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Classical mechanics describes the motion of bodies by assigning positions and velocities and specifying how their interactions change those quantities. A point mass is the simplest model: its size and internal structure are ignored, leaving a mass and a moving position. A rigid body retains orientation and rotational kinetic energy. Choosing a model means deciding which features affect the question being asked; a sliding block may be treated as a point mass, whereas predicting its rolling without slipping requires its moment of inertia.

Three complementary questions organize the subject. Newton's second law asks how the net force determines the instantaneous acceleration. The work-energy theorem asks how that force, acting through a displacement, changes kinetic energy. Conservation of mechanical energy asks which combinations of position and velocity are accessible when the interactions admit a time-independent potential energy and no other force supplies net power. These descriptions are compatible, but answer different questions: an energy calculation may find a maximum speed without first solving for position at every time.

The harmonic oscillator below makes this unity visible. A restoring force produces a repeating position trace; the corresponding phase plane has a closed ellipse; kinetic energy and potential energy exchange while their sum stays fixed. Lagrangian mechanics and Hamiltonian mechanics provide further ways to organize the same dynamics, especially when coordinates or constraints make individual forces awkward to track.

The scope of classical mechanics also matters. Newtonian mechanics approximates motion at speeds small compared with the speed of light; special relativity changes that description when this approximation fails. Quantum mechanics is needed for phenomena for which a classical assignment of simultaneous position and momentum is inadequate. These limitations do not prevent classical mechanics from being a useful predictive model within its regime. ^[ref]

A conserved quantity is a quantity that remains constant along the time evolution of a system. In Hamiltonian mechanics, a possibly time-dependent phase-space function $F$ is conserved exactly when ${\partial }_{t}F+\{F,H\}=0$.

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.

For a fixed-$N,V,T$ canonical ensemble, sample the normalized Boltzmann distribution in phase space:After integrating out the momenta for an ordinary separable Hamiltonian mechanics, configurational sampling uses $\pi (\mathbf{r})\propto e^{-\beta U(\mathbf{r})}$. The correct equilibrium distribution is ensemble-dependent; the Boltzmann form is the canonical answer, rather than a universal prescription for every possible ensemble.

In Lagrangian mechanics, let a one-parameter group on $Q$ have generator ${\xi }_Q={\xi }^i(q){\partial }_{q^i}$. Its tangent lift acts on $TQ$ asIf $L$ is invariant, ${\xi }_{TQ}L=0$. Along an Euler–Lagrange trajectory, the Noether conserved quantity for a mechanical point symmetry isbecauseThus the conserved quantity is explicitly derived, not merely attached to the name of a symmetry. If the change of $L$ is the total derivative $dB/dt$, then $p_i{\xi }^i-B$ is conserved. For a general point transformation with time component $\tau$, the conserved expression is $p_i{\xi }^i-E_L\tau -B$, provided the variation of $Ldt$ is $dB$. Time-translation invariance of an autonomous Lagrangian therefore yields conservation of energy.

In Hamiltonian mechanics, a cotangent-lifted action preserves $\theta$ and hence $\omega$. For its infinitesimal generator ${\xi }_{T^{\ast }Q}$, Cartan's magic formula giveswhere $J_{\xi }=p_i{\xi }^i$. These components define the canonical moment map $J:T^{\ast }Q\to {\mathfrak{g}}^{\ast }$, with $⟨J,\xi ⟩=J_{\xi }$. If $H$ is invariant, $\{H,J_{\xi }\}=0$, so ${\dot{J}}_{\xi }=\{J_{\xi },H\}=0$. This is the Hamiltonian form of Noether's theorem, and the Legendre transform in mechanics identifies its charges with the Lagrangian ones.

More generally, a Hamiltonian group action is an action whose infinitesimal symplectic generators have globally defined Hamiltonian functions. The existence of those functions is an assumption, not a consequence of every symplectic action. For example, translation on a symplectic two-torus preserves $dq\wedge dp$ but contraction with ${\partial }_q$ gives $dp$, which is closed and not globally exact. Such a symmetry need not have a globally single-valued moment-map component. On $T^{\ast }Q$, the cotangent lift does have the canonical components just derived. A Hamiltonian symmetry preserving $H$ gives a conserved generator; the global momentum-map hypotheses must be stated.

For a finite-dimensional mechanical system let $Q$ be its configuration space, a smooth manifold incorporating the holonomic constraints. Lagrangian mechanics starts with $L:TQ\to \mathbb{R}$; a curve $q(t)$ makes the action stationary under fixed-endpoint variations. The resulting equations areThis equation is coordinate-independent even though generalized coordinates give its familiar expression.

