Classical mechanics

A point mass constrained to a fixed-length massless arm under uniform gravitational acceleration is a simple pendulum. With no drag its angular equation is $\ddot{\theta }+(g/\ell )\sin \theta =0$. The small-angle approximation yields simple harmonic motion, whereas the full restoring force is nonlinear. Adding drag gives a damped pendulum.

In units where the small-amplitude natural frequency is one, linear velocity drag gives $\ddot{\theta }+c\dot{\theta }+\sin \theta =0$ with $c>0$. The even multiples of $\pi$ are attracting equilibria; the odd multiples are saddle equilibria. Small displacements obey a damped harmonic oscillator; the energy dissipation of a damped pendulum governs the global phase portrait.

For the normalized damped pendulum, $E={\dot{\theta }}^2/2+1-\cos \theta$ obeys $\dot{E}=-c{\dot{\theta }}^2\le 0$. Its mechanical energy is constant only along an equilibrium trajectory, and strictly decreases over every positive-length interval on a nonstationary trajectory. At an individual turning point its derivative vanishes. This excludes nonconstant periodic trajectories despite oscillatory approach to a stable focus.

A mechanical Lagrangian quadratic in velocities can be represented by geodesics of a Lorentzian metric with an extra null coordinate. For $L=\frac{1}{2}|\dot{\mathbf{x}}{|}^2-2\mathbf{A}\cdot \dot{\mathbf{x}}-U$, use $g=d{\mathbf{x}}^2+2dt(du-2\mathbf{A}\cdot d\mathbf{x}-Udt)$. The null Killing vector field ${\partial }_u$ gives the conserved momentum $p_u=t\prime$. Fixing $p_u=1$ projects the geodesic equation to the mechanical Euler-Lagrange equations. This is useful for relating mechanical forces to spacetime geometry.

A time-independent gauge transformation $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$ in an Eisenhart-Duval lift is the coordinate change $u\mapsto u+2\chi$. It preserves the metric one-form $du-2\mathbf{A}\cdot d\mathbf{x}-Udt$ and adds only $-2d\chi /dt$ to the reduced Lagrangian.

For an Eisenhart-Duval lift, reduction along the null isometry fixes a nonzero conserved $p_u=t\prime$ and projects geodesics to $(t,\mathbf{x})$. One never divides by $g_{uu}=0$. When $\mathbf{E}=-\nabla U$ and $\nabla \times \mathbf{A}=\mathbf{B}$, the reduced equation is $\ddot{\mathbf{x}}=\mathbf{E}+2\mathbf{B}\times \dot{\mathbf{x}}$. Every trajectory admits a null lift by choosing $\dot{u}=U+2\mathbf{A}\cdot \dot{\mathbf{x}}-\frac{1}{2}|\dot{\mathbf{x}}{|}^2$.

Newtonian mechanics describes motion using forces, masses and acceleration in inertial frames with absolute time. It is the low-speed limit of special relativity; for constant proper acceleration with $|at|\ll c$, relativistic hyperbolic motion reduces to $x\simeq a{t}^2/2$.

An equation of motion determines how a physical state evolves. A variational action gives Euler-Lagrange equations; a phase-space action instead gives canonical Hamiltonian mechanics equations together with any imposed constraints.

Time reversal in classical mechanics reverses the order of a trajectory's states and the sign of its velocities. A coordinate is time-even, while its velocity is time-odd. Microscopic reversibility compares a path with this physical reversal, including the correct equilibrium initial weights.

Newton's law of universal gravitation states that two point masses attract one another with a force proportional to the product of their masses and inversely proportional to the square of their separation.

Motion under a Newtonian gravitational field without other forces. The Newtonian radial free-fall speed from rest at infinity is $v=\sqrt{2GM/r}$.

In an Earth-fixed rotating reference frame, the horizontal displacement of a freely falling body receives contributions from Coriolis acceleration and centrifugal acceleration. A short-fall approximation substitutes the unperturbed downward velocity into the Coriolis term and the Earth's surface position into the centrifugal term. If gravitational acceleration excludes centrifugal acceleration, both contributions must be included explicitly.

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

For a Newtonian gravitational potential obeying the Poisson equation for Newtonian gravity, define $T_{ij}=[{\partial }_i\Phi {\partial }_j\Phi -{\delta }_{ij}|\nabla \Phi {|}^2/2]/(4\pi G)$. Commutation of mixed partial derivatives gives ${\partial }_j{T}_{ij}=\rho {\partial }_i\Phi$. Thus the gravitational force density can be put in conservative form as $-{\partial }_j{T}_{ij}$. The sign is chosen so that $T_{ij}$ is added to the momentum flux; reversing the stress convention reverses the divergence sign.

