Codex Wiki
classical-mechanics.bigb
= Classical mechanics
{wiki}
= Newtonian gravitational field
{parent=Classical mechanics}
{c}
{wiki=Newton%27s_law_of_universal_gravitation}
Outside a spherical mass $M$, the Newtonian gravitational field is radial and has magnitude
$$
g(r)=\frac{GM}{r^2}.
$$
= Degree of freedom
{parent=Classical mechanics}
{wiki=Degrees_of_freedom_(mechanics)}
A degree of freedom is an independent coordinate needed to specify a mechanical configuration after all <mechanical constraints> have been imposed.
= Degrees of freedom
{synonym}
= Mechanical constraint
{parent=Classical mechanics}
{wiki=Constraint_(classical_mechanics)}
A mechanical constraint restricts the allowed configurations or velocities of a system and reduces its <degrees of freedom>.
= Mechanical constraints
{synonym}
= Position
{parent=Classical mechanics}
{wiki=Position_(geometry)}
Position specifies the location of a particle relative to a coordinate system.
= Plane polar coordinates
{title2=$(r,\theta)$}
{parent=Position}
{wiki=Polar_coordinate_system}
Plane polar coordinates describe a point by its radial distance $r$ and polar angle $\theta$.
= Velocity
{parent=Position}
{wiki}
Velocity is the time <derivative> of <position>.
= Speed
{parent=Velocity}
{wiki}
Speed is the magnitude $|v|$ of <velocity>.
= Acceleration
{parent=Velocity}
{wiki}
Acceleration is the time <derivative> of <velocity>.
= Constant speed
{parent=Velocity}
{wiki=Speed}
A trajectory has constant speed when the magnitude of its velocity does not vary with time.
= Momentum
{parent=Classical mechanics}
{wiki}
For a nonrelativistic particle of mass $m$ and <velocity> $v$, the momentum is $p=mv$.
= Rocket equation
{parent=Momentum}
{wiki=Tsiolkovsky_rocket_equation}
For a body of varying mass $M$ ejecting material at fixed relative velocity, conservation of <momentum> gives a differential relation between $dM$ and the body's change $dv$. Integrating it produces a logarithm of the mass ratio.
= Forward-exhaust rocket equation
{parent=Rocket equation}
If a vehicle moving at speed $v$ ejects mass forward at relative speed $u>0$, then
$$
M\,dv=u\,dM.
$$
As $dM<0$, the vehicle slows down.
= Momenta
{synonym}
= Angular momentum
{title2=$L$}
{parent=Momentum}
{wiki}
The angular momentum of a particle about the origin is $L=r\times p$.
= Angular momenta
{synonym}
= Torque
{title2=$\tau$}
{parent=Angular momentum}
{wiki}
Torque is the moment of force, $\tau=r\times F$, and equals the rate of change of angular momentum.
= Areal velocity
{title2=$\dot A$}
{parent=Angular momentum}
{wiki=Kepler%27s_laws_of_planetary_motion#Second_law}
The areal velocity is the rate at which the radius vector sweeps area. For specific angular momentum $h$, it is constant and equals
$$
\dot A=\frac12r^2\dot\theta=\frac h2.
$$
= Energy
{parent=Classical mechanics}
{wiki}
Energy is a conserved scalar quantity associated with time-translation symmetry.
= Work
{parent=Energy}
{wiki=Work_(physics)}
The work done by a force along a path is $W=\int F\mathbin{\cdot}dr$.
= Work-energy theorem
{parent=Work}
{wiki=Work_(physics)#Work%E2%80%93energy_principle}
The net <work> on a particle equals its change in <kinetic energy>:
$$
\int_a^bF\mathbin{\cdot}dr=\Delta\left(\frac12mv^2\right).
$$
= Power
{parent=Work}
{wiki=Power_(physics)}
Power is the rate of energy transfer, $P=dW/dt$.
= Kinetic energy
{parent=Energy}
{wiki}
The nonrelativistic kinetic energy of a particle is $T=mv^2/2$.
= Rotational kinetic energy
{title2=$K=\frac12I\omega^2$}
{parent=Kinetic energy}
{wiki}
A rigid body rotating about a fixed axis with moment of inertia $I$ and angular speed $\omega$ has kinetic energy $K=I\omega^2/2$.
= Potential energy
{parent=Energy}
{wiki}
A conservative force is the negative gradient of a potential energy: $F=-\nabla V$.
= Conservative force
{parent=Potential energy}
{wiki}
A force is conservative when its work depends only on the endpoints. On a simply connected domain this is equivalent to $F=-\nabla V$ for a <potential energy> $V$.
= Inverse-square potential
{title2=$V(r)=-k/r$}
{parent=Potential energy}
{wiki=Inverse-square_law}
The potential $V(r)=-k/r$ generates the attractive inverse-square <central force> $F=-k r/r^3$.
= Exponential cusp potential
{parent=Potential energy}
The even potential
$$
V(x)=V_\infty(1-e^{-|x|/\lambda})
$$
has a cusp at the origin and approaches the finite value $V_\infty$. Motions below that value oscillate between symmetric turning points; motions above it escape.
= Escape velocity
{parent=Potential energy}
{wiki}
Escape velocity is the least initial speed whose total energy permits motion to spatial infinity.
