Classical Mechanics

Classical mechanics is the notebook’s foundation for reasoning about motion. The subject begins with forces and trajectories, but its deeper structure appears through action, symmetry, conservation laws, and the geometry of phase space.

From force to action; from trajectories to structure.

Study Compass

Newtonian Dynamics Motion, force, momentum, energy, and the initial-value description of mechanical systems. Lagrangian Mechanics Generalized coordinates, constraints, stationary action, and the Euler-Lagrange equations. Hamiltonian Mechanics Phase space, canonical equations, Poisson brackets, and transformations of mechanical description. Oscillations and Continuous Systems Normal modes, coupled motion, waves, and the passage from particles to fields.

Newtonian Dynamics

The Newtonian formulation treats a mechanical state through positions and velocities, with forces determining its evolution:

m \ddot{q} = F (q, \dot{q}, t) .

The central questions are how to choose coordinates, identify constraints, and recognize the quantities conserved by a given force law.

Lagrangian Mechanics

For generalized coordinates $q^{i}$ , the dynamics can be encoded by the Lagrangian $L = T - V$ . Stationary action yields

\frac{d}{d t} \frac{\partial L}{\partial q ˙ ^{i}} - \frac{\partial L}{\partial q ^{i}} = 0.

This language makes constraints and symmetries more visible, and it prepares the transition from mechanics to field theory.

Hamiltonian Mechanics

The Hamiltonian formulation replaces configuration-space trajectories with flow on phase space. Its canonical equations are

\overset{q}{˙}^{i} = \frac{\partial H}{\partial p _{i}}, \overset{p}{˙}_{i} = - \frac{\partial H}{\partial q ^{i}} .

Poisson brackets, canonical transformations, and Hamilton-Jacobi theory expose the structural ideas that later reappear in statistical and quantum mechanics.

Oscillations and Continuous Systems

Small oscillations provide the bridge from finite mechanical systems to normal modes and waves. Continuous media then extend the variational framework from finitely many coordinates to fields distributed over space.

Working Emphasis

The branch will keep derivations and assumptions visible: the choice of coordinates, the role of constraints, the origin of conserved quantities, and the relation between Newtonian, Lagrangian, and Hamiltonian descriptions.