In cylindrical coordinates, the Newtonian gravitational stress tensor gives radial angular-momentum flux $r{T}_{r\varphi }=r({\partial }_r\Phi )(r^{-1}{\partial }_{\varphi }\Phi )/(4\pi G)$. Nonaxisymmetric structure can correlate the field components and transport angular momentum. The isotropic negative-pressure part has zero off-diagonal component, so it gives no direct radial transport. This separates a shear flux from a contribution to normal force balance.

Self-gravity is the gravitational acceleration produced by a system's own mass density. It is obtained by solving the Poisson equation for Newtonian gravity for that density, rather than treating the Newtonian gravitational potential as an externally fixed field. It competes with pressure and rotational support in gravitational instability of an astrophysical disk.

The product $\mu =GM$ of the gravitational constant and a central mass. In the exact relative two-body problem, the central parameter is $G(M_1+M_2)$. The appropriate conservative parameter can change when inverse-square radiation pressure is included in a dust Kepler orbit.

The acceleration tidal tensor is $T_{jk}={\partial }_k{g}_j=-{\partial }_j{\partial }_k\Phi$ for gravitational acceleration $\mathbf{g}=-\nabla \Phi$. It describes differential acceleration $\Delta \mathbf{g}\simeq T\Delta \mathbf{r}$. Some authors instead call the positive Hessian matrix ${\partial }_j{\partial }_k\Phi$ the tidal tensor, reversing this sign convention. For a point mass and the positive potential convention $\mathbf{g}=\nabla \psi$, the tensor is $Gm(3\mathbf{R}{\mathbf{R}}^{\mathsf{T}}/R^5-1/R^3)$.

Subtracting the constant and uniform-acceleration terms from the point-mass relative potential gives ${\psi }_{\mathrm{tid}}=G{m}_p[3(\mathbf{r}\cdot {\hat{\mathbf{r}}}_p{)}^2-r^2]/(2r_p^3)$, with force convention $\nabla \psi$. Its tidal tensor is $G{m}_p(3{\hat{\mathbf{r}}}_p{\hat{\mathbf{r}}}_p^T-I)/r_p^3$. This is the leading term for target size small compared with separation.

Disruption of a self-gravitating body when differential gravitational acceleration across it exceeds its binding acceleration. For a body of radius $R$, mass $m$ at distance $r$ from a much more massive body $M$, the competing scales are $GMR/r^3$ and $Gm/R^2$; their equality gives $r\sim R(M/m{)}^{1/3}$, up to geometry and internal-structure factors. The tidal tensor gives the directional differential acceleration. A tidal disruption event near a black hole is a specific astronomical application.

Differential gravitational acceleration, multiplied by the affected mass, across a spatially extended body. In the linear approximation the acceleration difference is $T\Delta \mathbf{r}$, where $T$ is the tidal tensor. Stellar tidal forces limit the domain of a circumplanetary orbit; sufficiently strong ones can produce tidal disruption.

A Love number measures a body's tidal response to an applied potential. The degree-two potential response $k_2$ is sensitive to radial density structure and central concentration, and can complement mass and radius constraints on an exoplanet interior. Inferring it from orbital precession requires separating other contributions and assumptions about rotation and the tidal response.

For a point-mass companion at separation $a$, subtract the constant and linear terms of its Newtonian gravitational potential in a frame accelerating with the target's center. The leading residual isIt is degree two in spherical harmonics. In a circular orbit with angular frequency ${\omega }_o$, its time-dependent $m=\pm 2$ components force at $2{\omega }_o$ in a frame with no spin; there is also a static $m=0$ component. This is the leading approximation in $r/a$, and the tidal amplitude may be absorbed into a harmonic-dependent constant.

Gravitational acceleration is the acceleration of a freely falling test body in a gravitational field. Near Earth's surface its magnitude is approximately constant on shallow vertical scales, and is conventionally denoted by $g$.

The gravitational constant is the proportionality constant in Newton's law of universal gravitation.

For mass density $\rho$, the Newtonian gravitational potential satisfies the Poisson equation ${\nabla }^2\Phi =4\pi G\rho$. The gravitational acceleration is $-\nabla \Phi$. An isolated system also requires the exterior vacuum solution and decay at infinity; solving the interior equation alone does not fix a globally physical field.

The Newtonian gravitational potential is the potential energy per unit test mass. Outside a spherical mass $M$ it is $\Phi (r)=-GM/r$ after choosing zero at infinity, and the gravitational acceleration is $-\nabla \Phi$.

