Analytical Mechanics is an advanced course that extends classical Newtonian mechanics using powerful mathematical formulations, such as Lagrangian and Hamiltonian mechanics. These frameworks provide a deeper understanding of motion and dynamics, making them essential for fields like physics, engineering, and applied mathematics.

Key Topics Covered:

1. Variational Principles & Lagrangian Mechanics

• Principle of Least Action
• Generalized Coordinates & Degrees of Freedom
• Lagrange’s Equations of Motion
• Applications to constrained systems and central force motion

2. Hamiltonian Mechanics & Canonical Formulation

• Legendre Transform and Hamilton’s Equations
• Phase Space and Conserved Quantities
• Poisson Brackets & Canonical Transformations
• Liouville’s Theorem and Symplectic Geometry

3. Rigid Body Dynamics

• Euler Angles and Rotational Kinematics
• Moment of Inertia Tensor and Principal Axes
• Euler’s Equations of Motion
• Gyroscopic Motion and Stability

4. Non-Inertial Reference Frames

• Rotating Coordinate Systems
• Coriolis and Centrifugal Forces
• Foucault Pendulum & Earth’s Rotation Effects

5. Small Oscillations & Normal Modes

• Stability of Equilibrium Points
• Coupled Oscillators & Eigenvalue Problems
• Normal Mode Analysis in Multi-Particle Systems

6. Introduction to Hamilton-Jacobi Theory & Advanced Topics

• Hamilton’s Principal Function
• Action-Angle Variables
• Introduction to Classical Field Theory