hcju/proofwikips huggingface dataset:
Definition:Linear Measure/Thickness
'''Thickness''', like breadth, is used as a term for linear measure in a dimension perpendicular to both length and depth. However, whereas breadth has connotations of '''across''', '''thickness''' is used for distances '''through''' the solid figure.
Definition:Kinetic Energy
Let $B$ be a body. The '''kinetic energy''' of $B$ is defined as the energy appertaining to $B$ due to its motion. It is a scalar quantity.
Definition:Potential Energy
Let $B$ be a body. The '''potential energy''' of $B$ is the energy appertaining to $B$ as a result of its position in a force field or because of the internal configuration of its components.
Definition:Force Field
A '''force field''' is a field in which the physical quantity associated with every point is a force which applies to a body placed at that point.
Definition:Constant (Physics)
A physical quantity is defined as being '''constant''' if it does not change with the passage of time.
Definition:Pendulum/Ideals
The ideals of a pendulum are as follows: :The rod is of zero thickness, infinite stiffness and zero mass. :The pivot is immovable and  has zero friction. :The bob is of zero volume. :The gravitation field is (usually) taken to be constant and uniform. :The medium in which the pendulum moves offers no resistive force, unless otherwise specifically stated.
Definition:Pendulum/Bob
The point mass at the free end of the rod of a pendulum is called the '''(pendulum) bob'''.
Definition:Pendulum
A '''pendulum''' is an ideal body consisting of two parts: :$(1): \quad$ a rod attached at one end to a pivot :$(2): \quad$ a point mass at the other end of the rod.
Definition:Rod
A '''rod''' is a rigid straight body whose width and mass are approximated to zero.
Definition:Escape Velocity
Let a body be propelled vertically upwards from the surface of a planet Let the body keep travelling away from the planet indefinitely. Let resistance from the atmosphere of the planet be considered as negligible. The smallest initial speed at which this is possible is known as the '''escape velocity''' of the planet.
Definition:Terminal Velocity
Let a body travelling under a constant force through a medium which resists its progress. The '''terminal velocity''' of the body is the limiting speed through that medium for the given force.
Definition:Speed/Also known as
Some authors erroneously or carelessly refer to the '''speed''' of a body as its velocity. But this is technically wrong if the author does not specify its direction as well as its magnitude.
Definition:Sine/Real Function/Arch
Each section of the sine function between adjacent zeroes is called an '''arch''' of the sine function
Definition:Lemniscate of Bernoulli/Lobe
Each of the two loops that constitute the '''lemniscate''' can be referred to as a '''lobe''' of the lemniscate.
Definition:Brachistochrone
Let a point $A$ be joined by a wire to a lower point $B$. Let the wire be allowed to be bent into whatever shape is required. Let a bead be released at $A$ to slide down without friction to $B$. The shape of the wire so that the bead takes least time to descend from $A$ to $B$ is called the '''brachistochrone'''.
Definition:Wire
A '''wire''' is a rigid body modelled by a specific curve whose width and mass are approximated to zero.
Definition:Cycloid/Generating Circle
The circle which rolls along the straight line is called the '''generating circle''' of the cycloid.
Definition:Confocal Conics
Let $F$ be a set of conic sections. Then $F$ is a set of '''confocal conics'''  they all share the same foci.
Definition:Self-Orthogonal Trajectories
Let $\map f {x, y, c}$ define a one-parameter family of curves $F$. Let the family of orthogonal trajectories of $F$ be $F$ itself. Then $\map f {x, y, c}$ is '''self-orthogonal'''.
Definition:One-Parameter Family of Curves/Parameter
The value $c$ is the '''parameter''' of $F$.
Definition:T1/2 Space
Let $T = \left({S, \tau}\right)$ be a topological space. Then $T$ is a '''$T_{\frac 1 2}$ space''' : :$\forall A \subseteq S: A'$ is closed where $A'$ denotes the derivative of $A$.
Definition:Confocal Ellipses
Let $F$ be a set of ellipses. Then $F$ is a set of '''confocal ellipses'''  they all share the same foci.
Definition:Confocal Hyperbolas
Let $F$ be a set of hyperbolas. Then $F$ is a set of '''confocal hyperbolas'''  they all share the same foci.
