Lecture on Classical Mechanics

Introduction

• Classical mechanics is the basis of all physics.
• It describes the motion of objects (particles, mechanical systems, etc.).
• Basic principles include conservation of energy and momentum.

Simplest Systems and Laws of Nature

• Stroboscopic World: Time evaluated in discrete intervals.
• Simple systems imagined with primitive laws.

Example: Coin (Heads or Tails)

• States: Heads or Tails.
• Adding laws/rules to evolve states over time.

Laws of Nature for Coin

1. Law 1: Heads stays Heads, Tails stays Tails.
2. Law 2: Heads becomes Tails, Tails becomes Heads.

• Both laws are deterministic (outcome predictable if current state is known).
• Deterministic Laws: Allow consistent prediction of future states.
• Phase Space: Abstract space representing all possible states of a system.

Generalizing to More States

• Example: Die (6 sides):
  • Various laws possible (e.g., cyclic motion among states).
  • Graph representation of laws with arrows indicating state transitions.

Non-allowable Laws

• Laws that are irreversible (e.g., multiple arrows leading into a state) are not allowed in classical mechanics.
• Principles of Classical Mechanics: Every state must have a unique in and out path, maintaining determinism and reversibility.

Conservation Laws and Phase Space

• Conservation Law Example: Phase space with disconnected cycles preserving specific state characteristics.
• Information Conservation: Fundamental to classical physics (knowing current state provides knowledge of past and future states).

Continuous Time Systems

• Particle Motion:

  • Not enough to know position; velocity also needed (2D phase space: position + velocity).
  • State defined by both position and velocity.

• Newton's Equations:

  • Second order (involving accelerations) versus first order (involving velocities).

Equations of Motion

• Example of phony equation: f = m * velocity.
• Real equation: f = m * acceleration (Newton's Second Law).
• Implication: Position and velocity both needed to predict future state.

Conclusion

• Classical mechanics relies on deterministic laws with conservation of information.
• Equations of motion and phase space are central concepts.
• Key Takeaway: Phase space must include all necessary pieces of information to determine future states.

Questions and Clarifications

• Examples of needing multiple previously known states (heads/tails) to determine the next state.
• Transitional rules ensuring deterministic and reversible systems.
• Discussed experimental facts limiting classical mechanics to second order differential equations.