Introduction to Classical Mechanics

Key Points

• Mechanics is the basis for all of physics.
  • Describes motion of objects, particles, and mechanical systems.
  • Framework of all physics based on principles of classical mechanics.
  • Conservation of energy and momentum are central.

Simple Systems and Evolution of States

• Simplified systems to understand principles general to nature.
• Imaginary stroboscopic world: time evolves in intervals (e.g., 1-second beats).
• System configurations: heads or tails.
• Phase Space: Describes all possible states of a system.
  • Phase space for coin: two states (heads and tails).
  • Laws of nature determine how systems evolve from one interval to the next.
  • Deterministic laws: Knowing one state predicts future states.

Examples of Deterministic Laws

1. Consistency of State: Stays the same (heads -> heads).
2. Alternating States: Heads goes to tails and vice versa.

Generalization to Multiple States

• Example: Die with six states.
  • Variety of laws: sequential, cyclic, complex configurations.
  • Deterministic and reversible laws.

Non-allowable Laws in Classical Physics

1. Irreversible Laws: Can’t determine past states from current state.
2. Non-deterministic Laws: Multiple possible next states from one state.

Conservation Laws and Information Conservation

• Configuration conservation signifies a conservation law.
• Information conservation: Knowledge of current state predicts both the past and future states.
• Key Principle: One incoming and one outgoing arrow per state in phase space.

Continuous Systems and Position-Velocity Relationships

• In continuous physics, knowing position is insufficient; velocity is needed.
• Phase space: Two axes (position and velocity).
• Newton’s equations of motion: F = ma (Mass times acceleration).
  • Second-order equations: Need position and velocity to predict future motion.
  • Illustrated with simple equations and derivatives.

Practical Determinism vs. Theoretical Determinism

• Practical limitations in predicting due to imprecision in real-world measurements.
• Chaotic systems: Sensitivity to initial conditions.

Summary

• Classical mechanics deals with deterministic systems where knowledge of current states can map future and past states.
• Configuration space: Set of all necessary states to determine subsequent motion in deterministic equations.
• Continuous systems require more detailed phase spaces.

Remember, classical mechanics principles help derive predictable outcomes but are bound by the precision of initial state measurements, especially for complex or chaotic systems.