Definition:Asymptote
:500pxrightthumbAsymptotes to $y = x + \dfrac 1 x$ (blue): [[Definition:Y-Axis$y$-axis, and the line $x = y$ (red)]] Let $C$ be a curve in the plane. Let $L$ be a straight (infinite) line such that the distance between $C$ and $L$ approaches zero as they tend to infinity. Then $L$ is an '''asymptote to''' (or '''of''') $C$.
Definition:Conic Section/Center
The '''center''' of a section is the point midway between the foci. === Center of Circle === The circle is usually treated differently. From Circle has Two Coincident Foci, the foci and the center are all the same point. :300px {{:Definition:Circle/Center}} === Center of Ellipse === Let $K$ be an ellipse. :500px {{:Definition:Ellipse/Center}} === Center of Hyperbola === Let $K$ be a hyperbola. :500px {{:Definition:Hyperbola/Center}} === Center of Parabola === {{:Definition:Parabola/Center}}
Definition:Conic Section/Reduced Form
Let $K$ be a conic section. Let $K$ be embedded in a cartesian plane such that: :$(1)$ the center is at the origin :$(2)$ the foci are at $\left({\pm c, 0}\right)$ for some $c \in \R_{\ge 0}$. This can be interpreted in the contexts of the specific classes of conic section as follows: === Reduced Form of Ellipse === {{:Definition:Conic Section/Reduced Form/Ellipse}} === Reduced Form of Hyperbola === {{:Definition:Conic Section/Reduced Form/Hyperbola}} === Reduced Form of Circle === {{:Definition:Conic Section/Reduced Form/Circle}} === Reduced Form of Parabola === {{:Definition:Conic Section/Reduced Form/Parabola}}
Definition:Parabola/Center
The parabola has no center. From Parabola has One Focus, the definition of the center as being the point midway between the foci, it follows that for a parabola the definition is inapplicable.
Definition:Conic Section/Reduced Form/Ellipse
Let $K$ be an ellipse embedded in a cartesian plane. $K$ is in '''reduced form''' : :$(1)$ its major axis is aligned with the $x$-axis :$(2)$ its minor axis is aligned with the $y$-axis. :500px
Definition:Conic Section/Reduced Form/Hyperbola
Let $K$ be a hyperbola embedded in a cartesian plane. $K$ is in '''reduced form''' : :$(1)$ its major axis is aligned with the $x$-axis :$(2)$ its minor axis is aligned with the $y$-axis. :500px
Definition:Conic Section/Reduced Form/Circle
Let $K$ be a circle embedded in a cartesian plane. $K$ is in '''reduced form'''  its center is located at the origin. :500px
Definition:Conic Section/Reduced Form/Parabola
Let $K$ be a parabola embedded in a cartesian plane. As a Parabola has no Center, it is not possible to define the reduced form of a parabola in the same way as for the other classes of conic section. Instead, $K$ is in '''reduced form''' : :$(1)$ its focus is at the point $\tuple {c, 0}$ :$(2)$ its directrix is aligned with the line $x = -c$ for some $c \in \R_{> 0}$. :400px
Definition:Hyperbola/Equidistance
&nbsp; :400px Let $F_1$ and $F_2$ be two points in the plane. Let $d$ be a length less than the distance between $F_1$ and $F_2$. Let $K$ be the locus of points $P$ which are subject to the condition: :$\size {d_1 - d_2} = d$ where: :$d_1$ is the distance from $P$ to $F_1$ :$d_2$ is the distance from $P$ to $F_2$ :$\size {d_1 - d_2}$ denotes the absolute value of $d_1 - d_2$. Then $K$ is a '''hyperbola'''.
Definition:Set of Condensation Points
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. The '''set of condensation points''' of $A$ is the set of all condensation points of $A$, :$A^0 = \left\{{x \in S:x}\right.$ is condensation point of $\left.{A}\right\}$
Definition:Homogeneous Function/Real Space/Degree
The integer $n$ is known as the '''degree''' of $f$.