An axisymmetric logarithmic Newtonian gravitational potential has spheroidal equipotential surfaces and a flat galaxy rotation curve: $R{\Phi }_R(R,0)=V^2$. The singular model is defined away from the origin; a fixed reference length makes the logarithm dimensionless. Its central $r^{-2}$ mass density cusp is locally integrable, but its total mass diverges at large radius.

The Poisson equation for Newtonian gravity givesFor $V>0$, nonnegative mass density away from the origin is equivalent to $q^2\ge 1/2$. Strict positivity requires $q^2>1/2$. This condition does not by itself guarantee a nonnegative galactic distribution function.

A radial Newtonian gravitational potential $\Phi =-A{r}^{-\beta }$ produces inward acceleration $A\beta r^{-\beta -1}$. The point-mass case has $\beta =1$ and $A=GM$. Treating the power-law gravitational potential as prescribed does not require it to be sourced by the flowing gas.

For a binary star viewed far from its centre of mass, let $\mathbf{r}$ be the separation and $\mu$ the reduced mass, $M=M_1+M_2$ the total mass, $r=|\mathbf{r}|$ and $x=|\mathbf{x}|$. The Taylor expansion of the point-mass Newtonian gravitational potential has zero dipole because ${\sum }_A{M}_A{\mathbf{x}}_A=0$. Its second moment is ${\sum }_A{M}_A{x}_{Ai}{x}_{Aj}=\mu r_i{r}_j$, soThis holds for an instantaneous separation in any orbit, not only a circular orbit. The mass quadrupole moment convention $Q_{ij}=\mu (r_i{r}_j-r^2{\delta }_{ij}/3)$ gives ${\Phi }_2=-3G{Q}_{ij}{x}_i{x}_j/(2x^5)$.

A flattened power-law Newtonian gravitational potential has equipotential spheroids $R^2+{\lambda }^2{z}^2=\text{constant}$. For ${\Phi }_0<0$ and $0<\beta <1$, its midplane circular orbit angular frequency satisfies ${\Omega }^2=-\beta {\Phi }_0{R}_0^{\beta }{R}^{-\beta -2}$.

Applying the Poisson equation for Newtonian gravity to a flattened power-law gravitational potential yieldsFor $0<\beta <1$, nonnegative mass density everywhere outside the origin requires and is ensured by $\beta \le {\lambda }^2\le 2/(\beta +1)$. The central density singularity is locally integrable and carries no point mass, although the total mass diverges at large radius in this scale-free model.

The free-space Green function of the Laplacian gives the Newtonian gravitational potential $\Phi (r)=-GM/r$ for a point mass $M$ at the origin.

A spherically symmetric potential depends only on distance $r$ from one point. Its force is central, so angular momentum is conserved.

Two point masses $m_1,m_2$ separated by distance $r$ have potential energyThe corresponding attractive force is $-\nabla V$.

For positive masses, let $r$ be the actual minimum pairwise separation and use positive binding energy $U={\sum }_{j<k}G{m}_j{m}_k/r_{jk}$. If $m_{(1)},m_{(2)}$ are the smallest masses, then $G{m}_{(1)}{m}_{(2)}\le rU\le G{\sum }_{j<k}{m}_j{m}_k$. Thus $U^{-1}$ is comparable to $r$ with constants depending only on the masses.

With $E=T-U<0$ and $T\ge 0$, the positive binding energy satisfies $U\ge |E|$. The minimum-separation binding-energy bounds imply $r\le A_0/|E|$. At least one pair stays close; this does not bound every particle's distance or exclude a negative-energy gravitational collision.

The Cavendish experiment measures the weak Newtonian gravitational field between laboratory masses with a torsion balance, and can be used to determine the gravitational constant.

A Cavendish torsion balance measures the small gravitational torque exerted by source masses on test masses mounted on a suspended rod. The fiber's twist reveals the horizontal force.

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.

The common zero locus of constraint functions $C_a$ in phase space is the constraint surface. A physical trajectory lies on this surface, but off-surface identities are often needed when analyzing a constraint algebra.

A holonomic constraint is a relation $f(q,t)=0$ involving generalized coordinates and possibly time, rather than a genuinely nonintegrable velocity relation. Independent smooth constraints of full row rank define a local configuration submanifold. Ideal constraint forces are normal to its tangent virtual displacements and can be represented by Lagrange multipliers. A time-dependent holonomic constraint need not conserve mechanical energy.

A first-class constraint has Poisson brackets with all constraints that vanish on the constraint surface. For a regular constrained Hamiltonian system, first-class constraints generate gauge transformations; each independent constraint and its gauge direction remove one canonical pair.

