Classical Mechanics | Lecture 1 | Summary and Q&A

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December 15, 2011
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Stanford
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Classical Mechanics | Lecture 1

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Summary

This video introduces the concept of classical mechanics and explores the idea of rules and laws of motion. It discusses the specific laws for different types of systems and the general framework that encompasses these specific laws. The video also demonstrates simple examples of dynamical systems with their corresponding laws of motion, such as a coin and a die. It explains how these laws can be represented mathematically and emphasizes the deterministic and reversible nature of classical physics.

Questions & Answers

Q: What are the two types of questions in classical mechanics?

The first type of question is about the specific laws governing particular systems, such as a planet moving in the field of a heavy mass or an electrically charged particle moving in the field of a magnet. The second type of question concerns the general framework or rules that govern all the different specific laws in classical mechanics.

Q: What is a dynamical system?

A dynamical system is a system that undergoes changes over time. It can be described by its different states or configurations, and its evolution is governed by the laws of motion specific to that system.

Q: Can you give an example of a simple dynamical system?

One example is a coin on a table. The coin has two states - heads or tails - and its motion is governed by the law that it stays the same. If it starts with heads, it will remain heads indefinitely, and if it starts with tails, it will remain tails indefinitely.

Q: How can we represent the law of motion for the coin mathematically?

We can assign a variable, let's say Sigma, to represent the two states of a coin - 1 for heads and -1 for tails. The law of motion for the coin is that Sigma at time T+1 is equal to Sigma at time T, which means that the state of the coin remains unchanged over time.

Q: Are the laws of motion for a coin reversible?

Yes, the laws of motion for a coin are reversible because regardless of the initial state, the coin will return to that same state. For example, if it starts with heads, it will always go heads, heads, heads, and so on. If it starts with tails, it will always go tails, tails, tails, and so on.

Q: What is a die and how does its motion differ from that of a coin?

A die is a cube with six sides labeled with numbers. Unlike a coin, a die has more configurations or states. For example, a possible law for a die is that it cycles through the numbers - 1 goes to 2, 2 goes to 3, 3 goes to 4, and so on, until 6 goes back to 1. This creates a repeating cycle of states for the die.

Q: Are the laws of motion for a die reversible?

It depends on the specific law. Some laws of motion for a die are reversible, meaning that if you know the current state of the die, you can determine its previous state. However, there are other laws that are not reversible, where multiple states can lead to the same current state, making it impossible to determine the previous state with certainty.

Q: What is a conserved quantity?

A conserved quantity is a non-trivial quantity that remains constant over time in a system. For example, in a die with a single cycle, there is no conserved quantity because every state is visited. However, in a die with two cycles, there is a conserved quantity that can be assigned to each cycle, such as 0 and 1, and the system remains in that cycle indefinitely.

Q: Can dynamical systems have an infinite number of states?

Yes, dynamical systems can have an infinite number of states, even if they involve a single object. For example, a particle moving along an infinite line can be described by an infinite number of configurations based on the location of the particle.

Q: How is predictability affected by imperfect knowledge of initial conditions?

In classical mechanics, if the initial conditions are known precisely, the future evolution of a system can be predicted with certainty. However, if the initial conditions are not known exactly, there will be some ambiguity in predicting the future. The degree of predictability depends on the accuracy of the initial conditions and the complexity of the system itself.

Q: Are there laws of motion that are not allowed in classical mechanics?

Yes, there are laws of motion that are not allowed in classical mechanics. One example is a law that is completely predictive into the future but not retrodictive into the past. In other words, you can predict the future based on the current state, but you cannot determine the past state with certainty based on the current state. Such laws are considered irreversible and go against the rules of classical mechanics.

Takeaways

Classical mechanics is based on a set of rules or laws that describe the motion of objects. These laws can be specific to certain systems or part of a general framework that encompasses all different specific laws. In classical mechanics, laws of motion are deterministic, meaning that the future evolution of a system can be predicted with certainty given the initial conditions. Additionally, these laws are often reversible, allowing for both forward and backward prediction. However, there are laws that are not allowed in classical mechanics, such as those that only predict the future and cannot retrodict the past. These laws violate the principles of classical mechanics and are considered irreversible.

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