Something about configuration manifolds in classical mechanics

In summary, the conversation discusses a mechanical system consisting of a rotating circle with a mass M and a sliding bead with a mass m. The position of the bead is determined by the angle theta, while the rotation of the circle is controlled by the angle psi. The Lagrangian of the system is defined on either the tangent bundle T(T^2) if M>0 or on the tangent bundle T(S^2) if M=0. The set T is defined as T^n = S^1 x...x S^1, with T^1 being equal to S^1, but S^1 x S^1 does not equal S^2.
  • #1
wrobel
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I think it could be interesting.

Consider a mechanical system

Screenshot from 2021-02-25 09-30-39.png
A circle of mass M can rotate about the vertical axis. The angle of rotation is coordinated by the angle ##\psi##. A bead of mass m>0 can slide along this circle. The position of the bead relative the circle is given by the angle ##\theta##.

It is interesting to note that if M>0 the Lagrangian of this system is defined on the tangent bundle ##T\mathbb{T}^2##. But if M=0 then the Lagrangian is defined on the tangent bundle ##TS^2##.
 
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  • #2
how is define the set ##\mathbb{T}##?
 
  • #3
##\mathbb{T}^n=S^1\times\ldots\times S^1,\quad \mathbb{T}^1=S^1##
but
##S^1\times S^1\ne S^2##!
 
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Related to Something about configuration manifolds in classical mechanics

1. What are configuration manifolds in classical mechanics?

Configuration manifolds in classical mechanics refer to the set of all possible positions and orientations that a physical system can take. It is a mathematical concept used to describe the state of a system in terms of its position and orientation.

2. How are configuration manifolds used in classical mechanics?

Configuration manifolds are used in classical mechanics to describe the state of a physical system and to study its motion and dynamics. They provide a mathematical framework for understanding the behavior of physical systems and making predictions about their future state.

3. What is the difference between configuration manifolds and phase space in classical mechanics?

Configuration manifolds and phase space are closely related concepts in classical mechanics. While configuration manifolds describe the state of a system in terms of its position and orientation, phase space also takes into account the system's momentum. In other words, phase space is a higher-dimensional version of configuration space.

4. Can configuration manifolds be applied to all physical systems?

Yes, configuration manifolds can be applied to all physical systems, as long as they can be described in terms of position and orientation. This includes systems such as particles, rigid bodies, and even complex systems like fluids and gases.

5. How do configuration manifolds relate to Hamiltonian mechanics?

Configuration manifolds play a fundamental role in Hamiltonian mechanics, which is a mathematical framework for describing the dynamics of physical systems. In Hamiltonian mechanics, the configuration manifold is used to define the system's Hamiltonian, which is a function that describes the energy of the system in terms of its position and momentum.

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