An elastic beam is a slender body that supports bending moments and transverse loads. In the small-slope Euler-Bernoulli approximation its bending moment is $M=EI{y}_{xx}$, and transverse momentum balance gives $\rho A{y}_{tt}+EI{y}_{xxxx}=q$ for a uniform beam under distributed force per length $q$. Here $I$ is the second moment of area; boundary conditions specify the end constraints and applied moments and forces.

A symmetric stress tensor is in static elastic equilibrium when its divergence balances the prescribed body force. Its boundary traction is $t_i={\sigma }_{ij}{n}_j$. With zero body force, integration by parts gives ${\int }_{\Omega }\sigma :\mathrm{sym}\nabla v={\int }_{\partial \Omega }t\cdot v$. This is the weak principle of virtual work for a static elastic body.

A perfectly bonded elastic interface has continuous displacement field and continuous traction, with the same chosen normal on both sides. For piecewise affine displacement fields, the difference of their gradients annihilates every tangential vector, and hence has the form $a\otimes n$. Their infinitesimal strain tensors differ by $\mathrm{sym}(a\otimes n)$.

An elastic deformation is recoverable when the deforming loads are removed. In a local elastic constitutive model the stress depends on the present deformation. When this response also derives from a stored elastic energy, the material is a hyperelastic material. In the small-deformation approximation, the relevant kinematic variable is the infinitesimal strain tensor.

Einstein synchronisation sets clocks at rest in one inertial frame using exchanged light signals. If the sending clock records departure at $t_A$ and return at $t_A^{\prime }$, the distant clock's reflection reading is set to $t_B=(t_A+t_A^{\prime })/2$. This convention assigns equal outgoing and incoming travel times and defines the frame's distant simultaneity. It does not set the arrival time of an image equal to the time of the remote spacetime event.

The four-dimensional Majumdar–Papapetrou solution is a static multi-black-hole Einstein-Maxwell solution with $H=1+{\sum }_i{M}_i/|\mathbf{x}-{\mathbf{x}}_i|$ and positive geometrized masses $M_i$. The electromagnetic potential can be written $A=\pm (H^{-1}-1)dt$, with a common sign of charge. Its mass at infinity is ${\sum }_i{M}_i$. Each distinct centre is an extremal horizon of zero surface gravity rather than a pointlike naked metric singularity.

With the metric and stress-energy tensor fixed and zero cosmological constant, changing the sign convention for the Riemann curvature tensor changes the signs of the Ricci tensor, scalar curvature and Einstein tensor. The same physical field equation is therefore written with the opposite coupling sign. For signature $(+---)$, the convention $R^a{}_{bcd}={\partial }_d{\Gamma }_{bc}^a-{\partial }_c{\Gamma }_{bd}^a-{\Gamma }_{ec}^a{\Gamma }_{bd}^e+{\Gamma }_{ed}^a{\Gamma }_{bc}^e$ and $R_{ab}=R^c{}_{acb}$ gives $R_{00}\simeq -\Delta \Phi$ in a static Newtonian limit. The trace-reversed equation and $\Delta \Phi =4\pi G\rho$ then require $G_{ab}=-8\pi G{T}_{ab}$.

Assume a metric theory in four dimensions, a Levi-Civita connection, covariant stress-energy conservation and field equations linear in second metric derivatives. The contracted Bianchi identity selects $G_{ab}=R_{ab}-g_{ab}R/2$. Its trace-reversed equation is $R_{ab}=\kappa (T_{ab}-g_{ab}T/2)+\Lambda g_{ab}$. In the convention $R_{00}\simeq -{\nabla }^2\phi$ and signature $(+---)$, nonrelativistic matter gives $R_{00}\simeq \kappa \rho /2$. The Poisson equation ${\nabla }^2\phi =4\pi G\rho$ fixes $\kappa =-8\pi G$. Reversing the curvature convention reverses this displayed coupling sign without changing the physical theory. The strong equivalence principle alone does not establish uniqueness without the stated additional assumptions.

With signature $(+---)$, let $R^a{}_{bcd}={\partial }_d{\Gamma }^a{}_{bc}-{\partial }_c{\Gamma }^a{}_{bd}-{\Gamma }^a{}_{pc}{\Gamma }^p{}_{bd}+{\Gamma }^a{}_{pd}{\Gamma }^p{}_{bc}$ and $R_{ab}=R^c{}_{acb}$. This curvature is the negative of the convention in which $[{\nabla }_c,{\nabla }_d]X^a=R^a{}_{bcd}{X}^b$. The physical Einstein field equations are $G_{ab}+\Lambda g_{ab}=-8\pi G{T}_{ab}/c^4$. In a static weak field $R_{00}=-{\nabla }^2\Phi /c^2$; the trace-reversed equation then reproduces the Poisson equation ${\nabla }^2\Phi =4\pi G{\rho }_{\mathrm{mass}}$. Metric variation of $\int \sqrt{-g}(R-2\Lambda )/(2\kappa )+S_m$ gives the same sign when $\delta S_m=\int \sqrt{-g}{T}_{ab}\delta g^{ab}/2$.