The fibre derivative $\mathbb{F}L(q,v)=(q,p)$, with $p_i=L_{v^i}$, defines the Legendre transform in mechanics. The Cartan form of a regular mechanical Lagrangian isFor a regular Lagrangian, the velocity Hessian is invertible, ${\omega }_L$ is symplectic, and ${\iota }_{{\Gamma }_L}{\omega }_L=d{E}_L$ determines a second-order vector field on $TQ$. In coordinates this recovers the displayed Euler–Lagrange equations: the velocity components of the vector field are $v^i$, and its acceleration components are fixed by the invertible Hessian.

Hamiltonian mechanics takes place on $T^{\ast }Q$. Its tautological one-form is $\theta =p_{i}d{q}^i$, and its canonical symplectic form is $\omega =-d\theta =d{q}^i\wedge d{p}_i$. With ${\iota }_{X_H}\omega =dH$,The associated Poisson bracket is $\{f,g\}=f_{q^i}{g}_{p_i}-f_{p_i}{g}_{q^i}$, so $X_{H}f=\{f,H\}$. For a hyperregular Lagrangian, $\mathbb{F}L$ is a global diffeomorphism, $H=E_L\circ (\mathbb{F}L{)}^{-1}$ and $(\mathbb{F}L{)}^{\ast }\omega ={\omega }_L$. The two formulations then describe the same trajectories. Regularity gives only local equivalence; degenerate Lagrangians require constraint analysis instead. These geometric structures distinguish equations of motion from arbitrary coordinate formulae and make symmetry reduction precise.

For adjoint-valued quantities write $(X\times Y{)}_a={\epsilon }_{abc}{X}_b{Y}_c$. Substituting the infinitesimal transformation into the gauge field strength and using the Jacobi identity givesA useful way to organize the calculation is $[D_{\mu },D_{\nu }]X=g{F}_{\mu \nu }\times X$: the gauge field strength transforms covariantly because the gauge covariant derivative does. Thus $\delta (F_a^{\mu \nu }{F}_{\mu \nu a})=2g{F}_a^{\mu \nu }{\epsilon }_{abc}{F}_{\mu \nu b}{\lambda }_c=0$ by antisymmetry. This proves gauge invariance of the Yang-Mills action.

The Yang-Mills equations are $D_{\mu }{F}^{\mu \nu }=0$. They determine physical evolution only modulo gauge redundancy: gauge transformations with arbitrary spacetime-dependent parameters take solutions to equivalent solutions. In Hamiltonian mechanics, $A_0$ has no independent quadratic time-derivative term, and the corresponding equation is a Gauss law constraint in gauge theory. A gauge parameter can vanish with its necessary derivatives on the initial slice yet change the later potential energy, so the potential energy itself is not uniquely fixed by initial physical data.

In the uncorrected path integral, integration along gauge orbits overcounts equivalent fields. At the perturbative level the quadratic operator of the kinetic term has gauge zero modes and no inverse, so it does not supply a quantum field theory propagator. Gauge fixing together with the Faddeev-Popov determinant removes this obstruction in the local perturbative construction.

For the odd BRST transformation, products obey the graded Leibniz rule. The adjoint product with Grassmann variables satisfiesIn particular, $c\times c$ is not zero, and $c\times Dc=Dc\times c$. The supplied transformations givesince the even covariant derivative obeys $D(c\times c)=2(Dc)\times c$. For the ghost,The final equality is the graded Jacobi identity; in components it is the Jacobi identity for ${\epsilon }_{abc}$ contracted with the totally antisymmetric product of three ghosts. Also $s^2\bar{c}=-sb=0$ and $s^{2}b=0$. The square of an odd derivation is an even derivation, because its two mixed product terms cancel. Hence $s^2=0$ on every polynomial in the fields, off shell, without imposing field equations.

Let $\Psi =\int d^{d}x[{\partial }^{\mu }{\bar{c}}_a{A}_{\mu a}+\xi {\bar{c}}_a{b}_a/2]$ be the gauge-fixing fermion. Since $sF=gF\times c$, $s{\mathcal{L}}_{\mathrm{YM}}=0$. The BRST-exact operator in ${\mathcal{L}}_q={\mathcal{L}}_{\mathrm{YM}}-s\psi$ has zero variation by BRST nilpotence, soTo see the resulting kinetic terms and signs, expand the exact term with $s\bar{c}=-b$:After integration by parts,The auxiliary Nakanishi-Lautrup field $b$ can be completed into a square and integrated out, giving the gauge-fixing term $-(\partial \cdot A_a{)}^2/(2\xi )$ for $\xi \ne 0$. At $\xi =0$, retain $b$ as the multiplier enforcing the gauge condition rather than divide by $\xi$. The Grassmann Gaussian integral gives the determinant of the gauge-condition operator $-\partial \cdot D$. Field-independent normalizations cancel in the normalized integral; insertions involving $b$ can be handled before integration or through its source terms.