= Turning point
{parent=Energy}
{wiki=Classical_turning_point}
A turning point of one-dimensional motion is a position where the kinetic energy and instantaneous velocity vanish before the direction of motion reverses.
= Period of an oscillation
{title2=$T$}
{parent=Classical mechanics}
The period is the time required for a periodic motion to complete one full cycle.
= Oscillation period
{synonym}
= Force
{parent=Classical mechanics}
{wiki}
A force is an interaction that changes momentum; in Newtonian mechanics it satisfies $F=ma$ for constant mass.
= Newton's second law
{parent=Force}
{c}
{wiki=Newton%27s_laws_of_motion\#Second_law}
The net force equals the time derivative of momentum; for constant mass, $F=ma$.
= Newton's third law
{parent=Force}
{c}
{wiki=Newton%27s_laws_of_motion\#Third_law}
For an isolated pair of particles, the force of particle 1 on particle 2 is equal and opposite to the force of particle 2 on particle 1.
= Normal force
{parent=Force}
{wiki=Normal_force}
A normal force is a contact force perpendicular to the constraining surface.
= Circular motion
{parent=Classical mechanics}
{wiki}
Uniform circular motion of radius $a$ and angular frequency $\omega$ has centripetal acceleration of magnitude $a\omega^2$ toward the centre.
= Inverse-square force
{parent=Classical mechanics}
{wiki=Inverse-square_law}
An inverse-square central force has magnitude proportional to the reciprocal square of the separation.
= Displacement
{parent=Classical mechanics}
{wiki=Displacement_(geometry)}
Displacement is the change in position between two points.
= Time
{parent=Classical mechanics}
{wiki=Time_in_physics}
Time parametrizes physical change and orders events.
= Mass
{parent=Classical mechanics}
{wiki}
Mass measures inertia and acts as the source of Newtonian gravity.
= Rolling acceleration with rotational inertia
{parent=Classical mechanics}
For a round body with $I=\gamma Ma^2$ rolling without slipping down an incline, static friction gives
$$
a_{\rm cm}=\frac{g\sin\alpha}{1+\gamma}.
$$
= Angular momentum about the centre of mass
{parent=Classical mechanics}
The derivative of angular momentum about the moving centre of mass equals the total external torque about that centre, because total relative momentum vanishes and internal central torques cancel.
= Legendre transform in mechanics
{c}
{parent=Classical mechanics}
{wiki=Legendre_transformation}
For a regular Lagrangian, $p=\partial L/\partial\dot q$ and $H=p\dot q-L$. Regularity means the velocity Hessian is invertible.
= Singular Legendre transform of a degree-one velocity Lagrangian
{parent=Legendre transform in mechanics}
For $L=mc|\dot x|-V(x)$,
$$
p=mc\frac{\dot x}{|\dot x|},
\qquad |p|=mc,
\qquad H=V(x).
$$
The momentum loses the speed, so the velocity cannot be recovered from $(x,p)$ and the ordinary inverse Legendre transform does not exist.
= Hamiltonian of a charged particle
{c}
{parent=Classical mechanics}
For $L=m\dot r^2/2-e\phi+e\dot r\cdot A$, $p=m\dot r+eA$ and $H=|p-eA|^2/(2m)+e\phi$.
= Virial theorem
{parent=Classical mechanics}
{wiki}
For a confined Newtonian gravitational system, $\dot G=2T+V$ for $G=\sum p_i\cdot r_i$, hence $2\langle T\rangle=-\langle V\rangle$.
= Hamiltonian mechanics
{parent=Classical mechanics}
{c}
{wiki}
Hamiltonian mechanics describes phase-space evolution through a Hamiltonian and the symplectic structure.
= Phase space
{parent=Hamiltonian mechanics}
{wiki}
Phase space has one position coordinate $q_i$ and one conjugate momentum $p_i$ for every degree of freedom; a point $(q,p)$ specifies an instantaneous mechanical state.
= Symplectic form
{title2=$\omega$}
{parent=Phase space}
{wiki}
In canonical coordinates the symplectic form is $\omega=\sum_i dq_i\wedge dp_i$. A coordinate transformation is canonical precisely when it preserves this form.
= Hamiltonian
{parent=Hamiltonian mechanics}
{wiki=Hamiltonian_mechanics}
The Hamiltonian is a function of canonical position and momentum that generates time evolution through Hamilton's equations.
= Hamilton's equations
{c}
{parent=Hamiltonian}
{wiki}
For canonical coordinates $(q_i,p_i)$ and Hamiltonian $H$,
$$
\dot q_i=\frac{\partial H}{\partial p_i},
\qquad
\dot p_i=-\frac{\partial H}{\partial q_i}.
$$
= Poisson bracket
{parent=Hamiltonian mechanics}
{c}
{wiki}
In canonical coordinates,
$$
\{F,G\}=\sum_i\left(
\frac{\partial F}{\partial q_i}\frac{\partial G}{\partial p_i}
-\frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q_i}
\right).
$$
= Jacobi identity for the Poisson bracket
{parent=Poisson bracket}
{c}
The Poisson bracket satisfies
$$
\{F,\{G,H\}\}+\{G,\{H,F\}\}+\{H,\{F,G\}\}=0.
$$
= Poisson bracket of conserved quantities
{parent=Jacobi identity for the Poisson bracket}
If $F$ and $G$ have no explicit time dependence and satisfy $\{F,H\}=\{G,H\}=0$, the Jacobi identity gives $\{\{F,G\},H\}=0$. Their Poisson bracket is therefore also conserved.
