Classical mechanics

Outside a spherical mass $M$, the Newtonian gravitational field is radial and has magnitude

A degree of freedom is an independent coordinate needed to specify a mechanical configuration after all mechanical constraints have been imposed.

A mechanical constraint restricts the allowed configurations or velocities of a system and reduces its degrees of freedom.

Position specifies the location of a particle relative to a coordinate system.

Plane polar coordinates describe a point by its radial distance $r$ and polar angle $\theta$.

Velocity is the time derivative of position.

Speed is the magnitude $|v|$ of velocity.

Acceleration is the time derivative of velocity.

A trajectory has constant speed when the magnitude of its velocity does not vary with time.

For a nonrelativistic particle of mass $m$ and velocity $v$, the momentum is $p=mv$.

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.

If a vehicle moving at speed $v$ ejects mass forward at relative speed $u>0$, thenAs $dM<0$, the vehicle slows down.

The angular momentum of a particle about the origin is $L=r\times p$.

Torque is the moment of force, $\tau =r\times F$, and equals the rate of change of angular momentum.

The areal velocity is the rate at which the radius vector sweeps area. For specific angular momentum $h$, it is constant and equals

Energy is a conserved scalar quantity associated with time-translation symmetry.

The work done by a force along a path is $W=\int F\cdot dr$.

The net work on a particle equals its change in kinetic energy:

Power is the rate of energy transfer, $P=dW/dt$.

The nonrelativistic kinetic energy of a particle is $T=m{v}^2/2$.

A rigid body rotating about a fixed axis with moment of inertia $I$ and angular speed $\omega$ has kinetic energy $K=I{\omega }^2/2$.

A conservative force is the negative gradient of a potential energy: $F=-\nabla V$.

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 $V$.

The potential $V(r)=-k/r$ generates the attractive inverse-square central force $F=-kr/r^3$.

The even potentialhas 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 is the least initial speed whose total energy permits motion to spatial infinity.

A turning point of one-dimensional motion is a position where the kinetic energy and instantaneous velocity vanish before the direction of motion reverses.

The period is the time required for a periodic motion to complete one full cycle.

A force is an interaction that changes momentum; in Newtonian mechanics it satisfies $F=ma$ for constant mass.

The net force equals the time derivative of momentum; for constant mass, $F=ma$.

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.

A normal force is a contact force perpendicular to the constraining surface.

Uniform circular motion of radius $a$ and angular frequency $\omega$ has centripetal acceleration of magnitude $a{\omega }^2$ toward the centre.

An inverse-square central force has magnitude proportional to the reciprocal square of the separation.

Displacement is the change in position between two points.

Time parametrizes physical change and orders events.

Mass measures inertia and acts as the source of Newtonian gravity.

For a round body with $I=\gamma M{a}^2$ rolling without slipping down an incline, static friction gives

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.

For a regular Lagrangian, $p=\partial L/\partial \dot{q}$ and $H=p\dot{q}-L$. Regularity means the velocity Hessian is invertible.

For $L=mc|\dot{x}|-V(x)$,The momentum loses the speed, so the velocity cannot be recovered from $(x,p)$ and the ordinary inverse Legendre transform does not exist.

For $L=m{\dot{r}}^2/2-e\varphi +e\dot{r}\cdot A$, $p=m\dot{r}+eA$ and $H=|p-eA{|}^2/(2m)+e\varphi$.

For a confined Newtonian gravitational system, $\dot{G}=2T+V$ for $G=\sum p_i\cdot r_i$, hence $2⟨T⟩=-⟨V⟩$.

Hamiltonian mechanics describes phase-space evolution through a Hamiltonian and the symplectic structure.

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.

In canonical coordinates the symplectic form is $\omega ={\sum }_{i}d{q}_i\wedge d{p}_i$. A coordinate transformation is canonical precisely when it preserves this form.

The Hamiltonian is a function of canonical position and momentum that generates time evolution through Hamilton's equations.