At $g=0$ the gauge-fixed quadratic operator is invertible with the usual prescription, and the ghosts have a quadratic kinetic term. The remaining terms are cubic gauge interactions of order $g$, quartic gauge interactions of order $g^2$, and a ghost-gauge interaction of order $g$. Expanding them and using Wick contractions gives a perturbative expansion for any defined polynomial insertion $X$ of the fields. The exponent is the gauge-fixed action $S_q$, as printed in the PDF; the converted TeX incorrectly substitutes $S_A$.

At $\xi =1$, the free gauge action, up to a boundary term, is $\frac{1}{2}\int A_{\mu a}\square A_a^{\mu }$. Its Fourier kernel is $-p^2{\eta }_{\mu \nu }{\delta }_{ab}$. With the mostly-plus convention used above, the Feynman-gauge adjoint propagator isThis is consistent with the scalar-line convention in Question 2.

Finally define the odd functional $B=\frac{1}{2}\int d^{d}x{\bar{c}}_a{b}_a$. Then ${\partial }_{\xi }{S}_q=-sB$. Differentiating the normalized expectation, including its denominator, givesfor an insertion with no explicit $\xi$ dependence. A physical gauge-invariant insertion $X[A]$ is BRST-closed, since its infinitesimal gauge variation vanishes also when the parameter is replaced by $c$. The assumed BRST Ward identity gives $⟨sB⟩=0$ and $⟨s(XB)⟩=0$. For an even $X$ with $sX=0$, the latter says $⟨XsB⟩=0$. ThereforeThe same conclusion holds for any BRST-closed insertion. Mere color-singlet invariance of an arbitrary ghost-dependent expression need not imply $sX=0$; the physical-observable qualification is essential. The proof uses the invariant-measure assumptions encoded in the stated Ward identity.

The passage to Hamiltonian mechanics uses the Legendre transform in mechanics. Assume a regular Lagrangian: the velocity Hessian matrix $({\partial }^{2}L/\partial v^i\partial v^j)$ is invertible. Thendefines a locally invertible map from the tangent bundle to the cotangent bundle. Write its local inverse as $v=v(t,q,p)$ and define the HamiltonianDifferentiating this expression, all terms containing $dv$ cancel because $p_i=L_{v^i}$. HenceThe Euler-Lagrange equations become Hamilton's equations:Conversely, a solution of Hamilton's equations satisfies $\dot{q}=v(t,q,p)$, hence $p=L_v(t,q,\dot{q})$, and $\dot{p}=L_q$ recovers the Euler-Lagrange equations. This proves local equivalence of the two descriptions.

On the cotangent bundle, choose the canonical symplectic form ${\omega }_{\mathrm{can}}={\sum }_{i}d{q}^i\wedge d{p}_i=-d\lambda$ and the convention ${\iota }_{X_H}{\omega }_{\mathrm{can}}=dH$. Then $X_H={\sum }_i(H_{p_i}{\partial }_{q^i}-H_{q^i}{\partial }_{p_i})$, so its integral curves are exactly the phase-space equations above. This sign convention is used throughout these solutions. For a hyperregular Lagrangian, the Legendre transform in mechanics is globally invertible and gives global equivalence; regularity alone only gives local equivalence. A singular velocity Hessian matrix may instead produce constraints, so the ordinary unconstrained argument does not apply to every Lagrangian.

In Hamiltonian mechanics, the canonical momenta areHence

Complete the square in the velocity variable:Since $m>0$, the final term is nonpositive and vanishes at exactly one velocity. ThereforeThis is the Legendre transform in mechanics that constructs the Hamiltonian mechanics energy from the kinetic-minus-potential Lagrangian.

A symplectic basis organizes a nondegenerate alternating bilinear form into independent pairs. Let $(V,\omega )$ be a finite-dimensional vector space over a field $K$, with $\omega$ a nondegenerate bilinear form satisfying $\omega (v,v)=0$ for every vector $v$. Its dimension is even, say $2n$, and a symplectic basis is an ordered basis $\mathcal{B}=(e_1,\dots ,e_n,f_1,\dots ,f_n)$ such thatHere ${\delta }_{ij}$ is the Kronecker delta. Each $(e_i,f_i)$ is a hyperbolic pair; different pairs have zero mutual pairing. These conventions apply in every characteristic. In characteristic two, $-1=1$, but the condition $\omega (v,v)=0$ is still essential: a symmetric bilinear form need not be an alternating bilinear form. The existence theorem and the characteristic caveat are given in Bilinear Forms by Keith Conrad (Keith Conrad, Bilinear Forms, §1, Theorem 1.6; §5, Theorems 5.1 and 5.4 and Definition 5.3).

The benefit is a common coordinate language for every symplectic vector space of the same finite dimension over the same field. The span of all $e_i$, and separately of all $f_i$, is a Lagrangian subspace; each paired span is a symplectic subspace. In Hamiltonian mechanics, the corresponding pairing of position and momentum coordinates describes phase space and canonical transformations. On a symplectic manifold, a symplectic basis describes one tangent space; Darboux coordinates additionally describe a whole local neighborhood.^[ref]