Quote by Steven Strogatz: "Since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations."
Differential equations are used to describe change rather than absolute amounts.
The purpose of the lecture series is to provide a big picture view of differential equations and delve into specific examples.
Prerequisites
Assumes knowledge of basic calculus (derivatives and integrals).
Basic linear algebra may be needed later but not much beyond that.
Types of Differential Equations
Ordinary Differential Equations (ODEs): Involve functions with a single input (often time).
Partial Differential Equations (PDEs): Involves functions with multiple inputs.
Example given: temperature at every point in a solid body or velocity of a fluid.
Application in Physics
Motion described in terms of force, which determines acceleration (change).
Example: Throwing an object in the air affected by gravity.
Acceleration due to gravity: 9.8 m/s² (denoted as g).
Equation: y'' = -g (where y is the vertical position).
Solve by integrating to find velocity and position using initial conditions.
Complex Forces and Motion
More complex scenarios arise when forces depend on position (e.g., planetary motion).
The interplay of two variables leads to intricate differential equations.
Most common in physics: Second-order Differential Equations (highest derivative is second).
Example: The Pendulum
The angle θ changes as a function of time, demonstrating harmonic motion.
Period: 2π√(l/g) (l = length of pendulum).
Differential equation: θ'' = -g/l * sin(θ).
In real life, as the angle increases, the period deviates from simple harmonic motion.
Damping Term: To account for air resistance, modeled as -μθ'.
Solving Differential Equations
Difficult to find exact solutions for many equations (especially with damping).
Often, solutions require complex functions not easily interpreted.
Rather than finding analytic solutions, focus is on understanding behavior and making computations.
Phase Space and Visualization
Phase Space: Represents all possible states of a system (e.g., angle and angular velocity of a pendulum).
Vector Fields: Visual representation of how a system evolves.
Each point in phase space represents the state of the system.
Stability analysis involves examining fixed points in phase space (e.g., pendulum at rest).
Example of Phase Space in Personal Relationships
Strogatz’s quote applied to modeling dynamics of affection.
Affection dynamics can be modeled similarly to a pendulum's motion, reflecting attraction and repulsion.
Numerical Solutions
When exact solutions are infeasible, numerical simulations can approximate solutions using finite time steps.
Basic numerical method involves updating the state with small incremental steps based on the differential equation.
Challenges of Predicting Dynamics
Chaos theory reveals limits on predictability in systems (small changes lead to vastly different outcomes).
Example: Three-body problem is chaotic and sensitive to initial conditions.
Conclusion
While differential equations can be complex or chaotic, they provide a language to study dynamics in both physics and other areas.
Understanding these equations can help uncover patterns in complex systems.