Potential energy

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Potential energy
In the case of a bow and arrow, when the archer does work on the bow, drawing the string back, some of the chemical energy of the archer's body is transformed into elastic potential energy in the bent limb of the bow. When the string is released, the force between the string and the arrow does work on the arrow. The potential energy in the bow limbs is transformed into the kinetic energy of the arrow as it takes flight.
Common symbols	PE, U, or V
SI unit	joule (J)
Derivations from other quantities	U = m ⋅ g ⋅ h (gravitational) U = 1⁄2 ⋅ k ⋅ x2 (elastic) U = 1⁄2 ⋅ C ⋅ V2 (electric) U = −m ⋅ B (magnetic) U = $\int F(r)dr$

In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. The term potential energy was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to the ancient Greek philosopher Aristotle's concept of potentiality.

Common types of potential energy include the gravitational potential energy of an object, the elastic potential energy of an extended spring, and the electric potential energy of an electric charge in an electric field. The unit for energy in the International System of Units (SI) is the joule, which has the symbol J.

Potential energy is associated with forces that act on a body in a way that the total work done by these forces on the body depends only on the initial and final positions of the body in space. These forces, whose total work is path independent, are called conservative forces. If the force acting on a body varies over space, then one has a force field; such a field is described by vectors at every point in space, which is in-turn called a vector field. A conservative vector field can be simply expressed as the gradient of a certain scalar function, called a scalar potential.

Overview

There are various types of potential energy, each associated with a particular type of force. For example, the work of an elastic force is called elastic potential energy; work of the gravitational force is called gravitational potential energy; work of the Coulomb force is called electric potential energy; work of the strong nuclear force or weak nuclear force acting on the baryon charge is called nuclear potential energy; work of intermolecular forces is called intermolecular potential energy. Chemical potential energy, such as the energy stored in fossil fuels, is the work of the Coulomb force during rearrangement of configurations of electrons and nuclei in atoms and molecules. Thermal energy usually has two components: the kinetic energy of random motions of particles and the potential energy of their configuration.

Forces derivable from a potential are also called conservative forces. The work done by a conservative force is

$${\displaystyle W=-\mathrm{\Delta }U}$$

where ${\displaystyle \mathrm{\Delta }U}$ is the change in the potential energy associated with the force. The negative sign provides the convention that work done against a force field increases potential energy, while work done by the force field decreases potential energy. Common notations for potential energy are PE, U, V, and E_p.

Potential energy is the energy by virtue of an object's position relative to other objects. Potential energy is often associated with restoring forces such as a spring or the force of gravity. The action of stretching a spring or lifting a mass is performed by an external force that works against the force field of the potential. This work is stored in the force field, which is said to be stored as potential energy. If the external force is removed the force field acts on the body to perform the work as it moves the body back to the initial position, reducing the stretch of the spring or causing a body to fall.

Consider a ball whose mass is m and whose height is h. The acceleration g of free fall is approximately constant, so the weight force of the ball mg is constant. The product of force and displacement gives the work done, which is equal to the gravitational potential energy, thus

$${\displaystyle U_g=mgh.}$$

The more formal definition is that potential energy is the energy difference between the energy of an object in a given position and its energy at a reference position.

Work and potential energy

Potential energy is closely linked with forces. If the work done by a force on a body that moves from A to B does not depend on the path between these points (if the work is done by a conservative force), then the work of this force measured from A assigns a scalar value to every other point in space and defines a scalar potential field. In this case, the force can be defined as the negative of the vector gradient of the potential field.

If the work for an applied force is independent of the path, then the work done by the force is evaluated from the start to the end of the trajectory of the point of application. This means that there is a function U(x), called a "potential", that can be evaluated at the two points x_A and x_B to obtain the work over any trajectory between these two points. It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, that is

$${\displaystyle W={\int }_C\mathbf{F}\cdot d\mathbf{x}=U({\mathbf{x}}_{\text{A}})-U({\mathbf{x}}_{\text{B}})}$$

where C is the trajectory taken from A to B. Because the work done is independent of the path taken, then this expression is true for any trajectory, C, from A to B.