= Hamiltonian conservation law
{parent=Poisson bracket}
Along a trajectory generated by the <Hamiltonian> $H$,
$$
\frac{dF}{dt}=\frac{\partial F}{\partial t}+\{F,H\}.
$$
Thus a time-independent observable is conserved exactly when its <Poisson bracket> with $H$ vanishes.
= Phase-space dilation generator
{parent=Poisson bracket}
The function $D=q\cdot p$ generates the canonical scaling $(q,p)\mapsto(e^tq,e^{-t}p)$. An observable $F$ is invariant under this scaling exactly when
$$
\{F,D\}=q\cdot\nabla_qF-p\cdot\nabla_pF=0.
$$
= Variational derivative
{parent=Hamiltonian mechanics}
{wiki=Functional_derivative}
For a functional $H[u]$, its variational derivative is defined by
$$
\left.\frac d{d\epsilon}H[u+\epsilon v]\right|_{\epsilon=0}
=\int\frac{\delta H}{\delta u}v\,dx
$$
after integrating derivatives of $v$ by parts and discarding the boundary terms.
= Hamiltonian field equation
{parent=Variational derivative}
For a complex field $\psi$, the canonical Hamiltonian evolution convention
$$
i\psi_t=\frac{\delta H}{\delta\overline\psi}
$$
is accompanied by its complex conjugate equation.
= Canonical transformation
{parent=Hamiltonian mechanics}
{wiki}
A canonical transformation preserves the symplectic form and hence all Poisson brackets. In one degree of freedom, an invertible map $(q,p)\mapsto(Q,P)$ is canonical exactly when $\{Q,P\}_{q,p}=1$.
= Type-two generating function for a canonical transformation
{parent=Canonical transformation}
A function $S(q,P)$ generates a canonical transformation through
$$
p=\frac{\partial S}{\partial q},
\qquad
Q=\frac{\partial S}{\partial P}.
$$
= Type-one generating function for a canonical transformation
{parent=Canonical transformation}
A function $\Phi(q,Q)$ generates a canonical transformation through
$$
p=\frac{\partial\Phi}{\partial q},
\qquad
P=-\frac{\partial\Phi}{\partial Q}.
$$
= Integrable Hamiltonian system
{parent=Hamiltonian mechanics}
{wiki=Integrable_system}
Liouville integrability means having $n$ almost-everywhere independent first integrals in mutual Poisson involution.
= Arnold-Liouville theorem
{c}
{parent=Integrable Hamiltonian system}
{wiki=Liouville–Arnold_theorem}
A compact connected regular common level of $n$ commuting integrals is an $n$-torus with local action-angle coordinates.
= Action-angle variables
{parent=Integrable Hamiltonian system}
{wiki}
For a one-degree-of-freedom closed orbit, action-angle coordinates make the Hamiltonian a function $H(I)$ and give $\dot I=0$, $\dot\phi=dH/dI$.
= Action variable
{parent=Action-angle variables}
{wiki=Action-angle_coordinates\#Action_variables}
With angle period $2\pi$, the action is the enclosed phase-space area divided by $2\pi$:
$$I=(2\pi)^{-1}\oint p\,dq.$$
= Adiabatic invariance of the action
{parent=Action variable}
{wiki=Adiabatic_invariant}
When a Hamiltonian parameter varies slowly compared with the orbital period and no separatrix is crossed, the action changes only at higher adiabatic order.
= Adiabatic particle between moving parallel walls
{parent=Adiabatic invariance of the action}
For a one-dimensional particle of energy $E$ between walls separated by $L$, the unnormalized action is
$$
\mathcal I=\oint p\,dq=2L\sqrt{2mE}.
$$
Adiabatic conservation gives $E(t)L(t)^2=\text{constant}$.
= Adiabatic parameter scaling from an action
{parent=Adiabatic invariance of the action}
If an action has parameter dependence $I\propto\sqrt{m/|E|}$, adiabatic conservation implies $|E|\propto m$.
= Action of the one-dimensional Kepler collision orbit
{parent=Action variable}
For $H=p^2/(2m)-1/|q|=-|E|$, collision continuation between the two turning points gives $I=\sqrt{2m/|E|}$.
= Liouville theorem in Hamiltonian mechanics
{c}
{parent=Hamiltonian mechanics}
{wiki=Liouville%27s_theorem_(Hamiltonian)}
Hamiltonian flow has zero phase-space divergence and preserves phase volume.
= Poincare recurrence theorem
{c}
{parent=Classical mechanics}
{wiki=Poincaré_recurrence_theorem}
A finite measure-preserving dynamical system returns almost every point arbitrarily close to its start infinitely often.
= Particle dynamics
{parent=Classical mechanics}
{wiki}
Particle dynamics studies motion determined by forces through Newton’s laws.