A low-energy effective action retains the light degrees of freedom and describes the effects of heavier ones through local interactions and a derivative expansion. It can be obtained by matching an effective field theory or by a low-energy compactification. A Wilsonian effective action implements integration of high-energy modes with a scale cutoff. The quantum effective action that generates one-particle-irreducible vertices is a different construction; the two need not coincide.

An effective operator is a local interaction built from the retained fields in an effective field theory. Its coefficient records effects of heavier degrees of freedom, and allowed operators respect the relevant symmetries. In four dimensions an operator of mass dimension ${\Delta }_i$ has the displayed coefficient for dimensionless $C_i$ and a heavy mass scale $M$. For example a dimension-six proton-decay operator has a coefficient proportional to $M^{-2}$.

A nearly Hamiltonian system is a weak perturbation of a Hamiltonian system, writtenSince $J$ is skew-symmetric, its energy changes by $\dot{H}=\epsilon \nabla H\cdot R$. This identity is exact; it does not require replacing the perturbed trajectory by an unperturbed one.

For a smooth regular family of planar periodic Hamiltonian orbits labeled by energy $E$, the first-order change over one unperturbed period isAssume the family forms a regular annulus, the return section is transverse and $M(E_0)=0$, $M^{\prime }(E_0)\ne 0$. The return energy then has expansion $P_{\epsilon }(E)-E=\epsilon M(E)+O({\epsilon }^2)$. Dividing by $\epsilon$ and applying the implicit function theorem produces a fixed return energy $E(\epsilon )$ near $E_0$, hence a nearby periodic orbit for small perturbations.

These planar regular-orbit hypotheses are part of the persistence result. A single averaged energy equation is not automatically sufficient in a higher-dimensional phase space, and a multiple zero requires higher-order terms. Near an infinite-period separatrix, the Melnikov energy-balance method instead measures manifold splitting along a homoclinic orbit or a heteroclinic orbit.

A smooth flow on a smooth manifold is a smooth map $(t,x)\mapsto {\Phi }_t(x)$ withThe identity implies ${\Phi }_{-t}={\Phi }_t^{-1}$, so each time map is a diffeomorphism. Its generating vector field is $F(x)={\partial }_t{\Phi }_t(x){|}_{t=0}$; trajectories solve the autonomous differential equation $\dot{x}=F(x)$.

For example, ${\Phi }_t(x)=x+t$ is a smooth flow on the real line, while ${\Phi }_t(\theta )=\theta +t(\mathrm{mod}2\pi )$ is a rotation flow on a circle. The second has periodic trajectories, even though the same additive time parameter acts in both examples.

A smooth vector field initially gives only a local flow. The equation $\dot{x}=x^2$ has $x(t)=x_0/(1-x_{0}t)$, which blows up at $t=1/x_0$ for $x_0>0$. On a closed manifold, compactness prevents escape in finite time and local existence can be continued for every real time. Thus every smooth vector field there generates a global smooth flow. On a noncompact manifold, global existence is a separate property.

A hyperbolic toral automorphism is a toral automorphism $T_{A}x=Ax(\mathrm{mod}{\mathbb{Z}}^d)$ whose matrix has no eigenvalue on the unit circle. Here $A\in \mathrm{GL}(d,\mathbb{Z})$, so its determinant is $\pm 1$. Invertibility merely over the real numbers would not suffice: multiplication by two on a circle is a many-to-one endomorphism, not an automorphism.

The real stable and unstable subspaces come from the generalized eigenspaces with eigenvalue modulus less than and greater than one. They give the same splitting at every point of the real torus. A Jordan block can add a polynomial factor to the powers of its eigenvalue; choosing a slightly weaker exponential rate absorbs that factor into a constant. Thus contraction forward on the stable subspace and backward on the unstable subspace is uniform over the whole torus, proving that the map is an Anosov diffeomorphism.

For the cat map, $A=\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$ has eigenvalues $(3\pm \sqrt{5})/2$. One is inside and the other outside the unit circle, and their product is one. The toral automorphism mixing criterion applies as well, since a root of unity would lie on that circle.