Definition:Homogeneous Function/Real Space
Let $f: \R^2 \to \R$ be a real-valued function of two variables. $f \left({x, y}\right)$ is a '''homogeneous function of degree zero''' : :$\exists n \in \Z: \forall t \in \R: f \left({t x, t y}\right) = t^n f \left({x, y}\right)$
Definition:Homogeneous Function/Real Space/Zero Degree
$f \left({x, y}\right)$ is a '''homogeneous function of degree zero''' or '''of zero degree''' : :$\forall t \in \R: f \left({t x, t y}\right) = t^0 f \left({x, y}\right) = f \left({x, y}\right)$
Definition:Linear Ordinary Differential Equation
A '''linear ordinary differential equation''' is an '''ordinary differential equation''' where all dependent variables and their derivatives appear to the first power. Neither are products of dependent variables allowed.
Definition:Linear Second Order Ordinary Differential Equation
A '''linear second order ordinary differential equation''' is a differential equation which is in (or can be manipulated into) the form: :$\dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = \map R x$ where, as is indicated by the notation, $\map P x$, $\map Q x$ and $\map R x$ are functions of $x$ alone (or constants).
Definition:Niemytzki Plane
The '''Niemytzki plane''' is the topological space $T = \struct {S, \tau}$ defined as: {{begin-eqn}} {{eqn | l = S       | r = \set {\tuple {x, y} \in \R^2: y \ge 0} }} {{eqn | l = \map \BB {x, y}       | r = \set {\map {B_r} {x, y} \cap S: r > 0}       | c = if $x, y \in \R, y > 0$ }} {{eqn | l = \map \BB {x, 0}       | r = \set {\map {B_r} {x, r} \cup \set {\tuple {x, 0} }: r > 0}       | c = if $x \in \R$ }} {{eqn | l = \tau       | r = \set {\bigcup \GG: \GG \subseteq \bigcup_{\tuple {x, y} \mathop \in S} \map \BB {x, y} } }} {{end-eqn}} where $\map {B_r} {x, y}$ denotes the open $r$-ball of $\tuple {x, y}$ in the $\R^2$ Euclidean space.
Definition:Neighborhood System
Let $T = \struct {S, \tau}$ be a topological space. A '''neighborhood system''' is a family $\family {\NN_x}_{x \mathop \in S}$ indexed by points of $S$, such that $\NN_x$ is a local basis at $x$ for $x \in S$.
Definition:Tractrix
Let $S$ be a cord situated as a (straight) line segment whose endpoints are $P$ and $T$. Let $T$ be dragged in a direction perpendicular to the straight line in which $S$ is aligned. The curve along which $P$ travels is known as a '''tractrix'''. :400px
Definition:Tractrix/Linguistic Note
The word '''tractrix''' derives from the Latin '''traho (trahere, traxi, tractum)''' meaning '''to pull''' or '''to drag'''. The plural is '''tractrices'''.
Definition:Pursuit Curve
A '''pursuit curve''' is a curve constructed to model pursuers and pursuees. The '''pursuit curve''' is the curve traced by the pursuers. The pursuee is always on the pursuer's tangent.
Definition:Rate of Change
The '''rate of change''' of a physical quantity is its (first) derivative with respect to time. Loosely speaking, it means '''how fast something changes''', with a wider scope than change in physical displacement. === Dimension === {{:Definition:Rate of Change/Dimension}}
Definition:Steady-State
=== First Order ODE === {{:Definition:Steady-State/First Order ODE}} === Second Order ODE === {{:Definition:Steady-State/Second Order ODE}}
Definition:Steady-State/Electronics
The electric current $I$ in $K$ is given by the equation: :$I = \dfrac {E_0} R + \left({I_0 - \dfrac {E_0} R}\right) e^{-R t / L}$ The term $\dfrac {E_0} R$ is known as the '''steady-state''' part of $(1)$.
Definition:Transient/Electronics
The electric current $I$ in $K$ is given by the equation: :$I = \dfrac {E_0} R + \left({I_0 - \dfrac {E_0} R}\right) e^{-R t / L}$ The term: :$\left({I_0 - \dfrac {E_0} R}\right) e^{-R t / L}$ which dies away to zero over time, is known as the '''transient''' part.