= One-dimensional conservative motion
{parent=Particle dynamics}
{wiki}
For force minus the derivative of a potential, total kinetic plus potential energy is conserved.
= Period near a separatrix
{parent=One-dimensional conservative motion}
{wiki}
A trajectory approaching a <nondegenerate critical point>[nondegenerate] unstable maximum of its <potential energy> has a logarithmically divergent period.
= Simple harmonic motion
{parent=One-dimensional conservative motion}
{wiki}
Simple harmonic motion obeys $\ddot x+\omega^2x=0$ and has a period independent of its amplitude.
= Simple harmonic oscillator
{synonym}
= Harmonic oscillator equation
{synonym}
= Quartic oscillator
{parent=Simple harmonic motion}
{wiki=Quartic_oscillator}
A quartic oscillator adds an anharmonic term to the quadratic potential, for example $H=(p^2+q^2)/2+\epsilon q^4$.
= Isochronous harmonic motion
{parent=Simple harmonic motion}
Harmonic trajectories are isochronous: all nonconstant oscillations with a fixed frequency have the same period, regardless of amplitude.
= Collision mechanics
{parent=Particle dynamics}
{wiki}
Collision mechanics uses impulse together with conservation of momentum and, for elastic collisions, kinetic energy.
= Elastic collision
{parent=Collision mechanics}
{wiki}
An elastic collision conserves total momentum and total kinetic energy.
= Oblique collision
{parent=Collision mechanics}
{wiki}
In a frictionless oblique collision, impulse acts along the common normal and tangential velocity components remain unchanged.
= Impulse
{parent=Collision mechanics}
{wiki}
Impulse is the time integral of force and equals the change in linear momentum.
= Freely expanding rotating particle ring
{parent=Particle dynamics}
Particles initially on a circle of radius $R$ with common tangential speed $v$ move freely on tangent lines. They remain on a circle of radius
$$
R(t)=\sqrt{R^2+v^2t^2},
$$
rotated through $\theta(t)=\arctan(vt/R)$, while total angular momentum remains $mRv$.
= Celestial mechanics
{parent=Classical mechanics}
{wiki}
Celestial mechanics studies motion under gravitational forces.
= Kepler orbit
{parent=Celestial mechanics}
{c}
{wiki}
An inverse-square central force produces conic-section trajectories with the force centre at a focus.
= Binet equation
{parent=Kepler orbit}
{c}
{wiki}
Binet’s equation rewrites a central-force orbit using $u(\theta)=1/r$. If $\ell=mr^2\dot\theta$ is angular momentum and the potential is $V(r)$, then
$$
u''+u=\frac{m}{\ell^2u^2}\frac{dV}{dr}\bigg|_{r=1/u}.
$$
= Shifted inverse-radius potential
{parent=Binet equation}
For
$$
V(r)=-\frac{km}{r-r_0},
\qquad r>r_0,
$$
Binet’s equation is
$$
u''+u=\frac{K}{(1-r_0u)^2},
\qquad K=\frac{km^2}{\ell^2}.
$$
Its circular orbits solve $K=u(1-r_0u)^2$; the branch $u<1/(3r_0)$ is stable and the branch $u>1/(3r_0)$ is unstable.
= Orbital eccentricity
{parent=Kepler orbit}
{wiki}
Orbital eccentricity distinguishes circular, elliptic, parabolic, and hyperbolic conics.
= Hyperbolic Kepler orbit
{parent=Kepler orbit}
{wiki}
A positive-energy inverse-square orbit is a hyperbola with two scattering asymptotes.
= Gravitational scattering angle
{parent=Hyperbolic Kepler orbit}
{wiki}
The gravitational scattering angle is the change between incoming and outgoing asymptotic velocity directions.
= Parabolic Kepler orbit
{parent=Kepler orbit}
{wiki}
A zero-energy inverse-square orbit is parabolic and has eccentricity one.
= Barker equation
{parent=Parabolic Kepler orbit}
{c}
{wiki}
Barker’s equation relates time on a parabolic orbit to the tangent of half the true anomaly through a cubic.
= Periapsis
{parent=Kepler orbit}
{wiki}
The periapsis is the point of an orbit nearest its force centre.
= Apoapsis
{parent=Kepler orbit}
{wiki}
The apoapsis is the point of an orbit farthest from its force centre.
= Semi-major axis
{title2=$a$}
{parent=Kepler orbit}
{wiki=Semi-major_and_semi-minor_axes}
The semi-major axis is one half of an ellipse's longest diameter. For a Kepler ellipse with apsidal distances $r_1,r_2$, it is $a=(r_1+r_2)/2$.
= Semi-minor axis
{title2=$b$}
{parent=Kepler orbit}
{wiki=Semi-major_and_semi-minor_axes}
For an ellipse of eccentricity $e$, the semi-minor axis is $b=a\sqrt{1-e^2}$.
= Hohmann transfer
{parent=Kepler orbit}
{c}
{wiki=Hohmann_transfer_orbit}
A Hohmann transfer uses two tangential impulses and one half of an elliptic Kepler orbit to move between two coplanar circular orbits.
= Rigid body dynamics
{parent=Classical mechanics}
{wiki}
Rigid body dynamics studies translation and rotation of bodies whose internal distances are fixed.