An Anosov diffeomorphism is a continuously differentiable diffeomorphism of a compact smooth manifold with an invariant splitting of its tangent bundle into stable and unstable vector subbundles satisfying the displayed uniform exponential estimates for $n\ge 0$, with $C,\lambda >0$. Hyperbolic toral automorphisms give examples. An Anosov flow has a further one-dimensional invariant bundle along the flow direction; its definition is a distinct continuous-time version of uniform hyperbolicity.

For an equilibrium point $p$, a homoclinic orbit is a nonconstant trajectory $x(t)$ withAt a hyperbolic equilibrium point, the trajectory leaves a neighborhood of $p$ along its unstable manifold and returns along its stable manifold. It never reaches $p$ in finite time when local solution uniqueness holds. Its closure includes the equilibrium and can form a loop-shaped separatrix.

A completely explicit example iswith Hamiltonian function $H(x,v)=v^2/2-x^2/2+x^4/4$. Its derivative is zero along every trajectory, and the origin is a saddle point with eigenvalues $\pm 1$. The curvelies on $H=0$. Differentiating gives $\dot{x}=v$ and $\ddot{x}=x-x^3$, while the hyperbolic secant tends to zero at both time infinities. Thus it is a homoclinic orbit. Changing both signs gives the second loop.

Each loop bounds periodic orbits around one potential minimum. This is an integrable Hamiltonian system, so a homoclinic loop by itself does not establish chaos. A Shilnikov bifurcation and a simple-zero Melnikov energy-balance method calculation require additional dimensional, spectral or forcing hypotheses.

Coexistence of two stable long-time states at fixed parameters. Initial conditions and sufficiently strong perturbations can select different basins of attraction, allowing a dynamical system to retain a state-dependent memory.

Multistability is coexistence of several stable long-term behaviors at the same fixed parameter values. Those behaviors can be equilibrium points, periodic orbits or more complicated attractors. Different basins of attraction select which behavior a trajectory reaches from its initial state.

For example, $\dot{x}=x-x^3$ has attracting equilibria at $\pm 1$ and a repelling equilibrium at zero. A sufficiently large perturbation can move a state from one basin to the other, leaving a persistent change after the perturbation ends. This gives a simple mechanism for dynamical memory without changing the governing parameters. Bistability is the two-state case. Hysteresis additionally describes history-dependent switching under a changed control parameter; coexistence alone does not specify a complete hysteresis loop or its switching thresholds.

A Lyapunov exponent is the asymptotic exponential growth rate of an infinitesimal tangent displacement along a dynamical system. For a differentiable flow map, a starting point $x$ and a nonzero tangent vector $v$, one convention iswhen the limit exists. A discrete dynamical system uses the derivative of the $n$th iterate and divides by $n$ instead. The rate can depend on the point and the direction.

For the linear ordinary differential equation $\dot{\eta }=J\eta$, an eigenvector with real eigenvalue $a$ evolves as $e^{at}v$, so its exponent is $a$. Nonreal eigenvalues contribute their real parts to the growth rate. A polynomial factor from a Jordan block does not change the exponent because $t^{-1}\log t^m\to 0$.

In a smooth velocity field, the tangent displacement obeys $\dot{\eta }=(\nabla u)\eta$. Its instantaneous stretching rate depends on both the current Jacobian matrix and the current direction. Those can change along the trajectory, so averaging instantaneous largest eigenvalues need not give the actual largest Lyapunov exponent.

The infinitesimal qualification matters. Two trajectories at a fixed finite separation cannot grow apart exponentially forever inside a bounded region: their distance is bounded by the region's diameter. A positive tangent exponent instead concerns the derivative obtained by sending the initial perturbation to zero first. Finite errors follow the linear prediction only until they become large enough that nonlinear effects matter. Likewise, growth rates of moments of a random separation need not equal the exponent along a typical trajectory.

A reflection-symmetric planar cubic model can be expressed in polar variables $S=u^2+v^2$, $u=\sqrt{S}\cos \theta$, $v=\sqrt{S}\sin \theta$ as $\dot{S}=2S(\mu +\sin 2\theta -S/2)$ and $\dot{\theta }=\sigma +\cos 2\theta -S/2$. A nonzero equilibrium therefore has $\sin 2\theta =S/2-\mu$, $\cos 2\theta =S/2-\sigma$. Adding their squares gives $S^2/2-(\mu +\sigma )S+{\mu }^2+{\sigma }^2-1=0$. Each positive root $S=\mu +\sigma \pm \sqrt{2-(\mu -\sigma {)}^2}$ determines an antipodal equilibrium pair. The determinant of the polar linearization there is $2S(S-\mu -\sigma )$ and its trace is $2(\mu -S)$. These formulas distinguish folds of nonzero equilibria from symmetry-forced bifurcations at $S=0$.