Classical Mechanics

1. Lagrangian Mechanics
   1. Principle of Least Action
      1. Definition and Explanation
         1. Action integral concept
         2. Connection to energy principles
      2. Historical Background
         1. Origins of the principle
         2. Contributions by key figures like Maupertuis, Euler, and Lagrange
      3. Examples in Physical Systems
         1. Vibrational systems
         2. Motion of particles
      4. Role in Modern Physics
         1. Quantum mechanics
         2. Field theories
   2. Lagrange’s Equations
      1. Formulation
         1. Derivation from the principle of least action
         2. Use of generalized coordinates
      2. Comparison with Newton’s Equations
         1. Advantages in systems with constraints
         2. Coordinate independence
      3. Types of Lagrange’s Equations
         1. Euler-Lagrange equations
         2. Lagrange’s equations for systems with non-conservative forces
      4. Applications
         1. Simple pendulum
         2. Double pendulum and chaotic systems
         3. Electric circuits and electromagnetic systems
      5. Computational Methods
         1. Numerical solutions of Lagrange's equations
         2. Use in computer simulations and modeling
      6. Complex Systems
         1. Multibody systems
         2. Systems with coupled degrees of freedom
   3. Applications and Advantages Over Newtonian Mechanics
      1. Applications in Different Coordinates
         1. Cartesian, polar, and spherical coordinates
         2. Configuration space
      2. Handling Constraints
         1. Holonomic and non-holonomic constraints
         2. Use of Lagrange multipliers
      3. Systems with Symmetries
         1. Conservation laws derivation
         2. Noether’s theorem and implications
      4. Quantum Mechanics
         1. Path integral formulation by Feynman
         2. Transition to Hamiltonian mechanics for quantum systems
      5. Theoretical Physics
         1. Use in formulating theories of gravity
         2. Application in gauge theories and particle physics
      6. Extended Systems
         1. Application in continuous systems
         2. Use in fluid dynamics and elasticity theory