For canonical coordinates $(q_i,p_i)$ and Hamiltonian $H$,

In canonical coordinates,

The Poisson bracket satisfies

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.

Along a trajectory generated by the Hamiltonian $H$,Thus a time-independent observable is conserved exactly when its Poisson bracket with $H$ vanishes.

The function $D=q\cdot p$ generates the canonical scaling $(q,p)\mapsto (e^{t}q,e^{-t}p)$. An observable $F$ is invariant under this scaling exactly when

For a functional $H[u]$, its variational derivative is defined byafter integrating derivatives of $v$ by parts and discarding the boundary terms.

For a complex field $\psi$, the canonical Hamiltonian evolution conventionis accompanied by its complex conjugate equation.

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 $\{Q,P{\}}_{q,p}=1$.

A function $S(q,P)$ generates a canonical transformation through

A function $\Phi (q,Q)$ generates a canonical transformation through

Liouville integrability means having $n$ almost-everywhere independent first integrals in mutual Poisson involution.

A compact connected regular common level of $n$ commuting integrals is an $n$-torus with local action-angle coordinates.

For a one-degree-of-freedom closed orbit, action-angle coordinates make the Hamiltonian a function $H(I)$ and give $\dot{I}=0$, $\dot{\varphi }=dH/dI$.

With angle period $2\pi$, the action is the enclosed phase-space area divided by $2\pi$:

When a Hamiltonian parameter varies slowly compared with the orbital period and no separatrix is crossed, the action changes only at higher adiabatic order.

For a one-dimensional particle of energy $E$ between walls separated by $L$, the unnormalized action isAdiabatic conservation gives $E(t)L(t{)}^2=\text{constant}$.

If an action has parameter dependence $I\propto \sqrt{m/|E|}$, adiabatic conservation implies $|E|\propto m$.

For $H=p^2/(2m)-1/|q|=-|E|$, collision continuation between the two turning points gives $I=\sqrt{2m/|E|}$.

Hamiltonian flow has zero phase-space divergence and preserves phase volume.

A finite measure-preserving dynamical system returns almost every point arbitrarily close to its start infinitely often.

Particle dynamics studies motion determined by forces through Newton’s laws.

For force minus the derivative of a potential, total kinetic plus potential energy is conserved.

A trajectory approaching a nondegenerate unstable maximum of its potential energy has a logarithmically divergent period.

Simple harmonic motion obeys $\ddot{x}+{\omega }^{2}x=0$ and has a period independent of its amplitude.

A quartic oscillator adds an anharmonic term to the quadratic potential, for example $H=(p^2+q^2)/2+ϵq^4$.

Harmonic trajectories are isochronous: all nonconstant oscillations with a fixed frequency have the same period, regardless of amplitude.

Collision mechanics uses impulse together with conservation of momentum and, for elastic collisions, kinetic energy.

An elastic collision conserves total momentum and total kinetic energy.

In a frictionless oblique collision, impulse acts along the common normal and tangential velocity components remain unchanged.

Impulse is the time integral of force and equals the change in linear momentum.

Particles initially on a circle of radius $R$ with common tangential speed $v$ move freely on tangent lines. They remain on a circle of radiusrotated through $\theta (t)=\arctan (vt/R)$, while total angular momentum remains $mRv$.

Celestial mechanics studies motion under gravitational forces.

An inverse-square central force produces conic-section trajectories with the force centre at a focus.

Binet’s equation rewrites a central-force orbit using $u(\theta )=1/r$. If $\ell =m{r}^2\dot{\theta }$ is angular momentum and the potential is $V(r)$, then

ForBinet’s equation isIts circular orbits solve $K=u(1-r_{0}u{)}^2$; the branch $u<1/(3r_0)$ is stable and the branch $u>1/(3r_0)$ is unstable.

Orbital eccentricity distinguishes circular, elliptic, parabolic, and hyperbolic conics.

A positive-energy inverse-square orbit is a hyperbola with two scattering asymptotes.

The gravitational scattering angle is the change between incoming and outgoing asymptotic velocity directions.

