Outline & notation · Classical Mechanics

From constraints to orbital motion

Begin with Newtonian mechanics. Constraints identify which configurations and instantaneous directions are allowed. Degrees of freedom count independent choices. Generalized coordinates label those choices. Their transformation derivatives give actual velocities and virtual displacements. Virtual work projects force balance onto admissible directions. D’Alembert's principle applies that projection to dynamics.

In Unit 2, kinetic and potential energy organize these equations into the Lagrangian method. Generalized momentum and cyclic coordinates reveal conservation laws. Unit 3 replaces velocities by momenta and describes evolution in phase space. Unit 4 applies these ideas to central-force motion, effective potentials, conic orbits, Kepler's laws, and the Runge–Lenz vector.

Chapter dependencies

Chapter	Prerequisites	Subject
Chapter 1	Position and force	Classify constraints without guessing from wording
Chapter 2	Chapter 1	Count freedoms and explain configuration space
Chapter 3	Chapters 1–2; trigonometry	Construct transformations that satisfy constraints
Chapter 4	Chapter 3; chain rule	Derive velocity and virtual-displacement relations
Chapter 5	Chapter 4; dot product and work	Find generalized forces and equilibrium
Chapter 6	Chapter 5; Newton's law	Derive constrained equations of motion

Notation

Symbol	Meaning
$N$	Number of particles
$a$ as a subscript	Particle label, as in $\mathbf r_a$
$a$ without a subscript	Often a fixed length, explicitly defined in each example
$k$	Number of independent constraints in counting problems; spring stiffness only when explicitly defined
$n$	Number of configuration degrees of freedom
$q_i,\dot q_i,\ddot q_i$	Generalized coordinate, velocity, and acceleration
$\mathbf r_a,\mathbf v_a,\mathbf a_a$	Physical position, velocity, and acceleration of particle $a$
$\mathbf F_a$	Applied force
$\mathbf R_a$	Constraint reaction
$Q_i$	Generalized applied force conjugate to $q_i$
$\delta\mathbf r_a,\delta q_i$	Virtual changes at fixed time
$T,V$	Kinetic and potential energy
$\mathcal T$	String tension, distinct from kinetic energy $T$
$L$	Lagrangian in Unit 2
$\mathbf L,\ell$	Angular momentum vector and magnitude in later units
$\mu$	Reduced mass in Unit 4; friction coefficient is $\mu_f$

Coordinates and forces are defined anew in each worked problem where necessary. A subscript is a label unless a sum is written; this course uses explicit summation signs. Angles in differentiation and small-angle approximations are in radians.

Scope of the derivations

Unit 1 includes the development of virtual work and D’Alembert's principle. Unit 2 includes conservation-law proofs and kinetic-energy properties. Unit 4 includes full central-force property proofs, the orbit equation, Kepler's laws, and Runge–Lenz-vector proofs.

Full derivations of Lagrange's and Hamilton's equations are labeled optional, exactly as required by the supplied syllabus.

Dependencies in Units 2–4

Chapters	Background used	Development
7–8	Virtual work, independent coordinates, derivatives	Lagrange equations and remaining forces
9–11	Energy functions and generalized momentum	Conservation proofs, kinetic-energy properties, coupled systems
12–15	Momentum and invertible kinetic-energy coefficients	Hamiltonian construction, phase space, energy interpretation
16–17	Vector products, polar coordinates, energy	Central-force properties, two-body reduction, radial motion
18–20	Angular momentum, differential equations, local expansion	Binet equation, stability, closure, conics
21–22	Ellipse geometry, area law, vector identities	Kepler laws and Runge–Lenz vector

Orbital notation

In Unit 4, $\mu$ is the particle or reduced mass, $k>0$ the inverse-square attraction constant, $\mathbf L$ angular momentum, and $\ell=|\mathbf L|$ . Bold $\mathbf p=\mu\dot{\mathbf r}$ is linear momentum; italic $p=\ell^2/(\mu k)$ in the conic equation is semi-latus rectum. The orbital period is $\tau$ , leaving $T$ for kinetic energy. Primes in Binet's equation mean angle derivatives; primes on $V(r)$ mean radial derivatives, with the argument stated.