On The Problem of Matched Lagrangian Dynamics on Lie Algebroids

Abstract

Lie algebroids are geometric constructions generalizing both tangent bundles and Lie algebras. Lagrangian dynamics is possible on Lie algebroid frameworks in its most general form. In this work, we obtain the joint behaviour of two mutually interacting Lagrangian systems in a geometric and an algebraic way. Here, the interaction is decoded into linear representations (actions) of two Lie algebroids onto each other. By this means, mutally interacting two Lie algebroids those satisfying some certain compatibility condition are matched, in other words, they are recast as trivially intersecting Lie subalgebroids of a single Lie algebroid. Then, Lagrangian dynamics is recast on the matched Lie algebroid. In this framework, the equations involve both the dynamics of constitutive subsystems and the action terms. Along with the theory, we provide several examples.

Keywords

• Lie groupoid
• Lie algebroid
• Euler-Lagrange equations

Lie Cebiroidleri Üzerindeki Lagrange Dinamiğinin Eşlenmesi Problemi Üzerine

Abstract

Lie cebiroidleri, bir anlamda tanjant demetini ve Lie cebiri yapısını beraber ihtiva eden ve fakat daha genel olan geometrik inşaalardır. Lagrange dinamiğinin en genel ifadesi Lie cebiroidleri üzerinde mümkündür. Bu makalede, karşılıklı (Lie cebiroidi üzerinde tanımlı) etki içindeki iki Lagrange dinamiğinin beraber davranışı, geometrik ve cebirsel bir yol ile elde edilecektir. Bu bakış açısı ile etkileşim Lie cebiroidlerinin birbirleri üzerine olan lineer temsilleri (etkileri) ifade edilecektir. Bu sayede, belirli uyumluluk şartını sağlayan karşılıklı etki içindeki iki Lie cebiroidinin eşlenmesi, diğer bir ifade ile tek bir Lie cebiroidi olarak yazılması sağlanacaktır. Sonrasında ise eşlenmiş Lie cebiroidi üzerinde Lagrange dinamiği yazılacaktır. Elde edilecek kollektif (eşlenmiş) hareket denklemleri, bireysel davranışların gözlemlenmesinin yanı sıra karşılıklı etki terimlerinin de belirlenmesine olanak verecektir. Çalışmamız esnasında bir çok örnek sunularak teorik tanımların daha net anlatımı yakalanmaya çalışılacaktır.

Keywords

• Lie Cebiroidi
• Lie Grupoidi
• Euler-Lagrange denklemleri

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