= Symmetric top
{parent=Rigid body dynamics}
{wiki}
A symmetric top is a <rigid body dynamics>[rigid body] with two equal principal moments of inertia, conventionally $I_1=I_2$. Its third principal axis is its symmetry axis.
= Angular velocity
{title2=$\omega$}
{parent=Rigid body dynamics}
{wiki}
Angular velocity is the vector whose direction gives the instantaneous rotation axis and whose magnitude gives the angular speed.
= Fixed space frame
{parent=Rigid body dynamics}
A fixed space frame is an inertial orthonormal basis whose axes do not rotate with the body.
= Principal body frame
{parent=Rigid body dynamics}
A principal body frame is an orthonormal basis fixed in the body and aligned with the principal axes of its inertia tensor. In this frame the inertia tensor is diagonal with principal moments $I_1,I_2,I_3$.
= Derivative of a body-fixed basis vector
{parent=Principal body frame}
If $e_i$ is fixed in a rigid body with <angular velocity> $\omega$, then its derivative in the fixed space frame is
$$
\dot e_i=\omega\times e_i.
$$
= Euler equations for a torque-free rigid body
{parent=Rigid body dynamics}
{c}
{wiki=Euler%27s_equations_(rigid_body_dynamics)}
In principal-axis body coordinates, torque-free angular velocity satisfies
$$
I_1\dot\omega_1=(I_2-I_3)\omega_2\omega_3,
\quad
I_2\dot\omega_2=(I_3-I_1)\omega_3\omega_1,
\quad
I_3\dot\omega_3=(I_1-I_2)\omega_1\omega_2.
$$
= Torque-free rigid-body invariants
{parent=Euler equations for a torque-free rigid body}
The <Euler equations for a torque-free rigid body> conserve rotational kinetic energy and squared angular momentum,
$$
E=\frac12\sum_{i=1}^3I_i\omega_i^2,
\qquad
L^2=\sum_{i=1}^3I_i^2\omega_i^2.
$$
Their common level sets are the intersection of two ellipsoids in angular-velocity space.
= Coplanarity in a torque-free axisymmetric rigid body
{parent=Euler equations for a torque-free rigid body}
For principal moments $I_1=I_2=I$ and symmetry-axis unit vector $e_3$, angular momentum and angular velocity obey
$$
L=I\omega+(I_3-I)\omega_3e_3.
$$
Thus $L$, $\omega$, and the symmetry axis are coplanar at every instant.
= Intermediate axis theorem
{parent=Euler equations for a torque-free rigid body}
{wiki}
For $I_1<I_2<I_3$, steady torque-free rotation about the first or third principal axis is stable, while rotation about the intermediate principal axis is unstable.
= Intermediate-axis separatrix
{parent=Intermediate axis theorem}
On the invariant level $L^2=2EI_2$, the torque-free rigid-body orbit is a separatrix joining the two intermediate-axis rotations. With
$$
\mu=\sqrt{\frac{2E}{I_2}},
\qquad
\lambda=\mu\sqrt{\frac{(I_3-I_2)(I_2-I_1)}{I_1I_3}},
$$
one orientation has $\omega_2(t)=\mu\tanh(\lambda(t-t_0))$; the other components are constant multiples of $\operatorname{sech}(\lambda(t-t_0))$.
= Moment of inertia
{title2=$I$}
{parent=Rigid body dynamics}
{wiki}
Moment of inertia is the mass-weighted squared distance from a rotation axis and determines rotational kinetic energy.
= Inertia tensor
{title2=$I_{ij}$}
{parent=Moment of inertia}
{wiki}
The inertia tensor about a point is
$$
I_{ij}=\int\left(r^2\delta_{ij}-r_ir_j\right)\,dm.
$$
It maps <angular velocity> to <angular momentum>.
= Poinsot construction
{c}
{parent=Inertia tensor}
{wiki=Poinsot%27s_ellipsoid}
For torque-free rigid-body motion, the kinetic-energy ellipsoid fixed in the body rolls without slipping on the invariable plane perpendicular to the conserved angular momentum. Its point of contact is the angular-velocity vector.
= Parallel axis theorem
{parent=Moment of inertia}
{wiki}
The moment about a parallel axis equals the centre-of-mass moment plus mass times the squared axis separation.
= Perpendicular axis theorem
{parent=Moment of inertia}
{wiki}
For a planar lamina in the $xy$-plane, the moments about three mutually perpendicular axes through one point satisfy
$$
I_z=I_x+I_y.
$$
This follows pointwise from $x^2+y^2=y^2+x^2$ in the defining mass integrals.
= Angular rocket equation
{parent=Moment of inertia}
For a uniformly filled, negligibly light disc of fixed radius $R$, remaining mass $m(t)$, angular velocity $\omega(t)$, and tangential exhaust speed $u$ relative to its rim, angular-momentum balance gives
$$
m\dot\omega=\dot m\left(\omega+\frac{2u}{R}\right).
$$
At zero relative exhaust speed, $\omega/m$ is constant.
