Small Oscillations in Classical Mechanics - Goldstein

In summary, in the small oscillation problem of classical mechanics, the generalized force is zero (Eq. (6-1)) according to the Goldstein book. The suffix 'i' represents the particle number or coordinate, and the summation convention is used in Eq. (6-3) to simplify notation. Eq. (6-5) uses the mass of the system, m_{ij}, which varies with space and is expanded in terms of a Taylor series.
  • #1
saravanan13
56
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Dear friends,

In small oscillation problem of classical mechanics, the generalized force is zero (Eq. (6-1)) of the Goldstein book.
The suffix 'i' represent the the particle number or coordinate?
In Eq. (6-3) of the same problem the summation convention is used. what is the purpose of using it?
In Eq. (6-5), the mass of the system m_{ij} is used.
Whether it indicates that mass of the system varies with space?
Why the mass is expanded in terms of Taylor series?

Ref: Classical mechanics by Goldstein.
Thanks in well advance.
 
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  • #2
The suffix 'i' represents a particle number or coordinate, depending on the context. The summation convention is used to simplify the notation; it means that any variables used in the equation that occur twice should be added together. The mass of the system m_{ij} indicates that the mass of the system varies with respect to space, as the suffixes indicate. It is expanded in terms of a Taylor series to account for the variation of the mass with respect to the coordinates.
 

1. What are small oscillations in classical mechanics?

Small oscillations in classical mechanics refer to the motion of a system around a stable equilibrium point, where the displacement from the equilibrium is small. This type of motion can be described using linearized equations of motion and is often seen in simple harmonic motion.

2. What is the significance of small oscillations in classical mechanics?

Small oscillations are important in classical mechanics because they allow us to simplify the equations of motion and make predictions about the behavior of a system. They also provide a basis for understanding more complex systems and their behavior near equilibrium points.

3. How are small oscillations represented mathematically?

In classical mechanics, small oscillations are typically represented using the linearized equations of motion, which involve the first and second derivatives of the system's coordinates with respect to time. These equations can be solved to determine the system's motion and behavior.

4. What are some examples of systems that exhibit small oscillations?

Some common examples of systems that exhibit small oscillations include a mass-spring system, a pendulum, and a vibrating guitar string. These systems can be described using simple harmonic motion and are often used to model more complex systems in physics and engineering.

5. How do small oscillations differ from large oscillations?

The main difference between small and large oscillations is the amplitude of the motion. Small oscillations have a small amplitude and are typically described using linear equations, while large oscillations have a larger amplitude and may require more complex equations to describe their behavior.

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