The function U(x) is called the potential energy associated with the applied force. Examples of forces that have potential energies are gravity and spring forces.

Derivable from a potential

In this section the relationship between work and potential energy is presented in more detail. The line integral that defines work along curve C takes a special form if the force F is related to a scalar field U′(x) so that

$${\displaystyle \mathbf{F}=\mathrm{\nabla }{U}^{\prime }=\left(\frac{\mathrm{\partial }{U}^{\prime }}{\mathrm{\partial }x},\frac{\mathrm{\partial }{U}^{\prime }}{\mathrm{\partial }y},\frac{\mathrm{\partial }{U}^{\prime }}{\mathrm{\partial }z}\right).}$$

This means that the units of U′ must be this case, work along the curve is given by

$${\displaystyle W={\int }_C\mathbf{F}\cdot d\mathbf{x}={\int }_C\mathrm{\nabla }{U}^{\prime }\cdot d\mathbf{x},}$$

which can be evaluated using the gradient theorem to obtain

$${\displaystyle W=U^{\prime }({\mathbf{x}}_{\text{B}})-U^{\prime }({\mathbf{x}}_{\text{A}}).}$$

This shows that when forces are derivable from a scalar field, the work of those forces along a curve C is computed by evaluating the scalar field at the start point A and the end point B of the curve. This means the work integral does not depend on the path between A and B and is said to be independent of the path.

Potential energy U = −U′(x) is traditionally defined as the negative of this scalar field so that work by the force field decreases potential energy, that is

$${\displaystyle W=U({\mathbf{x}}_{\text{A}})-U({\mathbf{x}}_{\text{B}}).}$$

In this case, the application of the del operator to the work function yields,

$${\displaystyle \mathrm{\nabla }W=-\mathrm{\nabla }U=-\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }x},\frac{\mathrm{\partial }U}{\mathrm{\partial }y},\frac{\mathrm{\partial }U}{\mathrm{\partial }z}\right)=\mathbf{F},}$$

and the force F is said to be "derivable from a potential".^[7] This also necessarily implies that F must be a conservative vector field. The potential U defines a force F at every point x in space, so the set of forces is called a force field.

Computing potential energy

Given a force field F(x), evaluation of the work integral using the gradient theorem can be used to find the scalar function associated with potential energy. This is done by introducing a parameterized curve γ(t) = r(t) from γ(a) = A to γ(b) = B, and computing,

$${\displaystyle \begin{array}{rl}{\int }_{\gamma }\mathrm{\nabla }\mathrm{\Phi }(\mathbf{r})\cdot d\mathbf{r}& ={\int }_a^b\mathrm{\nabla }\mathrm{\Phi }(\mathbf{r}(t))\cdot {\mathbf{r}}^{\prime }(t)dt,\\ & ={\int }_a^b\frac{d}{dt}\mathrm{\Phi }(\mathbf{r}(t))dt=\mathrm{\Phi }(\mathbf{r}(b))-\mathrm{\Phi }(\mathbf{r}(a))=\mathrm{\Phi }\left({\mathbf{x}}_B\right)-\mathrm{\Phi }\left({\mathbf{x}}_A\right).\end{array}}$$

For the force field F, let v = dr/dt, then the gradient theorem yields,

$${\displaystyle \begin{array}{rl}{\int }_{\gamma }\mathbf{F}\cdot d\mathbf{r}& ={\int }_a^b\mathbf{F}\cdot \mathbf{v}dt,\\ & =-{\int }_a^b\frac{d}{dt}U(\mathbf{r}(t))dt=U({\mathbf{x}}_A)-U({\mathbf{x}}_B).\end{array}}$$

The power applied to a body by a force field is obtained from the gradient of the work, or potential, in the direction of the velocity v of the point of application, that is

$${\displaystyle P(t)=-\mathrm{\nabla }U\cdot \mathbf{v}=\mathbf{F}\cdot \mathbf{v}.}$$

Examples of work that can be computed from potential functions are gravity and spring forces.

