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Given a manifold $M$, Arnold's "Mathematical Methods of Classical Mechanics" defines a Lagrangian system as a pair $(M,L)$ where $L$ is some smooth function on the tangent bundle $TM$. The function $L$ is called the Lagrangian. In the case when $M$ is a Riemannian manifold and a particle in $M$ is moving under some conservative force field, taking the Lagrangian to be the kinetic minus the potential energy we recover Newton's second law.

I know that one of the main advantages to the Euler-Lagrange equations over Newtons is the way in which they simplify constrained systems. I know another is the coordinate independence of the equations. However, in all applications I've seen the manifold is always Riemannian and the Lagrangian is always $K-U$.

My questions are:

  1. Why do we have this abstract definition of a Lagrangian system and of an abstract Lagrangian?

  2. What are some of the cases in which $L$ is not $K-U$ that gives interesting results?

  3. Or cases in which the manifold is not Riemannian?

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