Concept connections · Classical Mechanics

Geometry, dynamics, and conserved quantities

The four units develop different parts of one argument. Constraints determine admissible configurations and virtual directions. Force projection then becomes an energy-based equation. Momentum and symmetry reduce the dynamics, and the central-force problem turns those reductions into a geometric orbit.

Course connections. Geometry feeds the energy formulation; central symmetry and the Hamiltonian reductions meet in orbital dynamics.

Connections to be able to explain

Earlier idea	Later use	What carries across
Independent generalized coordinates	Lagrange equations	One equation per independent virtual direction
Transformation derivatives	Kinetic-energy matrix	Geometry converts coordinate rates into physical velocity
Virtual work	Generalized non-conservative force	Work projection determines the force coefficient
D’Alembert principle	Lagrangian inertial terms	Acceleration projection is encoded by derivatives of $T$
Conjugate momentum	Hamiltonian construction	Invertible velocity–momentum relation changes state variables
Cyclic coordinate	Angular-momentum conservation	A missing polar angle gives constant $p_\theta$
Time independence	Energy balance	The energy function is conserved only with the required force conditions
Phase-space trajectory	Radial effective-potential motion	Position and momentum together distinguish inward and outward motion
Angular momentum	Planarity and equal areas	A fixed vector determines the plane and an area rate
Fixed angular momentum	Effective potential	Angular kinetic energy becomes a radial barrier term
Reciprocal radius	Inverse-square conic	The orbit equation becomes a forced harmonic equation
Conic geometry and area law	Kepler period law	Orbit area divided by area rate yields the period
Runge–Lenz vector	Fixed periapsis	An extra constant captures orientation inside the orbital plane

One pendulum, three descriptions

Newton's tangent equation is $ma\ddot\theta=-mg\sin\theta$ . The Lagrangian is $L=ma^2\dot\theta^2/2-mga(1-\cos\theta)$ . The Hamiltonian is $H=p_\theta^2/(2ma^2)+mga(1-\cos\theta)$ , with $p_\theta=ma^2\dot\theta$ . Each gives the same exact angular equation. What changes is how the geometric restriction and inertia are organized.

One orbit, three reductions

First, zero torque fixes the orbital plane. Second, angular-momentum conservation gives $\dot\theta=\ell/(\mu r^2)$ and the radial effective potential. Third, changing the independent variable from time to angle gives Binet's equation. Inverse-square attraction then produces a conic, while the area law restores its timing. The Runge–Lenz vector packages the same conic's eccentricity and orientation into one conserved object.