Potential energy for near-Earth gravity

A trebuchet uses the gravitational potential energy of the counterweight to throw projectiles over two hundred meters

For small height changes, gravitational potential energy can be computed using

$${\displaystyle U_g=mgh,}$$

where m is the mass in kilograms, g is the local gravitational field (9.8 metres per second squared on Earth), h is the height above a reference level in metres, and U is the energy in joules.

In classical physics, gravity exerts a constant downward force F = (0, 0, F_z) on the center of mass of a body moving near the surface of the Earth. The work of gravity on a body moving along a trajectory r(t) = (x(t), y(t), z(t)), such as the track of a roller coaster is calculated using its velocity, v = (v_x, v_y, v_z), to obtain

$${\displaystyle W={\int }_{t_1}^{t_2}\mathit{F}\cdot \mathit{v}dt={\int }_{t_1}^{t_2}{F}_z{v}_{z}dt=F_z\mathrm{\Delta }z.}$$

where the integral of the vertical component of velocity is the vertical distance. The work of gravity depends only on the vertical movement of the curve r(t).

Potential energy for a linear spring

Main article: Elastic potential energy

Springs are used for storing elastic potential energy

Archery is one of humankind's oldest applications of elastic potential energy

A horizontal spring exerts a force F = (−kx, 0, 0) that is proportional to its deformation in the axial or x direction. The work of this spring on a body moving along the space curve s(t) = (x(t), y(t), z(t)), is calculated using its velocity, v = (v_x, v_y, v_z), to obtain

$${\displaystyle W={\int }_0^t\mathbf{F}\cdot \mathbf{v}dt=-{\int }_0^{t}kx{v}_{x}dt=-{\int }_0^{t}kx\frac{dx}{dt}dt={\int }_{x(t_0)}^{x(t)}kxdx=\frac{1}{2}k{x}^2}$$

For convenience, consider contact with the spring occurs at t = 0, then the integral of the product of the distance x and the x-velocity, xv_x, is x²/2.

The function

$${\displaystyle U(x)=\frac{1}{2}k{x}^2,}$$

is called the potential energy of a linear spring.

Elastic potential energy is the potential energy of an elastic object (for example a bow or a catapult) that is deformed under tension or compression (or stressed in formal terminology). It arises as a consequence of a force that tries to restore the object to its original shape, which is most often the electromagnetic force between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into kinetic energy.

Potential energy for gravitational forces between two bodies

The gravitational potential function, also known as gravitational potential energy, is:

$${\displaystyle U=-\frac{GMm}{r},}$$

The negative sign follows the convention that work is gained from a loss of potential energy.

Derivation

The gravitational force between two bodies of mass M and m separated by a distance r is given by Newton's law of universal gravitation

$${\displaystyle \mathbf{F}=-\frac{GMm}{r^2}\hat{\mathbf{r}},}$$

where ${\displaystyle \hat{\mathbf{r}}}$ is a vector of length 1 pointing from M to m and G is the gravitational constant.

Let the mass m move at the velocity v then the work of gravity on this mass as it moves from position r(t₁) to r(t₂) is given by

$${\displaystyle W=-{\int }_{\mathbf{r}(t_1)}^{\mathbf{r}(t_2)}\frac{GMm}{r^3}\mathbf{r}\cdot d\mathbf{r}=-{\int }_{t_1}^{t_2}\frac{GMm}{r^3}\mathbf{r}\cdot \mathbf{v}dt.}$$

The position and velocity of the mass m are given by

$${\displaystyle \mathbf{r}=r{\mathbf{e}}_r,\mathbf{v}=\dot{r}{\mathbf{e}}_r+r\dot{\theta }{\mathbf{e}}_t,}$$

where e_r and e_t are the radial and tangential unit vectors directed relative to the vector from M to m. Use this to simplify the formula for work of gravity to,

$${\displaystyle W=-{\int }_{t_1}^{t_2}\frac{GmM}{r^3}(r{\mathbf{e}}_r)\cdot (\dot{r}{\mathbf{e}}_r+r\dot{\theta }{\mathbf{e}}_t)dt=-{\int }_{t_1}^{t_2}\frac{GmM}{r^3}r\dot{r}dt=\frac{GMm}{r(t_2)}-\frac{GMm}{r(t_1)}.}$$

