Show that the center of mass is well defined, i.e., does not depend on the choice of the origin of reference for radius vectors. The momentum of a system is equal to the momentum of a particle lying at the center of mass of the system and having mass \(\sum m_{i}\). In fact, \(\left(\sum m_{i}\right) \mathbf{r}=\sum\left(m_{i} \mathbf{r}_{i}\right)\), from which it follows that \(\left(\sum m_{i}\right) \dot{\mathbf{r}}=\sum m_{i} \dot{\mathbf{r}}_{i}\). We can now formulate the theorem about momentum as a theorem about the motion of the center of mass.

Classical mechanics is a branch of physics that deals with the motion of objects and the forces that affect them. It is grounded in Newton's laws of motion, which describe how objects move in various situations. One of the key principles is that an object will remain at rest or in motion in a straight line unless acted upon by a force. This fundamental framework allows us to predict the behavior of physical systems, from celestial bodies to everyday objects on Earth.

Understanding the center of mass is crucial in classical mechanics because it represents the average position of the system's mass, making it easier to analyze the system's motion. For example, when an object spins or moves through space, its center of mass follows a smoother path that can be described using the principles of mechanics, simplifying complex movements into more manageable equations.

Momentum in physics represents the quantity of motion an object possesses, which is directly tied to both its mass and velocity. It's a vector quantity, meaning it has both magnitude and direction. The momentum of a system can be calculated by summing the momentum of all its parts. The concept is central to the analysis of collisions and interactions between objects because the total momentum before and after such events remains constant if no external forces act on the system, a principle known as the conservation of momentum.

When solving problems involving the motion of multiple objects, the momentum of the center of mass provides a simplified way to understand the system's overall motion. The equation \(\sum m_{i}\) \(\dot{\mathbf{r}}\) = \(\sum m_{i} \dot{\mathbf{r}}_{i}\) highlights this relationship by equating the momentum of the entire system to that of a single particle located at the center of mass, simplifying calculations.

Invariance under translation is a concept in physics stating that the fundamental laws of physics are the same, regardless of where they are applied in space. In simpler terms, if you take a physical system and move it to a different location without altering its state, its physical behavior will remain unchanged.

When we say that the center of mass is invariant under translation, we mean that no matter where you place the origin of your coordinate system, the center of mass of an object remains constant. This is essential when analyzing physical phenomena because it assures us that our calculations will hold true in any reference frame, eliminating potential biases introduced by our choice of where to begin measuring.

A reference frame is a perspective from which we observe and measure the physical properties of a system. It can be thought of as a coordinate system that we use to define the positions and movements of objects. Reference frames are fundamental in mechanics because they define the 'stage' on which all physical processes occur.

There are two types of reference frames: inertial and non-inertial. Inertial frames move at a constant velocity and are not subject to any external forces, making them ideal for applying Newton's laws. A non-inertial frame, on the other hand, is accelerating, which means objects within it appear to be influenced by fictitious forces. The concept of the center of mass being independent of the choice of the origin of a reference frame is a key feature that ensures consistency in physical laws, as shown in the exercise above. By using the center of mass and understanding how it behaves in different reference frames, we can more accurately predict the motion of complex systems.