Definition:Transient
Consider the Decay Equation: :$\dfrac {\d y} {\d x} = k \paren {y_a - y}$ where: :$k \in \R: k > 0$ :$y = y_0$ at $x = 0$ which has the particular solution: :$(1): \quad y = y_a + \paren {y_0 - y_a} e^{-k x}$ The term: :$\paren {y_0 - y_a} e^{-k x}$ which tends to zero with increasing $x$, is known as the '''transient''' part. === Electronics === The term is often seen in the context of the current in an $LR$ electric circuit $K$: {{:Definition:Transient/Electronics}}
Definition:Electronics
'''Electronics''' is the branch of electromagnetism which studies how to control electricity.
Definition:Barrier
A complex function $\varphi \in \map \C {\overline \Omega}$ is a '''barrier''' for $\Omega$ at $z \in \partial \Omega$ : :$\varphi$ is subharmonic :$\map \varphi z = 0$ :$\varphi < 0$ on $\partial \Omega \setminus \set z$ {{explain|Define $\map \C {\overline \Omega}$}} {{explain|Define $\Omega$}} {{explain|Define $\partial \Omega$}} {{explain|What is the domain and range of $\varphi$?}} {{explain|If $\varphi$ is a complex function, its range is also complex, and so cannot be ordered, so the meaning of $\varphi < 0$ needs to be explained carefully}}
Definition:Surface of Revolution
Let $F$ be a plane figure. Let $L$ be a straight line in the same plane as $F$. Let $F$ be rotated one full turn about $L$ in $3$-dimensional space. The '''surface of revolution generated by $F$ around $L$''' is the surface of the solid of revolution generated by $F$ around $L$.
Definition:Riccati Equation
The '''Riccati equation''' is the first order ordinary differential equation: :$y' = p \left({x}\right) + q \left({x}\right) y + r \left({x}\right) y^2$
Definition:Special Riccati Equation
The '''special Riccati equation''' is the first order ordinary differential equation: :$y' + b y^2 = c x^m$ {{NamedforDef|Jacopo Francesco Riccati|cat = Riccati}}
Definition:Burnout Velocity
Let $R$ be a rocket which is propelled by backwards expulsion of the exhaust products of the burning of its internal fuel supply. Let $R$ be fired straight up from the surface of a planet. The '''burnout velocity''' of $R$ is the velocity attained by $R$ at the instant when its fuel has become exhausted.
Definition:Dynamics
'''Dynamics''' is the branch of the mechanics describing the motion of bodies under the influence of forces.
Definition:Burnout Height
Let $R$ be a rocket which is propelled by backwards expulsion of the exhaust products of the burning of its internal fuel supply. Let $R$ be fired straight up from the surface of a planet. The '''burnout height''' of $R$ is the height attained by $R$ at the instant when its fuel has become exhausted.
Definition:Homogeneous Linear Second Order ODE
A '''homogeneous linear second order ODE''' is a differential equation which is in (or can be manipulated into) the form: :$\dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ where, as is indicated by the notation, $\map P x$ and $\map Q x$ are functions of $x$ alone (or constants).
Definition:Nonhomogeneous Linear Second Order ODE
A '''nonhomogeneous linear second order ODE''' is a differential equation which is in (or can be manipulated into) the form: :$\dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = \map R x$ where: :$\map P x$, $\map Q x$ and $\map R x$ are functions of $x$ alone (or constants) :$\map R x$ is not the zero constant function. That is, a '''nonhomogeneous linear second order ODE''' is a linear second order ODE which is not homogeneous.
Definition:Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE
Let $C_1$ and $C_2$ be real numbers. Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ Then: :$C_1 \, \map {y_1} x + C_2 \, \map {y_2} x$ is known as a '''linear combination of particular solutions to $(1)$.
Definition:Linearly Dependent Real Functions
Let $f \left({x}\right)$ and $g \left({x}\right)$ be real functions defined on a closed interval $\left[{a \,.\,.\, b}\right]$. Let $f$ and $g$ be constant multiples of each other: :$\exists c \in \R: \forall x \in \left[{a \,.\,.\, b}\right]: f \left({x}\right) = c g \left({x}\right)$ or: :$\exists c \in \R: \forall x \in \left[{a \,.\,.\, b}\right]: g \left({x}\right) = c f \left({x}\right)$ Then $f$ and $g$ are '''linearly dependent'''.