= Moment of inertia of a solid cone about its vertex
{parent=Moment of inertia}
For a uniform solid cone of height $l$, base radius $R$, and mass $M$, whose symmetry axis points from vertex to base, the centre of mass lies at $3l/4$ from the vertex. The principal moments at the vertex are
$$I_1=I_2=\frac{3M}{20}(R^2+4l^2),
\qquad I_3=\frac{3MR^2}{10}.$$
= Euler angles for a symmetric top
{parent=Rigid body dynamics}
{c}
{wiki=Euler_angles}
With the convention in which $\theta$ is inclination, $\phi$ is precession, and $\psi$ is body-axis spin, an axisymmetric body's kinetic energy is
$$T=\frac I2(\dot\theta^2+\sin^2\theta\dot\phi^2)
+\frac J2(\dot\psi+\cos\theta\dot\phi)^2.$$
= Heavy symmetric top Lagrangian
{parent=Euler angles for a symmetric top}
{wiki=Lagrange,_Euler,_and_Kovalevskaya_tops}
For a fixed point a distance $a$ from the centre of mass and inclination measured from the upward vertical,
$$L=T-Mga\cos\theta.$$
The spin and precession angles are cyclic.
= Heavy symmetric top Hamiltonian
{parent=Heavy symmetric top Lagrangian}
With transverse and axial moments $I,J$, the Hamiltonian is
$$H=\frac{p_\theta^2}{2I}
+\frac{(p_\phi-p_\psi\cos\theta)^2}{2I\sin^2\theta}
+\frac{p_\psi^2}{2J}+Mga\cos\theta.$$
= Heavy symmetric top reduction
{parent=Heavy symmetric top Hamiltonian}
Fixing the cyclic momenta $p_\psi=\lambda$ and $p_\phi=\mu$ leaves the canonical pair $(\theta,p_\theta)$ and a one-degree-of-freedom Hamiltonian with effective potential
$$V_{\lambda,\mu}(\theta)
=\frac{(\mu-\lambda\cos\theta)^2}{2I\sin^2\theta}
+\frac{\lambda^2}{2J}+Mga\cos\theta.$$
= Gyroscopic stabilization of an inverted symmetric top
{parent=Heavy symmetric top reduction}
At the upright spinning state, $p_\phi=p_\psi=\lambda$. The reduced potential is locally minimized at zero inclination when
$$\lambda^2>4IMga,$$
so sufficiently large axial angular momentum stabilizes the gravitationally inverted top.
= Rolling without slipping
{parent=Rigid body dynamics}
{wiki}
Rolling without slipping imposes that the contact point is instantaneously at rest, giving v=R omega.
= Lagrangian mechanics
{parent=Classical mechanics}
{c}
{wiki}
Lagrangian mechanics derives motion from the stationary action of $L=T-V$ through the Euler--Lagrange equations.
= Action
{title2=$S$}
{parent=Lagrangian mechanics}
{wiki=Action_(physics)}
The action of a path is the time integral of its <Lagrangian>. The <principle of stationary action> requires its first variation to vanish for every admissible fixed-endpoint variation.
= Principle of stationary action
{parent=Action}
{wiki}
The principle of stationary action says that the physical path between fixed endpoints is a <stationary point> of the <action>.
= Total-time-derivative invariance of a Lagrangian
{parent=Lagrangian mechanics}
Replacing a Lagrangian by
$$
L'(q,\dot q,t)=L(q,\dot q,t)+\frac d{dt}F(q,t)
$$
adds only the endpoint term $F(q(t_2),t_2)-F(q(t_1),t_1)$ to the action. Fixed-endpoint variations therefore give the same Euler--Lagrange equations.
= Charged-particle electromagnetic Lagrangian
{parent=Lagrangian mechanics}
For electromagnetic scalar and vector potentials $\phi$ and $A$,
$$
L=\frac12m\dot r^2-q\phi+q\dot r\cdot A.
$$
With
$$
E=-\nabla\phi-\partial_tA,
\qquad
B=\nabla\times A,
$$
the Euler--Lagrange equations give the Lorentz force $m\ddot r=q(E+\dot r\times B)$, while the canonical momentum is $p=m\dot r+qA$.
= Gauge variation of the charged-particle Lagrangian
{parent=Charged-particle electromagnetic Lagrangian}
Under $\phi\mapsto\phi-\partial_tf$ and $A\mapsto A+\nabla f$, the charged-particle Lagrangian changes by
$$
L' - L=q\bigl(\partial_tf+\dot r\cdot\nabla f\bigr)
=q\frac{df(r(t),t)}{dt}.
$$
It therefore has unchanged equations of motion by total-time-derivative invariance.
= Canonical momentum parallel to a uniform magnetic field
{parent=Charged-particle electromagnetic Lagrangian}
For a constant uniform magnetic field $B$, choose the symmetric gauge $A=\tfrac12B\times r$. The coordinate parallel to $B$ is cyclic, so its canonical momentum is conserved. Since $A\cdot B=0$, this component is also the mechanical momentum $m\dot r\cdot\widehat B$.
= Generalized momentum
{parent=Lagrangian mechanics}
{wiki=Generalized_momenta}
The generalized, or conjugate, momentum associated with a coordinate $q_i$ is
$$
p_i=\frac{\partial L}{\partial\dot q_i}.
$$
= Ignorable coordinate
{parent=Generalized momentum}
{wiki=Cyclic_coordinate}
A coordinate $q_i$ is ignorable, or cyclic, when $\partial L/\partial q_i=0$. The Euler--Lagrange equation then gives $\dot p_i=0$, so its generalized momentum is conserved.