A zero-energy inverse-square orbit is parabolic and has eccentricity one.

Barker’s equation relates time on a parabolic orbit to the tangent of half the true anomaly through a cubic.

The periapsis is the point of an orbit nearest its force centre.

The apoapsis is the point of an orbit farthest from its force centre.

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 $a=(r_1+r_2)/2$.

For an ellipse of eccentricity $e$, the semi-minor axis is $b=a\sqrt{1-e^2}$.

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 studies translation and rotation of bodies whose internal distances are fixed.

A symmetric top is a rigid body with two equal principal moments of inertia, conventionally $I_1=I_2$. Its third principal axis is its symmetry axis.

Angular velocity is the vector whose direction gives the instantaneous rotation axis and whose magnitude gives the angular speed.

A fixed space frame is an inertial orthonormal basis whose axes do not rotate with the body.

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 $I_1,I_2,I_3$.

If $e_i$ is fixed in a rigid body with angular velocity $\omega$, then its derivative in the fixed space frame is

In principal-axis body coordinates, torque-free angular velocity satisfies

The Euler equations for a torque-free rigid body conserve rotational kinetic energy and squared angular momentum,Their common level sets are the intersection of two ellipsoids in angular-velocity space.

For principal moments $I_1=I_2=I$ and symmetry-axis unit vector $e_3$, angular momentum and angular velocity obeyThus $L$, $\omega$, and the symmetry axis are coplanar at every instant.

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.

On the invariant level $L^2=2E{I}_2$, the torque-free rigid-body orbit is a separatrix joining the two intermediate-axis rotations. Withone orientation has ${\omega }_2(t)=\mu \tanh (\lambda (t-t_0))$; the other components are constant multiples of $\mathrm{sech}(\lambda (t-t_0))$.

Moment of inertia is the mass-weighted squared distance from a rotation axis and determines rotational kinetic energy.

The inertia tensor about a point isIt maps angular velocity to angular momentum.

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.

The moment about a parallel axis equals the centre-of-mass moment plus mass times the squared axis separation.

For a planar lamina in the $xy$-plane, the moments about three mutually perpendicular axes through one point satisfyThis follows pointwise from $x^2+y^2=y^2+x^2$ in the defining mass integrals.

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 givesAt zero relative exhaust speed, $\omega /m$ is constant.

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

With the convention in which $\theta$ is inclination, $\varphi$ is precession, and $\psi$ is body-axis spin, an axisymmetric body's kinetic energy is

For a fixed point a distance $a$ from the centre of mass and inclination measured from the upward vertical,The spin and precession angles are cyclic.

With transverse and axial moments $I,J$, the Hamiltonian is

Fixing the cyclic momenta $p_{\psi }=\lambda$ and $p_{\varphi }=\mu$ leaves the canonical pair $(\theta ,p_{\theta })$ and a one-degree-of-freedom Hamiltonian with effective potential

At the upright spinning state, $p_{\varphi }=p_{\psi }=\lambda$. The reduced potential is locally minimized at zero inclination whenso sufficiently large axial angular momentum stabilizes the gravitationally inverted top.

Rolling without slipping imposes that the contact point is instantaneously at rest, giving v=R omega.

Lagrangian mechanics derives motion from the stationary action of $L=T-V$ through the Euler--Lagrange equations.

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.

The principle of stationary action says that the physical path between fixed endpoints is a stationary point of the action.

Replacing a Lagrangian byadds 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.

For electromagnetic scalar and vector potentials $\varphi$ and $A$,Withthe Euler--Lagrange equations give the Lorentz force $m\ddot{r}=q(E+\dot{r}\times B)$, while the canonical momentum is $p=m\dot{r}+qA$.

Under $\varphi \mapsto \varphi -{\partial }_{t}f$ and $A\mapsto A+\nabla f$, the charged-particle Lagrangian changes byIt therefore has unchanged equations of motion by total-time-derivative invariance.