This calculation uses the fact that

$${\displaystyle \frac{d}{dt}{r}^{-1}=-r^{-2}\dot{r}=-\frac{\dot{r}}{r^2}.}$$

Potential energy for electrostatic forces between two bodies

The electrostatic force exerted by a charge Q on another charge q separated by a distance r is given by Coulomb's Law

$${\displaystyle \mathbf{F}=\frac{1}{4\pi {\epsilon }_0}\frac{Qq}{r^2}\hat{\mathbf{r}},}$$

where ${\displaystyle \hat{\mathbf{r}}}$ is a vector of length 1 pointing from Q to q and ε₀ is the vacuum permittivity.

The work W required to move q from A to any point B in the electrostatic force field is given by the potential function

$${\displaystyle U(r)=\frac{1}{4\pi {\epsilon }_0}\frac{Qq}{r}.}$$

Reference level

The potential energy is a function of the state a system is in, and is defined relative to that for a particular state. This reference state is not always a real state; it may also be a limit, such as with the distances between all bodies tending to infinity, provided that the energy involved in tending to that limit is finite, such as in the case of inverse-square law forces. Any arbitrary reference state could be used; therefore it can be chosen based on convenience.

Typically the potential energy of a system depends on the relative positions of its components only, so the reference state can also be expressed in terms of relative positions.

Gravitational potential energy

Main articles: Gravitational potential, Gravitational energy, and Gravity field

Gravitational energy is the potential energy associated with gravitational force, as work is required to elevate objects against Earth's gravity. The potential energy due to elevated positions is called gravitational potential energy, and is evidenced by water in an elevated reservoir or kept behind a dam. If an object falls from one point to another point inside a gravitational field, the force of gravity will do positive work on the object, and the gravitational potential energy will decrease by the same amount.

Gravitational force keeps the planets in orbit around the Sun

Consider a book placed on top of a table. As the book is raised from the floor to the table, some external force works against the gravitational force. If the book falls back to the floor, the "falling" energy the book receives is provided by the gravitational force. Thus, if the book falls off the table, this potential energy goes to accelerate the mass of the book and is converted into kinetic energy. When the book hits the floor this kinetic energy is converted into heat, deformation, and sound by the impact.

The factors that affect an object's gravitational potential energy are its height relative to some reference point, its mass, and the strength of the gravitational field it is in. Thus, a book lying on a table has less gravitational potential energy than the same book on top of a taller cupboard and less gravitational potential energy than a heavier book lying on the same table. An object at a certain height above the Moon's surface has less gravitational potential energy than at the same height above the Earth's surface because the Moon's gravity is weaker. "Height" in the common sense of the term cannot be used for gravitational potential energy calculations when gravity is not assumed to be a constant. The following sections provide more detail.

Local approximation

The strength of a gravitational field varies with location. However, when the change of distance is small in relation to the distances from the center of the source of the gravitational field, this variation in field strength is negligible and we can assume that the force of gravity on a particular object is constant. Near the surface of the Earth, for example, we assume that the acceleration due to gravity is a constant g = 9.8 m/s² (standard gravity). In this case, a simple expression for gravitational potential energy can be derived using the W = Fd equation for work, and the equation

$${\displaystyle W_F=-\mathrm{\Delta }{U}_F.}$$

The amount of gravitational potential energy held by an elevated object is equal to the work done against gravity in lifting it. The work done equals the force required to move it upward multiplied with the vertical distance it is moved (remember W = Fd). The upward force required while moving at a constant velocity is equal to the weight, mg, of an object, so the work done in lifting it through a height h is the product mgh. Thus, when accounting only for mass, gravity, and altitude, the equation is:

$${\displaystyle U=mgh}$$

where U is the potential energy of the object relative to its being on the Earth's surface, m is the mass of the object, g is the acceleration due to gravity, and h is the altitude of the object.