= Conserved quantities from cyclic coordinates
{parent=Lagrangian mechanics}
{wiki=Cyclic_coordinate}
If a Lagrangian is independent of a coordinate, the conjugate momentum to that coordinate is conserved by the Euler--Lagrange equations.
= Axially symmetric isotropic-mass Lagrangian
{parent=Lagrangian mechanics}
If $L=|\dot q|^2/2-V(q)$ and $V$ depends only on $|q|^2$ and $(q\cdot n)^2$, then rotations about the axis $n$ leave $L$ invariant. The quantity supplied by <Noether theorem> is the axial angular momentum
$$
J_n=n\cdot(q\times\dot q).
$$
If $L$ also has no explicit time dependence, the energy $|\dot q|^2/2+V(q)$ is conserved independently.
= Double pendulum
{parent=Lagrangian mechanics}
{wiki}
A double pendulum consists of two pendula coupled because the second pivot moves with the first mass.
= Double pendulum Lagrangian
{parent=Double pendulum}
The kinetic energy contains the coupling $ml_1l_2\cos(\theta_1-\theta_2)\dot\theta_1\dot\theta_2$, while gravity supplies one cosine potential for each mass height.
= Double pendulum equations of motion
{parent=Double pendulum}
The nonlinear Euler--Lagrange equations couple both angular accelerations and contain centrifugal terms quadratic in angular velocity.
= Kinetic energy from Cartesian position vectors
{parent=Lagrangian mechanics}
For point masses with generalized coordinates, kinetic energy follows by differentiating their Cartesian position vectors and summing $m_i|\dot r_i|^2/2$.
= Small oscillation
{parent=Lagrangian mechanics}
{wiki=Small_oscillations}
Near a stable equilibrium, retaining quadratic Lagrangian terms gives a linear system $\mathsf M\ddot z+\mathsf Kz=0$.
= Stable equilibrium in Lagrangian mechanics
{parent=Small oscillation}
At an equilibrium, a positive-definite <Hessian matrix> of the potential gives stable <small oscillations>. If a continuous symmetry supplies zero modes, positive definiteness on the directions transverse to the symmetry gives stability of the relative configuration.
= Angular frequency
{parent=Small oscillation}
{wiki}
Angular frequency measures phase advance per unit time. For $x(t)=A\cos(\omega t+\varphi)$ it is $\omega=2\pi/T$, where $T$ is the period.
= Angular frequencies
{synonym}
= Two vertically suspended springs
{parent=Small oscillation}
For two masses hung in series from springs of natural length $\ell$, the lower equilibrium spring tension is $m_2g$ and the upper tension is $(m_1+m_2)g$. Hence their equilibrium lengths are
$$
L_2=\ell+\frac{m_2g}{k_2},
\qquad
L_1=\ell+\frac{(m_1+m_2)g}{k_1}.
$$
= Vertical normal modes of two equal suspended masses
{parent=Two vertically suspended springs}
For equal masses $m$ and spring constants $k$, vertical displacements from equilibrium have stiffness and mass matrices
$$
K=k\begin{pmatrix}2&-1\\-1&1\end{pmatrix},
\qquad M=mI.
$$
The two squared frequencies and displacement ratios are
$$
\omega_\pm^2=\frac{k}{2m}(3\pm\sqrt5),
\qquad
\frac{\eta_2}{\eta_1}=\frac{1\mp\sqrt5}{2}.
$$
Gravity shifts the equilibrium but does not enter these vertical frequencies.
= Small-oscillation mass and stiffness matrices
{parent=Small oscillation}
The Hessians of kinetic and potential energy at equilibrium give the symmetric mass matrix $\mathsf M$ and stiffness matrix $\mathsf K$.
= Generalized eigenvalue problem for small oscillations
{parent=Small oscillation}
Normal frequencies satisfy $\det(\mathsf K-\Omega^2\mathsf M)=0$, and each null vector gives the fixed amplitude ratio of a mode.
= Mass-matrix orthogonality of normal modes
{parent=Generalized eigenvalue problem for small oscillations}
For symmetric stiffness matrix $K$ and positive-definite mass matrix $M$, eigenvectors $a_i,a_j$ of
$$
Ka=\omega^2Ma
$$
with distinct squared frequencies satisfy
$$
a_i^TMa_j=0.
$$
Thus normal modes are orthogonal in the kinetic-energy inner product.
= Normal modes of a double pendulum
{parent=Generalized eigenvalue problem for small oscillations}
For equal lengths and mass ratio $\mu=m/M$, the squared frequencies are $\omega_0^2(1+\mu\pm\sqrt{\mu(1+\mu)})$.
= Square-root splitting of nearly degenerate frequencies
{parent=Generalized eigenvalue problem for small oscillations}
A weakly coupled degenerate pair can split by order $\sqrt\mu$ when the small parameter enters a singular mass or stiffness matrix.