For a constant uniform magnetic field $B$, choose the symmetric gauge $A=\frac{1}{2}B\times r$. 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 $m\dot{r}\cdot \hat{B}$.

The generalized, or conjugate, momentum associated with a coordinate $q_i$ is

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.

If a Lagrangian is independent of a coordinate, the conjugate momentum to that coordinate is conserved by the Euler--Lagrange equations.

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 momentumIf $L$ also has no explicit time dependence, the energy $|\dot{q}{|}^2/2+V(q)$ is conserved independently.

A double pendulum consists of two pendula coupled because the second pivot moves with the first mass.

The kinetic energy contains the coupling $m{l}_1{l}_2\cos ({\theta }_1-{\theta }_2){\dot{\theta }}_1{\dot{\theta }}_2$, while gravity supplies one cosine potential for each mass height.

The nonlinear Euler--Lagrange equations couple both angular accelerations and contain centrifugal terms quadratic in angular velocity.

For point masses with generalized coordinates, kinetic energy follows by differentiating their Cartesian position vectors and summing $m_i|{\dot{r}}_i{|}^2/2$.

Near a stable equilibrium, retaining quadratic Lagrangian terms gives a linear system $\mathsf{M}\ddot{z}+\mathsf{K}z=0$.

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 measures phase advance per unit time. For $x(t)=A\cos (\omega t+\phi )$ it is $\omega =2\pi /T$, where $T$ is the period.

For two masses hung in series from springs of natural length $\ell$, the lower equilibrium spring tension is $m_{2}g$ and the upper tension is $(m_1+m_2)g$. Hence their equilibrium lengths are

For equal masses $m$ and spring constants $k$, vertical displacements from equilibrium have stiffness and mass matricesThe two squared frequencies and displacement ratios areGravity shifts the equilibrium but does not enter these vertical frequencies.

The Hessians of kinetic and potential energy at equilibrium give the symmetric mass matrix $\mathsf{M}$ and stiffness matrix $\mathsf{K}$.

Normal frequencies satisfy $\det (\mathsf{K}-{\Omega }^2\mathsf{M})=0$, and each null vector gives the fixed amplitude ratio of a mode.

For symmetric stiffness matrix $K$ and positive-definite mass matrix $M$, eigenvectors $a_i,a_j$ ofwith distinct squared frequencies satisfyThus normal modes are orthogonal in the kinetic-energy inner product.

For equal lengths and mass ratio $\mu =m/M$, the squared frequencies are ${\omega }_0^2(1+\mu \pm \sqrt{\mu (1+\mu )})$.

A weakly coupled degenerate pair can split by order $\sqrt{\mu }$ when the small parameter enters a singular mass or stiffness matrix.

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.

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 withThe two vibrational modes are degenerate by the permutation symmetry.

For equal masses $m$, wall-spring constants $k_1,k_2$, and coupling-spring constant $k_3$, the small-oscillation mass and stiffness matrices areIf $k_1=k(1+\epsilon \delta )$, $k_2=k(1-\epsilon \delta )$, and $k_3=k\epsilon$, the squared normal frequencies are

A Lagrangian with no explicit time dependence conserves $E={\sum }_i{\dot{q}}_i\partial L/\partial {\dot{q}}_i-L$.

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.

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 separates translational motion from internal coordinates by using the conserved or externally forced center coordinate.

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

For Jacobi coordinates $\mathbf{a},\mathbf{b},\mathbf{c}$ and their corresponding reduced masses $\alpha ,\beta ,\gamma$, total angular momentum is

For three ordered angular positions $q_1,q_2,q_3$, define consecutive gapsand 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 $r$,Thus the ignorable coordinate $\Phi$ decouples from the relative gap dynamics whenever the potential depends only on the gaps.

For masses $m_1,m_2$, the relative coordinate obeys a one-body equation with reduced mass

In a uniform gravitational field, every freely falling center of mass has constant acceleration $-g$.

A normal coordinate diagonalizes the linearized kinetic and potential forms so that its motion decouples as a harmonic oscillator.

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.