Graph representation of laws with arrows indicating state transitions.

Various laws possible (e.g., cyclic motion among states).

Phase Space: Abstract space representing all possible states of a system.

Deterministic Laws: Allow consistent prediction of future states.

Both laws are deterministic (outcome predictable if current state is known).

Law 2: Heads becomes Tails, Tails becomes Heads.

Law 1: Heads stays Heads, Tails stays Tails.

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

State defined by both position and velocity.

Not enough to know position; velocity also needed (2D phase space: position + velocity).

Example: Die (6 sides):

Implication: Position and velocity both needed to predict future state.

Real equation: f = m * acceleration (Newton's Second Law).

Example of phony equation: f = m * velocity.

Newton's Equations:

Particle Motion:

Principles of Classical Mechanics: Every state must have a unique in and out path, maintaining determinism and reversibility.

Laws that are irreversible (e.g., multiple arrows leading into a state) are not allowed in classical mechanics.

Generalizing to More States

Laws of Nature for Coin

Discussed experimental facts limiting classical mechanics to second order differential equations.

Transitional rules ensuring deterministic and reversible systems.

Examples of needing multiple previously known states (heads/tails) to determine the next state.

Key Takeaway: Phase space must include all necessary pieces of information to determine future states.

Equations of motion and phase space are central concepts.

Classical mechanics relies on deterministic laws with conservation of information.

Equations of Motion

Continuous Time Systems

Non-allowable Laws

Example: Coin (Heads or Tails)

Basic principles include conservation of energy and momentum.

It describes the motion of objects (particles, mechanical systems, etc.).

Classical mechanics is the basis of all physics.

Questions and Clarifications

Conclusion

Conservation Laws and Phase Space

Simplest Systems and Laws of Nature

Introduction

Lecture on Classical Mechanics