Definition:Linearly Independent Real Functions
Let $f \left({x}\right)$ and $g \left({x}\right)$ be real functions defined on a closed interval $\left[{a \,.\,.\, b}\right]$. Let $f$ and $g$ be such that they are not linearly dependent. Then $f$ and $g$ are '''linearly independent'''.
Definition:Wronskian/General Definition
Let $\map {f_1} x, \map {f_2} x, \dotsc, \map {f_n} x$ be real functions defined on a closed interval $\closedint a b$. Let $f_1, f_2, \ldots, f_n$ be $n - 1$ times differentiable on $\closedint a b$. The '''Wronskian''' of $f_1, f_2, \ldots, f_n$ on $\closedint a b$ is defined as: :$\map W {f_1, f_2, \dotsc, f_n} = \begin {vmatrix} \map {f_1} x & \map {f_2} x & \cdots & \map {f_n} x \\ \map { {f_1}'} x & \map { {f_2}'} x & \cdots & \map { {f_n}'} x \\ \vdots & \vdots & \ddots & \vdots \\ \map { {f_1}^{\paren {n - 1} } } x & \map { {f_2}^{\paren {n - 1} } } x & \cdots & \map { {f_n}^{\paren {n - 1} } } x \\ \end{vmatrix}$ where: :$\begin{vmatrix} \cdots \end{vmatrix}$ denotes the determinant :$\map { {f_1}^{\paren {n - 1} } } x$ denotes the $n - 1$th derivative of $f_1$.
Definition:Wronskian
Let $\map f x$ and $\map g x$ be real functions defined on a closed interval $\closedint a b$. Let $f$ and $g$ be differentiable on $\closedint a b$. The '''Wronskian''' of $f$ and $g$ is defined as: :$\map W {f, g} = \begin {vmatrix} \map f x & \map g x \\ \map {f'} x & \map {g'} x \\ \end {vmatrix} = \map f x \, \map {g'} x - \map g x \, \map {f'} x$ === General Definition === {{:Definition:Wronskian/General Definition}}
Definition:Homogeneous Linear Second Order ODE with Constant Coefficients
A '''homogeneous linear second order ODE with constant coefficients''' is a second order ODE which can be manipulated into the form: :$y'' + p y' + q y = 0$ where $p$ and $q$ are real constants. Thus it is a homogeneous linear second order ODE: :$y'' + \map P x y' + \map Q x y = 0$ where $\map P x$ and $\map Q x$ are constant functions.
Definition:Auxiliary Equation
Let: :$(1): \quad y'' + p y' + q y = 0$ be a constant coefficient homogeneous linear second order ODE. The '''auxiliary equation''' of $(1)$ is the quadratic equation: :$m^2 + p m + q = 0$
Definition:Cauchy-Euler Equation
The linear second order ordinary differential equation: :$x^2 \dfrac {\d^2 y} {\d x^2} + p x \dfrac {\d y} {\d x} + q y = 0$ is the '''Cauchy-Euler equation'''.
Definition:Cauchy-Euler Equation/General Form
Let $n \in \Z_{>0}$ be a strictly positive integer. The linear ordinary differential equation: :$a_n x^n \, \map {y^{\paren n} } x + \dotsb + a_1 x \, \map {y'} x + a_0 \, \map y x = 0$ is the '''$n$th order Cauchy-Euler equation'''.
Definition:Directed Subset
Let $\struct {S, \precsim}$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is a '''directed subset''' of $S$ : :$\forall x, y \in H: \exists z \in H: x \precsim z$ and $y \precsim z$
Definition:Filtered Subset
Let $\left({S, \precsim}\right)$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is a '''filtered subset''' of $S$ : :$\forall x, y \in H: \exists z \in H: z \precsim x$ and $z \precsim y$
Definition:Spring/Ideal
An '''ideal spring''' obeys Hooke's Law: :$\mathbf F = -k \mathbf x$ where: : $\mathbf F$ is the force caused by a displacement $\mathbf x$ : $k$ is the constant of proportion. The negative sign indicates that the force pulls in the opposite direction to that of the displacement imposed.
