Unit 4 · Review · Classical Mechanics

From centrality to conic motion

The central-force unit separates general consequences of radial direction from the special consequences of inverse-square magnitude. Zero torque gives constant angular momentum, a fixed orbital plane for nonzero angular momentum, and constant areal velocity. Time-independent radial dependence gives a potential and conserved energy. Fixing angular momentum produces an effective radial problem.

\mathbf L=\mathbf r\times\mu\dot{\mathbf r},\quad \dot{\mathbf L}=0,\quad \dot S=\frac\ell{2\mu},\quad V_{\rm eff}=V+\frac{\ell^2}{2\mu r^2},

(P.1)

E=\frac12\mu\dot r^2+V_{\rm eff},\qquad u=\frac1r,\quad \frac{d^2u}{d\theta^2}+u=-\frac{\mu F(1/u)}{\ell^2u^2}.

(P.2)

The reciprocal-radius symbol in the last equation is $u$ throughout the course. The equation assumes $\ell\ne0$ ; radial motion is treated directly from its energy equation.

Proofs and their assumptions

Development	Assumptions to state	Location
Torque, angular momentum, plane, area	Radial force; nonzero angular momentum for a unique plane	16
Potential and energy	Force depends on radius only, no explicit time	16
Two-body reduction	Isolated pair, constant masses, interaction depends on separation	16
Effective potential and turning radii	Fixed angular momentum, time-independent potential	17
Binet equation	Nonzero angular momentum, positive radius	18
Circular stability	Small radial disturbance at fixed angular momentum	17–19
Bertrand theorem	Statement and local argument; no claim of a complete necessity proof	19
Conics and energy–eccentricity	Attractive inverse-square force, potential zero at infinity	20
Kepler laws	Bound relative two-body ellipse; second law is more general	21
Runge–Lenz conservation and geometry	Exact inverse-square force and consistent vector convention	22

Orbital constants

For inverse-square attraction,

p=\frac{\ell^2}{\mu k},\quad r=\frac p{1+e\cos\theta},\quad e^2=1+\frac{2E\ell^2}{\mu k^2},\quad E=-\frac k{2a}\ \text{for an ellipse},

(P.3)

\tau^2=\frac{4\pi^2\mu a^3}{k},\quad \mathbf A=\mathbf p\times\mathbf L-\mu k\hat{\mathbf r},\quad \mathbf A\cdot\mathbf L=0,\quad A=\mu ke.

(P.4)

Italic $p$ without a subscript is semi-latus rectum here, and bold $\mathbf p$ is relative linear momentum. $\ell$ is the magnitude of the angular-momentum vector, whereas $L$ without boldface earlier denoted the Lagrangian.

Questions connecting the unit

Explain why an orbit can be bounded without being closed, and why a circle can be unstable. Reconstruct Binet's equation from the polar radial equation. Derive the conic equation both by solving Binet's equation and by taking a dot product with the Runge–Lenz vector. Derive Kepler's third law without assuming constant orbital speed. These connected arguments are more useful than memorizing isolated orbit formulas.

The separate solutions cover the exercises in all seven chapters. The master formula sheet retains the conditions under which each orbital expression may be used.