# A Newtonian system mathematically always a trivial bundle?

• pellman

#### pellman

The phase space of a Newtonian system is a cotangent bundle, where the base space is the manifold M of the configuration space (the positions) and the typical fiber is the cotangent space T*M (the momenta). Is it always the case that this cotangent bundle is the trivial bundle M x TM?

No. The cotangent bundle is homeomorphic to the tangent bundle and tangent bundles are not always trivial. M cross TM is the manifold cross the tangent bundle, which is not what you want to say there.

pellman

## 1. What is a Newtonian system?

A Newtonian system is a physical system that follows the laws of classical mechanics, as described by Sir Isaac Newton's laws of motion.

## 2. What is a trivial bundle?

A trivial bundle is a mathematical concept that refers to a product space where each point in one space is paired with a unique point in another space. In other words, it is a set of points that can be described as a direct product of two spaces.

## 3. How are Newtonian systems and trivial bundles related?

In the context of classical mechanics, a Newtonian system can be mathematically represented as a trivial bundle, with the base space being the configuration space and the fiber space being the velocity space.

## 4. Why is a Newtonian system always a trivial bundle?

This is because the mathematical description of classical mechanics, including Newton's laws of motion, is based on the concept of a trivial bundle. Therefore, any system that follows these laws will also be represented as a trivial bundle.

## 5. Are there any exceptions to a Newtonian system being a trivial bundle?

Yes, there are some cases where a Newtonian system may not be represented as a trivial bundle, such as when dealing with non-inertial reference frames or systems with constraints. However, in most cases, a Newtonian system can be mathematically described as a trivial bundle.