Definition:Spring/Force Constant
The constant of proportion: :$k \in \R_{>0}$ is known as the '''(spring) force contant'''. Its value depends on the particular spring under investigation. The SI unit of the '''force constant''' is newtons per metre: $\mathrm N \ \mathrm m^{-1}$.
Definition:Wall
A '''wall''' is an immovable vertical plane in space to which other physical objects may be attached.
Definition:Cart
A '''cart''' is a body with a given mass which is free to move with zero friction in a single dimension, that is, back and forth. It is usually imagined as having wheels on which it can roll.
Definition:Immovable
An '''immovable''' body is one whose position cannot be changed by applying a force to it. It can be considered as having an infinite mass and zero momentum.
Definition:Spring/Equilibrium Position
The '''equilibrium position''' of a body $B$ attached to a spring $S$ is the position it occupies when $S$ is exerting no force upon $B$. For an ideal spring obeying Hooke's Law $\mathbf F = -k \mathbf x$, the '''equilibrium position''' is set to be the point $\mathbf x = \bszero$.
Definition:Simple Harmonic Motion/Amplitude
The parameter $A$ is known as the '''amplitude''' of the motion.
Definition:Simple Harmonic Motion
Consider a physical system $S$ whose motion can be expressed in the form of the following equation: :$x = A \map \sin {\omega t + \phi}$ where $A$ and $\phi$ are constants. Then $S$ is in a state of '''simple harmonic motion'''. === Amplitude === {{:Definition:Simple Harmonic Motion/Amplitude}} === Period === {{:Definition:Simple Harmonic Motion/Period}} === Frequency === {{:Definition:Simple Harmonic Motion/Frequency}}
Definition:Simple Harmonic Motion/Period
The '''period''' $T$ of the motion of $S$ is the time required for one complete cycle: :$T = \dfrac {2 \pi} \omega$
Definition:Simple Harmonic Motion/Frequency
The '''frequency''' $\nu$ of the motion of $S$ is the number of complete cycles per unit time: :$\nu = \dfrac 1 T = \dfrac \omega {2 \pi}$
Definition:Periodic Function/Frequency
The '''frequency''' $\nu$ of $f$ is the reciprocal of the period $L$ of $f$: :$\nu = \dfrac 1 L$
Definition:Fluid
A '''fluid''' is a substance which has no defined shape, but which is free to flow according to external forces. === Ideal Fluid === {{:Definition:Fluid/Ideal}}
Definition:Medium
A '''medium''' is a fluid $F$ through which a body $B$ can pass.
Definition:Damping Force
Consider a physical system consisting of a body $B$ passing through a medium $M$. In order for $B$ to move, it needs to push $M$ out of the way to make room for it. Hence $M$ exerts a force upon $B$ which opposes the motion of $B$. This force opposing the motion of $B$ is referred to as a '''damping force'''.
Definition:Viscosity
The viscosity of a fluid is a measure of how large a damping force it exerts upon a body moving through it. {{stub|Establish its formal definition: how great the damping force against how large a body at what velocity. Fluid mechanics has not really been started on this site yet. I have a degree semester to infodump.}}
Definition:Fluid/Ideal
An '''ideal fluid''' is a fluid which delivers a damping force whose magnitude is proportional to the velocity of a body passing through it.
Definition:Overdamped
Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b > a$, so that the solution of $(1)$ is in the form: :$x = C_1 e^{m_1 t} + C_2 e^{m_1 t}$ for $m_1, m_2 < 0$. Then $S$ is described as being '''overdamped'''. :500px
Definition:Critically Damped
Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b = a$, so that the solution of $(1)$ is in the form: :$x = C_1 e^{-a t} + C_2 t e^{-a t}$ Then $S$ is described as being '''critically damped'''. :500px
Definition:Underdamped
Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so that the solution of $(1)$ is in the form: :$x = e^{-b t} \paren {C_1 \cos \alpha t + C_2 \sin \alpha t}$ where $\alpha = \sqrt {a^2 - b^2}$. Then $S$ is described as being '''underdamped'''. :600px