Classical Mechanics

We begin with the Newtonian picture: particles, forces, and laws. We practice energy and momentum methods and learn when each is more efficient than summing forces. Then we translate problems into the Lagrangian approach, where dynamics come from a single scalar called the action and constraints are handled cleanly. Finally, we enter Hamiltonian mechanics and phase space, where symmetries and conserved quantities appear as geometry. You will see the same system solved three ways. Start simple, then grow the vocabulary as the problems demand it.

Newton’s second law relates acceleration to force: $F = m a$. The content is deeper than the compact form suggests. It assumes an inertial frame and a constant mass during the motion. Forces can be contact (springs, constraints, friction) or at-a-distance (gravity, electrostatics in appropriate regimes). The goal is to translate a physical situation into $F(x,v,t)$ and integrate to obtain motion; in one dimension, $m \, \ddot{x} = F(x,\dot{x},t)$.

Two shortcuts dominate in practice: the work–energy theorem and momentum methods. The work–energy theorem trades vectors for a scalar: the net work done equals the change in kinetic energy, $\Delta K = \int \mathbf F \cdot d\mathbf r$. It shines when forces are parallel to motion or when $ \mathbf F = -\nabla V$. Momentum methods remove details of internal forces and let you track the motion of a whole system under external impulses: $\mathbf P_{\text{final}} = \mathbf P_{\text{initial}} + \int \mathbf F_{\text{ext}} \, dt$. Use energy when forces are conservative and direction changes complicate the vector sum; use momentum when interactions are brief or internal and symmetry makes the vector story simple.

A small catalog helps. Springs: $F = -k x$. Low-speed drag: $F = -b v$. Gravity near Earth: $\mathbf F = - m g\, \hat{\mathbf y}$. Planetary gravity: $\mathbf F = - \dfrac{G M m}{r^{2}} \, \hat{\mathbf r}$. Normal forces enforce constraints by canceling motion into a surface; ideal normal forces do no work. Static friction adjusts up to a maximum; kinetic friction has near-constant magnitude opposite motion. Describe the allowable motion first (kinematics), then let unspecified forces enforce it (constraints).

The Lagrangian method begins with a single scalar function $L(q,\dot q,t)=T-V$, kinetic minus potential energy, in a set of generalized coordinates $q$ that encode only the true degrees of freedom. The equations of motion follow from the Euler–Lagrange equation:

Constraints reduce the number of independent variables. For a bead sliding on a rigid wire in space, a single parameter like the arc length along the curve may be the only coordinate you need. Forces of constraint never appear explicitly; the geometry handles them. Symmetries of $L$ imply conserved quantities: if $L$ has no explicit time dependence, the energy is conserved; if it has no dependence on a coordinate (a cyclic coordinate), the corresponding momentum $p_i=\partial L/\partial \dot q_i$ is conserved. These are early glimpses of Noether’s theorem.

Example: the simple pendulum of length $\ell$ with angle $\theta$. The kinetic energy is $T=\tfrac12 m \ell^{2}\dot{\theta}^{2}$, and potential energy is $V= m g \ell (1-\cos\theta)$. Thus $L=T-V$. The Euler–Lagrange equation gives $\ell\, \ddot{\theta} + g \sin\theta = 0$. In the small-angle limit, $\sin\theta \approx \theta$, we recover simple harmonic motion with period $2\pi\sqrt{\ell/g}$. The method needed no tension force; the constraint was built into the coordinate choice.

Hamiltonian mechanics lifts the description to phase space: coordinates $q$ and conjugate momenta $p=\partial L/\partial \dot q$. The Hamiltonian $H(q,p,t)$ is usually the total energy in familiar systems. Time evolution is given by Hamilton’s equations:

The Poisson bracket for functions $F(q,p)$ and $G(q,p)$ is $$ \{F,G\} = \sum_i \left(\frac{\partial F}{\partial q_i}\frac{\partial G}{\partial p_i} - \frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q_i}\right). $$ A quantity $C$ is conserved if $\{C,H\} + \partial C/\partial t = 0$. Canonical transformations preserve the bracket and let you simplify problems (e.g., action–angle variables in integrable systems). Liouville’s theorem states that Hamiltonian flow preserves phase-space volume.

Small oscillations. Expand the potential around equilibrium, keep quadratic terms, and diagonalize the mass and stiffness matrices to find normal modes. Central forces. Reduce to an effective one-dimensional radial problem with an effective potential $V_{\text{eff}}(r)=V(r)+\ell^2/(2 m r^2)$ that includes the angular momentum barrier. Driven/damped motion. Use linear response and complex impedance methods for steady-state amplitudes and phases. Constrained systems. Choose generalized coordinates so constraints are identities; if needed, add Lagrange multipliers to handle non-integrable constraints.

Noether in practice. If your Lagrangian is unchanged by a continuous transformation, there is a conserved quantity. Spatial translation $\Rightarrow$ linear momentum; rotation $\Rightarrow$ angular momentum; time translation $\Rightarrow$ energy. Write the transformation, compute the variation of $L$, and read off the current. When stuck, look for a hidden symmetry (e.g., the Laplace–Runge–Lenz vector in the Kepler problem) that explains special degeneracies and closed orbits.

Mass–spring–dashpot $(m,k,b)$. Newton: $m\ddot{x}+b\dot{x}+kx = F_0 \cos(\omega t)$. Solve via trial solution or complex impedance; resonance near $\omega \approx \sqrt{k/m}$ with width set by $b$. Lagrange: $L=\tfrac12 m\dot{x}^2-\tfrac12 k x^2$, include Rayleigh dissipation $R=\tfrac12 b\dot{x}^2$ to model damping; Euler–Lagrange with damping term yields the same ODE. Hamilton: define $p=m\dot{x}$, $H=p^2/2m + \tfrac12 k x^2$; for driving/damping treat nonconservative effects as external.

Central force orbit. Lagrange in polar coordinates gives conservation of angular momentum $\ell=m r^{2}\dot{\theta}$; reduce to radial motion in $V_{\text{eff}}(r)$. For inverse-square, bounded motion is an ellipse with one focus at the center. Hamilton’s equations show the closed orbits correspond to an extra conserved quantity beyond energy and angular momentum (the Runge–Lenz vector), revealing the problem’s integrability.

Symplectic geometry. Phase space is not just axes; it carries a two-form that Hamiltonian flow preserves. Symplectic integrators respect this structure numerically and dramatically reduce long-time drift in energy. Action–angle variables. For integrable systems, label tori by actions and evolve angles linearly in time—almost trivial motion in a curved coordinate system. From classical to quantum. Many quantum commutators mirror classical Poisson brackets; path integrals mirror actions. Learning the classical structures pays dividends later.

Ready to branch. For a deeper Newtonian toolkit, practice multi-body constraints and collision problems with momentum and energy together. For Lagrangian fluency, work small oscillation problems and bead-on-wire constraints in multiple coordinates. For Hamiltonian insight, sketch phase portraits, compute Poisson brackets, and try action–angle variables for the harmonic oscillator and the planar pendulum (below separatrix). Each lens is the same system told in a different grammar.