Hence, the potential difference is

$${\displaystyle \mathrm{\Delta }U=mg\mathrm{\Delta }h.}$$

General formula

However, over large variations in distance, the approximation that g is constant is no longer valid, and we have to use calculus and the general mathematical definition of work to determine gravitational potential energy. For the computation of the potential energy, we can integrate the gravitational force, whose magnitude is given by Newton's law of gravitation, with respect to the distance r between the two bodies. Using that definition, the gravitational potential energy of a system of masses m₁ and M₂ at a distance r using the Newtonian constant of gravitation G is

$${\displaystyle U=-G\frac{m_1{M}_2}{r}+K,}$$

where K is an arbitrary constant dependent on the choice of datum from which potential is measured. Choosing the convention that K = 0 (i.e. in relation to a point at infinity) makes calculations simpler, albeit at the cost of making U negative; for why this is physically reasonable, see below.

Given this formula for U, the total potential energy of a system of n bodies is found by summing, for all $\frac{n(n-1)}{2}$ pairs of two bodies, the potential energy of the system of those two bodies.

Gravitational potential summation ${\displaystyle U=-m\left(G\frac{M_1}{r_1}+G\frac{M_2}{r_2}\right)}$

Considering the system of bodies as the combined set of small particles the bodies consist of, and applying the previous on the particle level we get the negative gravitational binding energy. This potential energy is more strongly negative than the total potential energy of the system of bodies as such since it also includes the negative gravitational binding energy of each body. The potential energy of the system of bodies as such is the negative of the energy needed to separate the bodies from each other to infinity, while the gravitational binding energy is the energy needed to separate all particles from each other to infinity.

$${\displaystyle U=-m\left(G\frac{M_1}{r_1}+G\frac{M_2}{r_2}\right)}$$

therefore,

$${\displaystyle U=-m\sum G\frac{M}{r},}$$

Negative gravitational energy

As with all potential energies, only differences in gravitational potential energy matter for most physical purposes, and the choice of zero point is arbitrary. Given that there is no reasonable criterion for preferring one particular finite r over another, there seem to be only two reasonable choices for the distance at which U becomes zero: ${\displaystyle r=0}$ and ${\displaystyle r=\mathrm{\infty }}$. The choice of ${\displaystyle U=0}$ at infinity may seem peculiar, and the consequence that gravitational energy is always negative may seem counterintuitive, but this choice allows gravitational potential energy values to be finite, albeit negative.

The singularity at ${\displaystyle r=0}$ in the formula for gravitational potential energy means that the only other apparently reasonable alternative choice of convention, with ${\displaystyle U=0}$ for ${\displaystyle r=0}$, would result in potential energy being positive, but infinitely large for all nonzero values of r, and would make calculations involving sums or differences of potential energies beyond what is possible with the real number system. Since physicists abhor infinities in their calculations, and r is always non-zero in practice, the choice of ${\displaystyle U=0}$ at infinity is by far the more preferable choice, even if the idea of negative energy in a gravity well appears to be peculiar at first.

The negative value for gravitational energy also has deeper implications that make it seem more reasonable in cosmological calculations where the total energy of the universe can meaningfully be considered; see inflation theory for more on this.

Uses

Further information: Gravitational potential energy storage

Gravitational potential energy has a number of practical uses, notably the generation of pumped-storage hydroelectricity. For example, in Dinorwig, Wales, there are two lakes, one at a higher elevation than the other. At times when surplus electricity is not required (and so is comparatively cheap), water is pumped up to the higher lake, thus converting the electrical energy (running the pump) to gravitational potential energy. At times of peak demand for electricity, the water flows back down through electrical generator turbines, converting the potential energy into kinetic energy and then back into electricity. The process is not completely efficient and some of the original energy from the surplus electricity is in fact lost to friction.

Gravitational potential energy is also used to power clocks in which falling weights operate the mechanism.

It is also used by counterweights for lifting up an elevator, crane, or sash window.

Roller coasters are an entertaining way to utilize potential energy – chains are used to move a car up an incline (building up gravitational potential energy), to then have that energy converted into kinetic energy as it falls.

