Definition index

A–C

Apoapsis. A local maximum of distance from the force center on a bounded orbit. For a Kepler ellipse, $r_a=a(1+e)$.

Apsidal angle. An angle between specified successive radial extrema. The course states whether it means periapsis-to-periapsis or periapsis-to-apoapsis, which differ by a factor of two for a radial cycle.

Areal velocity. Area swept by the radius vector per unit time, $\dot S=\ell/(2\mu)$ for central-force motion.

Bertrand's theorem. The theorem identifying inverse-square attraction and isotropic linear attraction as the smooth central-force laws for which every bounded non-collision orbit is closed, within the stated stable-orbit setting.

Bilateral constraint. An equality restriction imposing both sides of an admissible displacement condition, such as a fixed-length rigid rod.

Binet equation. The orbit differential equation for $u=1/r$ as a function of polar angle, valid for nonzero angular momentum.

Bounded orbit. An orbit remaining inside a finite spatial region. It need not close.

Canonical variables. Generalized coordinates and their conjugate momenta $(q_i,p_i)$ in the Hamiltonian description.

Central force. In this course, a force $F(r)\hat{\mathbf r}$ whose direction is radial and whose signed magnitude depends only on distance from the center.

Closed orbit. A trajectory that repeats after a finite cycle; for regular bounded central motion this requires commensurate radial and angular cycles.

Configuration. A complete specification of the system's positions consistent with the constraints, without specifying velocities.

Configuration space. The set of all admissible configurations; locally it is described by independent generalized coordinates.

Conjugate momentum. The quantity $p_i=\partial L/\partial\dot q_i$ belonging to coordinate $q_i$; it need not be ordinary linear momentum.

Conservative force. A position force expressible as the negative gradient of a time-independent potential, with path-independent work. An explicitly time-dependent potential requires a separate energy-exchange qualification.

Constraint. A restriction on admissible configurations or motions, imposed by the physical model.

Constraint reaction. A force enforcing a constraint. It may do actual work even when its virtual work vanishes for a moving ideal constraint.

Cyclic coordinate. A coordinate absent explicitly from $L$ or the corresponding regular $H$. Its conjugate momentum is conserved when no remaining generalized force acts in that coordinate.

D–H

D’Alembert's principle. The virtual-work projection of applied force minus momentum rate onto all admissible virtual displacements, with ideal reactions eliminated.

Degree of freedom. An independent local positional choice needed to specify a configuration. Velocity constraints and coordinate singularities require care in simple counting rules.

Eccentric anomaly. An auxiliary angle parametrizing an ellipse by $x=a(\cos\psi-e)$, $y=b\sin\psi$ relative to one focus; distinct from the true polar angle.

Eccentricity. A nonnegative parameter measuring conic shape: zero for a circle, between zero and one for an ellipse, one for a parabola, and above one for a hyperbola.

Eccentricity vector. The Runge–Lenz vector divided by $\mu k$; its magnitude is $e$ and its direction points toward periapsis when nonzero.

Effective potential. The radial function $V_{\rm eff}=V+\ell^2/(2\mu r^2)$ obtained by fixing angular momentum and incorporating angular kinetic energy into a one-dimensional radial description.

Energy function. The Lagrangian combination $\mathcal E=\sum_i\dot q_ip_i-L$. After velocity inversion it becomes $H(q,p,t)$; its equality with physical energy requires further conditions.

Equilibrium. A state with no acceleration and no motion for a static mechanical equilibrium; in phase space a fixed point has all state derivatives zero. Steady relative motions should be specified separately.

Generalized coordinate. A variable in an independent coordinate description of admissible configurations; it may be a distance, angle, or another suitable quantity.

Generalized force. The coefficient of a generalized virtual displacement in virtual work, $\delta W=\sum_iQ_i\delta q_i$.

Generalized velocity. The time derivative $\dot q_i$ of a generalized coordinate. Its units depend on the coordinate.

Hamiltonian. The regular Legendre transform $H=\sum_ip_i\dot q_i-L$, expressed entirely in coordinates, momenta, and time.

Holonomic constraint. A constraint expressible as a relation among positions and possibly time; an integrable velocity relation can also represent one.

