A New Framework For Minimizing Functionals

The Problem With Functionals

Introductory Concepts

Math Example 2 Calculus of Variations Recap & Geodesics Introduction

Math Example 1 Determining The Shortest Distance Between Two Points

Deriving The Euler

Math Example 2

Math Example 2

Physics Example 3

Physics Example 3

Physics Example 3

Quiz 1

Hamiltons Principle The Principle of Least Action

The Lagrangian and Lagranges Equations

Formal Proof

Formal Proof

Major Advantages of Lagrangian Mechanics & Coordinate Invariance

Lagrangian Mechanics Vs Newtonian Mechanics A Quick Comparison

A General Procedure for Lagrangian Mechanics Analyses

A Particle in Polar Coordinates Demonstrating Coordinate Invariance

Physics Example 1 Simple Harmonic Motion Revisited

Example 1

Example 2

Example 2

Physics Example 2 The Atwood Machine Revisited

Quiz 2 Solution Part 3

Quiz 2 Solution Part 2

Quiz 2 Solution Part 5

Obtaining Analytical Solutions to Verify Physics Simulations

Linearizing Equations of Motion to Obtain Analytical Solutions

The Finite Difference Method Forward Difference Approximations

The Finite Difference Method Part 2 Backward & Central Difference Approximation

Applying the Finite Difference Method to Simple Harmonic Motion

Simulating SHM

Simulating Simple Harmonic Motion Introduction & Setting Up the Simulation

Exploring Our First Physics Simulation Balancing Accuracy & Computation Time

Problem Introduction and Deriving the Systems Kinetic Energy

Determining the Double Pendulum Lagrangian

Deriving the Double Pendulums First Lagrange Equation

Deriving the Double Pendulums Second Lagrange Equation

Eliminating the Systems Non

The Art of Solving Differential Equations Setting up an Analytical Framework

Determining the Natural Frequencies for Each Normal Mode

Figuring Out How Each Normal Modes Amplitudes Are Related

Constructing the Full Generalized Solution

Programming the Simulation in MATLAB

Assessing the Physics Simulations Level of Accuracy

Using ODE45 to Computationally Solve Coupled 2nd Order Differential Equations

Exploring the Simulation Chaotic Motion Phase Portraits & Real

Problem Introduction and Geometry Analysis

Deriving the Systems Total Kinetic Energy

Determining The Systems Lagrangian

Utilizing the Lagrangian to Derive Each Cases Equation of Motion

Case 2 Why the Analytical Solution Consists of Transient & Stationary Solutions

Case 1 Eliminating Non

The Method of Undetermined Coefficients & Why it Needs to be Modified

Case 2 Determining the Steady

Case 2 Using Superposition to Yield the Full Analytical Solution

Physics Simulation Developing Algorithms for Rendering the Geoemtry

Case 2 Modifying the Physics Simulation & Verifying Results

Case 1 Programming the Finite Difference Algorithm & Verifying the Solution

Case 2 Exploring the Effects of Centripetal Acceleration & Gravity

Determing Each Rigid Bodys Center of Mass

Problem Introduction & Geometry Analysis

Determining Each Rigid Bodys Center of Mass

The Center of Mass Trick

The Difference Between Translational & Rotational Kinetic Energy

Determining the Systems Total Kinetic Energy Fail

Simplifying Our Kinetic Energy Derivation A More Efficient Approach

Determining the Equation of Motion

Deriving the Full Systems Lagrangian

Determining the Equation of Motion

Finding the Transient & Steady

Eliminating Non

Constructing the Full Analytical Solution

Developing an Algorithm for Rendering the Geometry and Motion

Developing the ODE45 Computational Algorithm for Our Simulation

Exploring the Requirements for Static Equilibrium

Verifying the Computational Solution & Assessing its Accuracy

Implementing Collision Detection Algorithms into the Simulation

Demonstrating How Non

Visually Experimenting with Each Case & Exploring Static Equilibrium Conditions

Concluding Remarks