Another practical use is utilizing gravitational potential energy to descend (perhaps coast) downhill in transportation such as the descent of an automobile, truck, railroad train, bicycle, airplane, or fluid in a pipeline. In some cases the kinetic energy obtained from the potential energy of descent may be used to start ascending the next grade such as what happens when a road is undulating and has frequent dips. The commercialization of stored energy (in the form of rail cars raised to higher elevations) that is then converted to electrical energy when needed by an electrical grid, is being undertaken in the United States in a system called Advanced Rail Energy Storage (ARES).

Chemical potential energy

Main article: Chemical energy

Chemical potential energy is a form of potential energy related to the structural arrangement of atoms or molecules. This arrangement may be the result of chemical bonds within a molecule or otherwise. Chemical energy of a chemical substance can be transformed to other forms of energy by a chemical reaction. As an example, when a fuel is burned the chemical energy is converted to heat, same is the case with digestion of food metabolized in a biological organism. Green plants transform solar energy to chemical energy through the process known as photosynthesis, and electrical energy can be converted to chemical energy through electrochemical reactions.

The similar term chemical potential is used to indicate the potential of a substance to undergo a change of configuration, be it in the form of a chemical reaction, spatial transport, particle exchange with a reservoir, etc.

Electric potential energy

Main article: Electric potential energy

An object can have potential energy by virtue of its electric charge and several forces related to their presence. There are two main types of this kind of potential energy: electrostatic potential energy, electrodynamic potential energy (also sometimes called magnetic potential energy).

Plasma formed inside a gas filled sphere

Electrostatic potential energy

Electrostatic potential energy between two bodies in space is obtained from the force exerted by a charge Q on another charge q which is given by

$${\displaystyle {\mathbf{F}}_e=-\frac{1}{4\pi {\epsilon }_0}\frac{Qq}{r^2}\hat{\mathbf{r}},}$$

where ${\displaystyle \hat{\mathbf{r}}}$ is a vector of length 1 pointing from Q to q and ε₀ is the vacuum permittivity.

If the electric charge of an object can be assumed to be at rest, then it has potential energy due to its position relative to other charged objects. The electrostatic potential energy is the energy of an electrically charged particle (at rest) in an electric field. It is defined as the work that must be done to move it from an infinite distance away to its present location, adjusted for non-electrical forces on the object. This energy will generally be non-zero if there is another electrically charged object nearby.

The work W required to move q from A to any point B in the electrostatic force field is given by

$${\displaystyle \mathrm{\Delta }{U}_{AB}(\mathbf{r})=-{\int }_A^B{\mathbf{F}}_{\mathbf{e}}\cdot d\mathbf{r}}$$

typically given in J for Joules. A related quantity called electric potential (commonly denoted with a V for voltage) is equal to the electric potential energy per unit charge.

Magnetic potential energy

The energy of a magnetic moment ${\displaystyle \mathit{\mu }}$ in an externally produced magnetic B-field B has potential energy

$${\displaystyle U=-\mathit{\mu }\cdot \mathbf{B}.}$$

The magnetization M in a field is

$${\displaystyle U=-\frac{1}{2}\int \mathbf{M}\cdot \mathbf{B}dV,}$$

where the integral can be over all space or, equivalently, where M is nonzero. Magnetic potential energy is the form of energy related not only to the distance between magnetic materials, but also to the orientation, or alignment, of those materials within the field. For example, the needle of a compass has the lowest magnetic potential energy when it is aligned with the north and south poles of the Earth's magnetic field. If the needle is moved by an outside force, torque is exerted on the magnetic dipole of the needle by the Earth's magnetic field, causing it to move back into alignment. The magnetic potential energy of the needle is highest when its field is in the same direction as the Earth's magnetic field. Two magnets will have potential energy in relation to each other and the distance between them, but this also depends on their orientation. If the opposite poles are held apart, the potential energy will be higher the further they are apart and lower the closer they are. Conversely, like poles will have the highest potential energy when forced together, and the lowest when they spring apart.

Nuclear potential energy

Nuclear potential energy is the potential energy of the particles inside an atomic nucleus. The nuclear particles are bound together by the strong nuclear force. Weak nuclear forces provide the potential energy for certain kinds of radioactive decay, such as beta decay.