Homogeneous quadratic kinetic energy. Kinetic energy satisfying $T(q,\lambda\dot q,t)=\lambda^2T(q,\dot q,t)$ at fixed coordinates and time. Stationary ordinary coordinate transformations give this form.

I–P

Ideal constraint. A constraint whose reactions perform zero total virtual work for every admissible virtual displacement. “Ideal” is a force property, not the same classification as “holonomic.”

Ignorable coordinate. Another name for a cyclic coordinate.

Impact parameter. The perpendicular distance between the force center and the incoming asymptotic straight-line trajectory in a scattering problem.

Inertia matrix. The symmetric matrix multiplying quadratic velocity terms in kinetic energy, also the velocity Hessian for an ordinary mechanical Lagrangian.

Lagrangian. For ordinary mechanical systems treated here, $L=T-V$, the scalar function used in Lagrange's equations. It is not the mechanical-energy sum.

Legendre transformation. The replacement of velocities by their conjugate momenta through $p_i=L_{\dot q_i}$ and $H=\sum_ip_i\dot q_i-L$, assuming local invertibility.

Non-holonomic constraint. A motion restriction not reducible to position-and-time equalities; non-integrable velocity restrictions are important examples. An equation containing velocity is not automatically non-holonomic.

Open orbit. Often used for an unbounded path. The notes use “non-closed” for bounded precessing paths when needed to avoid ambiguity.

Periapsis. A local minimum of distance from the force center; for a regular Kepler conic, $r_p=p/(1+e)$.

Phase space. The space of mechanical states described locally by $(q_1,\ldots,q_n,p_1,\ldots,p_n)$, of dimension $2n$ for a regular $n$-coordinate system.

Phase trajectory. The path traced by the full mechanical state through phase space as time evolves.

Potential energy. A configuration function whose negative gradient gives the conservative force, with an arbitrary additive reference constant.

Principle of virtual work. For static equilibrium with ideal constraints, total virtual work of the applied forces vanishes for every admissible virtual displacement, with appropriate contact qualifications.

R–V

Rayleigh dissipation function. A function of generalized velocities whose negative velocity derivatives give linear viscous generalized forces in the quadratic model; twice it is the dissipative power magnitude in stationary coordinates.

Reduced mass. $\mu=m_1m_2/(m_1+m_2)$, the inertial coefficient of relative motion in an isolated two-body system.

Regular coordinates. A local coordinate description whose independent changes produce independent allowed positional changes. A singular chart can fail at isolated configurations.

Rheonomous constraint. A constraint with explicit time dependence.

Runge–Lenz vector. The conserved vector $\mathbf A=\mathbf p\times\mathbf L-\mu k\hat{\mathbf r}$ for inverse-square attraction; it records eccentricity and orbital orientation.

Scleronomous constraint. A constraint with no explicit time dependence.

Semi-latus rectum. The conic length $p$ in $r=p/(1+e\cos\theta)$, equal to the radius at a right angle to the periapsis axis.

Semimajor axis. Half the longest diameter of an ellipse; for relative Kepler motion it determines bound energy and period.

Separatrix. A phase trajectory or boundary separating qualitatively different motions, such as pendulum oscillations and rotations.

Stable circular orbit. A circular orbit with restoring radial behavior under small perturbations; positive curvature of the effective potential is a sufficient linear criterion.

State. The information needed to determine local mechanical evolution under the given model, including coordinates and velocities or equivalent momenta, and time for a non-autonomous system.

Turning point. A location where the relevant coordinate velocity vanishes and changes sign on an ordinary motion branch. A radial turning point need not have zero total speed.

Unilateral constraint. An inequality restriction, such as non-penetration of a surface or the fact that a string can pull but cannot push.

Unstable circular orbit. A circle admitting arbitrarily small radial perturbations with growing departures; negative effective-potential curvature produces exponential growth in the linear approximation.

Virtual displacement. An infinitesimal comparison between neighboring admissible configurations at the same instant, with time held fixed; it is not an actual short-time trajectory segment.

Virtual work. The force–virtual-displacement scalar sum, $\delta W=\sum_a\mathbf F_a\cdot\delta\mathbf r_a$, used to project mechanics onto admissible directions.