Master formula sheet · Classical Mechanics

Coordinates, constraints, and virtual work

Symbols follow the course notation. A formula's conditions are part of the result. In particular, distinguish fixed-time virtual variations from actual time evolution, and distinguish remaining forces from forces already included in a potential.

Formula	Meaning	Symbols	Conditions
$f_\alpha(\mathbf r_1,\ldots,\mathbf r_N,t)=0$	Holonomic equality constraint	$\alpha$ labels constraints	A regular independent set is needed for simple counting
$n=3N-k$	Configuration degrees of freedom	$N$ particles, $k$ independent equalities	Three-dimensional point particles; local regular constraints
$\mathbf r_a=\mathbf r_a(q,t)$	Coordinate transformation	$a$ particle, $q_i$ independent coordinates	A valid local coordinate patch
$\mathbf v_a=\sum_i\partial_{q_i}\mathbf r_a\dot q_i+\partial_t\mathbf r_a$	Actual velocity	Last term is support transport	Smooth transformation
$\delta\mathbf r_a=\sum_i\partial_{q_i}\mathbf r_a\delta q_i$	Virtual displacement	$\delta t=0$	Holonomic coordinates; admissible variation
$Q_i=\sum_a\mathbf F_a\cdot\partial_{q_i}\mathbf r_a$	Generalized force	Forces appropriate to the stated work	In Units 2–3, $\mathbf F_a$ here denotes remaining forces
$\delta W=\sum_iQ_i\delta q_i$	Virtual work	$Q_i\delta q_i$ has energy units	Same chosen force set
$\sum_a\mathbf R_a\cdot\delta\mathbf r_a=0$	Ideality of constraints	$\mathbf R_a$ reactions	Zero total virtual work for all admissible variations
$\sum_a\mathbf F_a\cdot\delta\mathbf r_a=0$	Virtual-work equilibrium	Applied forces only	Static equilibrium, ideal constraints; unilateral qualifications apply
$\sum_a(\mathbf F_a-m_a\mathbf a_a)\cdot\delta\mathbf r_a=0$	D’Alembert principle	$\mathbf a_a$ physical acceleration	Inertial frame, constant masses, ideal constraints

Lagrangian mechanics

Formula	Meaning	Symbols	Conditions
$L=T-V$	Ordinary mechanical Lagrangian	$T$ kinetic, $V$ potential energy	Velocity-independent position potential in the main treatment
$dL_{\dot q_i}/dt-L_{q_i}=Q_i$	Lagrange equation	Subscripts denote partial derivatives	Independent holonomic coordinates, ideal eliminated reactions
$p_i=L_{\dot q_i}$	Conjugate momentum	$p_i$ belongs to $q_i$	Defined by the chosen $L$
$L_{q_i}=0,\ Q_i=0\Rightarrow\dot p_i=0$	Cyclic-coordinate conservation	$q_i$ may still vary	The remaining force must vanish
$T=\tfrac12\sum_{ij}M_{ij}\dot q_i\dot q_j+\sum_ib_i\dot q_i+c$	General kinetic-energy structure	$M_{ij}$ inertia matrix, $b_i,c$ transport terms	Ordinary particle kinetic energy and smooth transformations
$M_{ij}=\sum_am_a\partial_{q_i}\mathbf r_a\cdot\partial_{q_j}\mathbf r_a$	Symmetric inertia coefficients	$M_{ij}=M_{ji}$	Positive definite in regular independent coordinates with positive masses
$\sum_i\dot q_iT_{\dot q_i}=2T_2+T_1$	Euler homogeneity identity	$T_j$ has velocity degree $j$	Coefficients held fixed in velocity derivatives
$dT_{\dot q_i}/dt-T_{q_i}=\sum_am_a\mathbf a_a\cdot\partial_{q_i}\mathbf r_a$	Inertial projection identity	Same coordinates as the transformation	Constant masses, smooth mixed derivatives
$\mathcal E=\sum_i\dot q_ip_i-L=T_2-T_0+V$	Lagrangian energy function	$\mathcal E$ need not be laboratory energy	Last equality assumes $L=T-V(q,t)$
$\dot{\mathcal E}=\sum_iQ_i\dot q_i-L_t$	Energy-function balance	$L_t$ is explicit-time derivative	Along solutions
$\mathcal R=\tfrac12\sum_{ij}c_{ij}\dot q_i\dot q_j$	Rayleigh function	Symmetric damping matrix	Linear viscous damping; nonnegative matrix for dissipation
$Q_i^{(d)}=-\mathcal R_{\dot q_i},\quad P_d=-2\mathcal R$	Damping force and power	$P_d$ dissipative power	Quadratic $\mathcal R$ ; stationary coordinates for full physical power

Hamiltonian mechanics

Formula	Meaning	Symbols	Conditions
$H(q,p,t)=\sum_ip_i\dot q_i-L$	Legendre transformation	Eliminate all dynamical $\dot q_i$	Nonsingular velocity Hessian
$\dot q_i=H_{p_i},\quad\dot p_i=-H_{q_i}$	Canonical Hamilton equations	Other phase variables held fixed	No remaining generalized forces
$\dot q_i=H_{p_i},\quad\dot p_i=-H_{q_i}+Q_i$	Forced Hamilton equations	$Q_i$ expressed in $q,p,t$	Same regular underlying $L$
$\dot H=H_t+\sum_iQ_i\dot q_i$	Hamiltonian balance	Explicit-time plus force terms	Along forced motion
$H=T+V$	Mechanical-energy identification	Velocity replaced by momentum	Sufficient: stationary transformations, $T=T_2$ , $V$ velocity-independent
$H_{q_i}=0,\ Q_i=0\Rightarrow\dot p_i=0$	Hamiltonian cyclicity	Conserved conjugate momentum	Unforced in that coordinate
$H=p_r^2/(2\mu)+p_\theta^2/(2\mu r^2)+V(r)$	Polar central-force Hamiltonian	$p_r=\mu\dot r$ , $p_\theta=\mu r^2\dot\theta$	Regular region $r>0$