Nuclear particles like protons and neutrons are not destroyed in fission and fusion processes, but collections of them can have less mass than if they were individually free, in which case this mass difference can be liberated as heat and radiation in nuclear reactions (the heat and radiation have the missing mass, but it often escapes from the system, where it is not measured). The energy from the Sun is an example of this form of energy conversion. In the Sun, the process of hydrogen fusion converts about 4 million tonnes of solar matter per second into electromagnetic energy, which is radiated into space.

Forces and potential energy

Potential energy is closely linked with forces. If the work done by a force on a body that moves from A to B does not depend on the path between these points, then the work of this force measured from A assigns a scalar value to every other point in space and defines a scalar potential field. In this case, the force can be defined as the negative of the vector gradient of the potential field.

For example, gravity is a conservative force. The associated potential is the gravitational potential, often denoted by ${\displaystyle \varphi }$ or ${\displaystyle V}$, corresponding to the energy per unit mass as a function of position. The gravitational potential energy of two particles of mass M and m separated by a distance r is

$${\displaystyle U=-\frac{GMm}{r}.}$$

The gravitational potential (specific energy) of the two bodies is

$${\displaystyle \varphi =-\left(\frac{GM}{r}+\frac{Gm}{r}\right)=-\frac{G(M+m)}{r}=-\frac{GMm}{\mu r}=\frac{U}{\mu }}$$

where ${\displaystyle \mu }$ is the reduced mass.

The work done against gravity by moving an infinitesimal mass from point A with ${\displaystyle U=a}$ to point B with ${\displaystyle U=b}$ is ${\displaystyle (b-a)}$ and the work done going back the other way is ${\displaystyle (a-b)}$ so that the total work done in moving from A to B and returning to A is

$${\displaystyle U_{A\to B\to A}=(b-a)+(a-b)=0.}$$

If the potential is redefined at A to be ${\displaystyle a+c}$ and the potential at B to be ${\displaystyle b+c}$, where ${\displaystyle c}$ is a constant (i.e. ${\displaystyle c}$ can be any number, positive or negative, but it must be the same at A as it is at B) then the work done going from A to B is

$${\displaystyle U_{A\to B}=(b+c)-(a+c)=b-a}$$

as before.

In practical terms, this means that one can set the zero of ${\displaystyle U}$ and ${\displaystyle \varphi }$ anywhere one likes. One may set it to be zero at the surface of the Earth, or may find it more convenient to set zero at infinity (as in the expressions given earlier in this section).

A conservative force can be expressed in the language of differential geometry as a closed form. As Euclidean space is contractible, its de Rham cohomology vanishes, so every closed form is also an exact form, and can be expressed as the gradient of a scalar field. This gives a mathematical justification of the fact that all conservative forces are gradients of a potential field.

Manipulation (psychology)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Manipulation_(psychology)

In psychology, manipulation is defined as subterfuge designed to influence or control another, usually in a manner which facilitates one's personal aims. The methods used distort or orient the interlocutor's perception of reality, in particular through seduction, suggestion, persuasion and non-voluntary or consensual submission. Definitions for the term vary in which behavior is specifically included, influenced by both culture and whether referring to the general population or used in clinical contexts. Manipulation is generally considered a dishonest form of social influence as it is used at the expense of others.

Manipulative tendencies may derive from cluster B personality disorders such as narcissistic or antisocial personality disorder. Manipulation is also correlated with higher levels of emotional intelligence, and is a chief component of the personality construct dubbed Machiavellianism.

Manipulation differs from general influence and persuasion. Non-manipulative influence is generally perceived to be harmless and it is not seen as unduly coercive to the individual's right of acceptance or rejection of influence. Persuasion is the ability to move others to a desired action, usually within the context of a specific goal. Persuasion often attempts to influence ones beliefs, religion, motivations, or behavior. Influence and persuasion are neither positive nor negative, unlike manipulation which is strictly negative. Manipulation is often seen as negative, though some argue that it has positive aspects. Positive manipulation is a form of practice where an individual can turn any aspect that may not be going well into a positive experience. Ultimately, one's goal is to not be manipulated but if the situation does arise, the individual is able to manifest for the best. Self-development provides the opportunity for an individual to grow, and help influence the behaviors of others as well. Individuals who behave in prosocial behavior manners can be manipulated to have positive mood reactions. Alongside showing encouragement during a time where an individual is feeling down can result in improvements in mood.