= Degenerate normal modes from permutation symmetry
{parent=Generalized eigenvalue problem for small oscillations}
If the mass and stiffness matrices commute with a <permutation representation>, each symmetry subspace is invariant under the <generalized eigenvalue problem for small oscillations>. On an irreducible subspace, <Schur lemma> forces a commuting operator to be scalar, producing a degenerate <normal mode> multiplet.
= Normal modes of three equal masses with a symmetric gap potential
{parent=Degenerate normal modes from permutation symmetry}
Let three equal masses move on a circle of radius $r$, and let the potential be $U(\alpha)+U(\beta)+U(\gamma)$ with $\alpha+\beta+\gamma=2\pi$. At equal spacing $s_0=2\pi/3$, angular displacements have a zero-frequency rigid-rotation mode $(1,1,1)$ and a two-dimensional sum-zero eigenspace with
$$
\omega^2=\frac{3U''(s_0)}{mr^2}.
$$
The two vibrational modes are degenerate by the permutation symmetry.
= Normal modes of two equal masses between three springs
{parent=Generalized eigenvalue problem for small oscillations}
For equal masses $m$, wall-spring constants $k_1,k_2$, and coupling-spring constant $k_3$, the <small-oscillation mass and stiffness matrices> are
$$
\mathsf M=mI,
\qquad
\mathsf K=
\begin{pmatrix}
k_1+k_3&-k_3\\
-k_3&k_2+k_3
\end{pmatrix}.
$$
If $k_1=k(1+\varepsilon\delta)$, $k_2=k(1-\varepsilon\delta)$, and $k_3=k\varepsilon$, the squared normal frequencies are
$$
\omega_\pm^2=\frac{k}{m}
\left[1+\varepsilon\left(1\pm\sqrt{1+\delta^2}\right)\right].
$$
= Conservation of energy from time-translation invariance
{parent=Lagrangian mechanics}
{wiki=Noether%27s_theorem}
A Lagrangian with no explicit time dependence conserves $E=\sum_i\dot q_i\,\partial L/\partial\dot q_i-L$.
= Center of mass
{parent=Classical mechanics}
{wiki=Center_of_mass}
For masses $m_i$ at positions $\mathbf r_i$, the center of mass is $\mathbf R=(\sum_i m_i\mathbf r_i)/(\sum_i m_i)$, with the analogous mass-density integral for a continuous body.
= Center-of-mass motion
{parent=Center of mass}
{wiki=Center_of_mass}
The center of mass moves as if the total external force acted on the total mass; internal forces cancel in its equation.
= Center-of-mass reduction
{parent=Center-of-mass motion}
Center-of-mass reduction separates translational motion from internal coordinates by using the conserved or externally forced center coordinate.
= Jacobi coordinates for three particles
{c}
{parent=Center-of-mass reduction}
{wiki=Jacobi_coordinates}
For three particles, choose the relative position of particles one and two, the position of their center of mass relative to particle three, and the total center of mass. The kinetic energy diagonalizes with masses
$$
\frac{m_1m_2}{m_1+m_2},
\qquad
\frac{(m_1+m_2)m_3}{m_1+m_2+m_3},
\qquad
m_1+m_2+m_3.
$$
= Angular momentum decomposition in three-body Jacobi coordinates
{parent=Jacobi coordinates for three particles}
For Jacobi coordinates $\mathbf a,\mathbf b,\mathbf c$ and their corresponding reduced masses $\alpha,\beta,\gamma$, total angular momentum is
$$
\mathbf L
=\alpha\mathbf a\times\dot{\mathbf a}
+\beta\mathbf b\times\dot{\mathbf b}
+\gamma\mathbf c\times\dot{\mathbf c}.
$$
= Collective rotation and gap coordinates on a circle
{title2=$\Phi$}
{parent=Center-of-mass reduction}
For three ordered angular positions $q_1,q_2,q_3$, define consecutive gaps
$$
\alpha=q_2-q_1,
\qquad
\beta=q_3-q_2,
\qquad
\gamma=2\pi-q_3+q_1
$$
and the collective angle $\Phi=(q_1+q_2+q_3)/3$. Then $\alpha+\beta+\gamma=2\pi$. For three equal masses on a circle of radius $r$,
$$
T=\frac{3mr^2}{2}\dot\Phi^2
+\frac{mr^2}{3}
\left(\dot\alpha^2+\dot\alpha\dot\beta+\dot\beta^2\right).
$$
Thus the <ignorable coordinate> $\Phi$ decouples from the relative gap dynamics whenever the potential depends only on the gaps.
= Reduced mass
{title2=$\mu$}
{parent=Center of mass}
{wiki}
For masses $m_1,m_2$, the relative coordinate obeys a one-body equation with reduced mass
$$
\mu=\frac{m_1m_2}{m_1+m_2}.
$$
= Uniform gravitational field
{parent=Classical mechanics}
In a uniform gravitational field, every freely falling center of mass has constant acceleration $-g$.
= Normal coordinate
{parent=Classical mechanics}
{wiki=Normal_mode}
A normal coordinate diagonalizes the linearized kinetic and potential forms so that its motion decouples as a harmonic oscillator.
= Antisymmetric molecular vibration
{parent=Normal coordinate}
In an antisymmetric vibration of a symmetric linear triatomic molecule, the outer atoms move oppositely and the central atom does not couple to that mode.