For $n$ independent regular first-class constraints in a $2N$-dimensional phase space, imposing the equations removes $n$ dimensions and quotienting independent gauge orbits removes another $n$. Reducibility, singular strata or vanishing gauge directions can invalidate that elementary count. Closure of the constraint algebra is necessary but does not by itself establish regularity.

A first-class constraint generates an infinitesimal canonical transformation $\delta F=\{F,{ϵ}^i{\phi }_i\}$. For the phase-space action $I=\int (p\dot{q}-{\lambda }^i{\phi }_i)dt$, the multiplier transformation $\delta {\lambda }^k={\dot{ϵ}}^k-{\lambda }^i{ϵ}^j{C}_{ij}{}^k$ cancels the bulk variation when $\{{\phi }_i,{\phi }_j\}=C_{ij}{}^k{\phi }_k$. The remaining temporal boundary term vanishes for parameters that vanish at the endpoints.

The Poisson brackets of first-class constraints close on the constraints. When the constraints are independent and the coefficients are constant, the Jacobi identity for the Poisson bracket makes the coefficients structure constants of a Lie algebra.

The coefficients $C_{ij}{}^k(q,p)$ in $\{{\phi }_i,{\phi }_j\}=C_{ij}{}^k{\phi }_k$ are structure functions. A finite real span of first-class constraints is a Lie algebra when it closes with constant coefficients. General structure functions instead describe closure with phase-space-dependent coefficients, while still permitting canonical gauge transformations.

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 $\theta$.

Differentiating $\mathbf{r}=r\hat{\mathbf{r}}$ twice and using $\dot{\hat{\mathbf{r}}}=\dot{\theta }\hat{\mathbf{\theta }}$ and $\dot{\hat{\mathbf{\theta }}}=-\dot{\theta }\hat{\mathbf{r}}$ gives the displayed acceleration. For a central force, its angular component vanishes, so $r^2\dot{\theta }$ is constant and expresses conservation of angular momentum.

Velocity is the time derivative of position.

The normal velocity is the scalar component of a velocity in a specified unit-normal direction. Its sign depends on the orientation of the normal. At a material interface, no penetration equates the fluid and interface normal velocities; it does not require their tangential components to agree. Surface acoustic impedance uses a fixed inward normal-velocity convention to determine the sign of energy absorbed by a boundary.

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

Acceleration is the time derivative of velocity.

The kinematic jerk is the time derivative of acceleration. For a Kepler orbit with relative position $\mathbf{r}$, relative velocity $\mathbf{v}$, $r=|\mathbf{r}|$, $\dot{r}=\mathbf{r}\cdot \mathbf{v}/r$ and total mass $M$, differentiation of Newton's law of universal gravitation gives $\mathbf{j}=-GM(\mathbf{v}-3\dot{r}\mathbf{r}/r)/r^3$.

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

Terminal velocity is the constant limiting velocity reached when a velocity-dependent resistive force balances the applied force.

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

For a charged particle minimally coupled to a magnetic vector potential, the mechanical momentum is $\mathbf{\pi }=\mathbf{p}-q\mathbf{A}$, with $\mathbf{p}$ the canonical momentum. It is the mass times velocity in nonrelativistic mechanics.

Momentum conservation states that the total momentum of an isolated system is constant. In a translation-invariant field theory, integrating over the spacetime zero mode enforces momentum conservation at interactions and scattering amplitudes.

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.

When matter leaves a moving body with zero instantaneous relative velocity, it carries the body's current momentum per unit mass but creates no additional recoil term. A short-interval momentum balance including the newly released material gives $q\dot{v}=F_{\mathrm{ext}}$ for the remaining body's mass $q(t)$. The expression $\frac{d}{dt}(qv)=F_{\mathrm{ext}}$ omits the outgoing momentum flux. Nonzero relative ejection velocity instead produces the thrust term of the rocket equation.

A classical rocket ejecting fuel at rate $\alpha >0$ and relative exhaust speed $u>0$, while collecting stationary dust of mass $\beta dx$ per traveled distance, obeysThese are mass conservation and momentum conservation for the rocket, incoming dust and outgoing gas. The dust-loading term supplies the momentum needed to accelerate incoming stationary material. Put $\lambda =\sqrt{\alpha /(\beta u)}$ and $V=\lambda u$. While the engines operate and $0\le v<V$, an initially stationary rocket satisfiesFor $\lambda >1$, continuing to $V$ would drive total mass to zero. For $0<\lambda <1$, total mass instead grows without bound and $V$ is approached only after infinite time under indefinitely maintained thrust. In both cases a finite fuel supply runs out before that limiting speed. Total stored mass must not be confused with remaining fuel.