechanical systems on Lie algebroids, generalizing previous models on $$TQ\times \mathbb {R}$$ T Q × R and introducing new ones as for instance for reduced systems on Lie algebras, semidirect products (action Lie algebroids) and Atiyah bundles.

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