Central forces and radial motion

Formula	Meaning	Symbols	Conditions
$\mathbf F=F(r)\hat{\mathbf r}=-V'(r)\hat{\mathbf r}$	Central interaction	$F$ signed radial magnitude	Time-independent radial position force
$\mu=m_1m_2/(m_1+m_2)$	Reduced mass	$m_1,m_2$ body masses	Isolated two-body relative motion
$\mathbf L=\mathbf r\times\mathbf p,\quad\dot{\mathbf L}=0$	Angular momentum	$\mathbf p=\mu\dot{\mathbf r}$	Radial force; constant mass
$\ell=\mu r^2\dot\theta,\quad\dot S=\ell/(2\mu)$	Angular momentum and area rate	Choose positive orbital orientation	Nonradial motion; area magnitude
$a_r=\ddot r-r\dot\theta^2,\quad a_\theta=r\ddot\theta+2\dot r\dot\theta$	Polar acceleration components	Radial and tangential basis	Plane polar coordinates
$V_{\rm eff}=V+\ell^2/(2\mu r^2)$	Effective radial potential	Angular momentum fixed	Central conservative problem
$E=\mu\dot r^2/2+V_{\rm eff}$	Radial energy equation	$E$ relative energy	Allowed radii satisfy $E\geq V_{\rm eff}$
$V'_{\rm eff}(r_c)=0$	Circular radius	$r_c$ constant radius	Also $E=V_{\rm eff}(r_c)$
$\omega_r^2=V''_{\rm eff}(r_c)/\mu$	Small radial oscillation frequency squared	$\omega_r$ local frequency	Positive: linear stability; negative: instability; zero: inconclusive
$u''+u=-\mu F(1/u)/(\ell^2u^2)$	Binet orbit equation	$u=1/r$ , prime $d/d\theta$	$\ell\ne0$ , $r>0$
$E=\ell^2(u'^2+u^2)/(2\mu)+V(1/u)$	Energy in orbit variables	Same $u$ convention	Same nonradial assumptions
$\omega_r^2/\Omega^2=3+r_cV''/V'$	Near-circular frequency ratio	$\Omega=\ell/(\mu r_c^2)$	Stable circle; derivatives at $r_c$

Inverse-square orbits

In this table $k>0$ , $V=-k/r$ , $V(\infty)=0$ , and $\ell>0$ . Italic $p$ denotes semi-latus rectum; bold $\mathbf p$ denotes momentum.

Formula	Meaning	Symbols	Conditions
$r=p/(1+e\cos\theta)$	Conic orbit	$\theta=0$ toward periapsis	Positive denominator on the physical branch
$p=\ell^2/(\mu k)$	Semi-latus rectum	Length $p$	Attractive inverse-square orbit
$e^2=1+2E\ell^2/(\mu k^2)$	Energy–eccentricity relation	$e\geq0$	Excludes degenerate radial classification
$r_p=p/(1+e)$	Periapsis radius	$r_p$ minimum radius	Nonradial conic
$r_a=p/(1-e)$	Apoapsis radius	$r_a$ maximum radius	Ellipse $0\leq e<1$
$p=a(1-e^2),\quad b=a\sqrt{1-e^2}$	Ellipse geometry	$a,b$ semiaxes	Ellipse
$E=-k/(2a)$	Elliptic relative energy	$a$ relative semimajor axis	Bound orbit
$v^2=(k/\mu)(2/r-1/a)$	Elliptic vis-viva	$v$ relative speed	For hyperbola with positive $a_h$ , replace $-1/a$ by $+1/a_h$
$v_c=\sqrt{k/(\mu r)}$	Circular relative speed	Radius $r$	Exact circle
$v_{\rm esc}=\sqrt{2k/(\mu r)}$	Escape threshold speed	Zero energy at infinity	Ideal inverse-square model; collision restrictions separate
$\tau^2=4\pi^2\mu a^3/k$	Kepler third law	$\tau$ period	Relative ellipse
$\tau^2=4\pi^2a^3/[G(m_1+m_2)]$	Gravitational form	$k=Gm_1m_2$	Isolated Newtonian pair
$\mathbf A=\mathbf p\times\mathbf L-\mu k\hat{\mathbf r}$	Runge–Lenz vector	Fixed periapsis direction when nonzero	Exact attractive inverse-square force
$\mathbf A\cdot\mathbf L=0,\quad A^2=\mu^2k^2+2\mu E\ell^2$	LRL constraints	$A=\|\mathbf A\|$	Same model
$\mathbf e=\mathbf A/(\mu k)$	Eccentricity vector	Magnitude $e$ , periapsis direction	Direction undefined for a circle
$\tan(\chi/2)=k/(\mu b_{\rm imp}v_\infty^2)$	Attractive deflection angle	Impact parameter and incoming speed	Hyperbolic encounter, $b_{\rm imp}>0$

The formula sheet is a reference after studying the chapters. The derivation index links each important argument to its assumptions and worked development.