Elements of manipulation

The motivation for manipulation can be self-serving or it can be intended to help or benefit others. Antisocial manipulation is using "skills to advance personal agendas or self-serving motives at the expense of others", pro-social behavior is a voluntary act intended to help or benefit another individual or group of individuals and is an important part of empathy.

Different measures of manipulativeness focus on different aspects or expressions of manipulation, and tend to paint slightly different pictures of its predictors. Features such as low empathy, high narcissism, use of self-serving rationalisations, and an interpersonal style marked by high agency (dominance) and low communion (i.e. coldheartedness) are consistent across measures.

Manipulative behaviors typically exploit the following vulnerabilities:

Vulnerability	Description
Naïveté or immaturity	People who find it too hard to accept the idea that some people are cunning, devious and ruthless or are "in denial" if they are being taken advantage of. They will acknowledge the fact of being manipulated only if it occurs too often.
Over-conscientiousness	People who are much harder on themselves than on others often are too willing to give another the benefit of the doubt and see their side of things while blaming themselves for hurting the manipulator.
Low self-esteem	People who struggle with self-doubting, lacking in confidence and assertiveness, or chronically unsure of their right to pursue their legitimate wants and needs. They are likely to go on the defensive too easily when challenged by an aggressive personality.
Over-intellectualization	People who believe that others only do hurtful things when there's some legitimate, understandable reason for manipulation. They might delude themselves into believing that uncovering and understanding all the reasons for the manipulator's behavior will be sufficient to make things different.
Emotional dependency	People who have a submissive or dependent personality. The more emotionally dependent a person is, the more vulnerable they are to being exploited and manipulated.

Manipulation and mental illnesses

Individuals with the following mental health issues are often prone to manipulative behavior:

• Antisocial personality disorder,
• Borderline personality disorder,
• Conduct disorder,
• Factitious disorder,
• Histrionic personality disorder,
• Narcissistic personality disorder.

Deceitfulness and exceptional manipulative abilities are the most common traits among antisocial personality disorder and narcissistic personality disorder. It is the major feature found in the dark triad personality traits, particularly Machiavellianism.

Antisocial personality disorder or sociopathy refers to individuals who will not realize the rights and wrongs of their action and the ability to neglect others emotionally. People with this disorder may not feel that they are doing anything wrong and therefore feel free to manipulate others. This mental disorder relies on features of deceitfulness and arrogance acts.

Borderline Personality Disorder is unique in the grouping as "borderline" manipulation is characterized as unintentional and dysfunctional manipulation. Marsha M. Linehan has stated that people with borderline personality disorder often exhibit behaviors which are not truly manipulative, but are erroneously interpreted as such. According to Linehan, these behaviors often appear as unthinking manifestations of intense pain, and are often not deliberate as to be considered truly manipulative. In the DSM-V, manipulation was removed as a defining characteristic of borderline personality disorder.

Conduct disorder is where behavioral and age appropriate actions are taken advantage of, primarily occurring in children and adolescents. Individuals with this are characterized as "lack of empathy, sense of guilt, and shallow emotion". These behaviors are shown in connection to manipulation by tying in narcissistic traits. Aggression and violence are two factors pursued by individuals with this disorder. In order for this disorder to be consistent and shown, the progression must be made for at least 12 months.

Factitious disorder is a mental illness in which individuals who purposely forge symptoms, physically or psychologically. Fabricating illnesses allows individuals to feel a thrill and receive free aid in hospital admissions and treatment. Feelings of persistence, abuse in early childhood, and excessive thoughts were common for these individuals who connected to Borderline Personality Disorder.

Histrionic personality disorder foresee individuals who seek scrutinizing behaviors, inappropriate alluring tactics, and irregular emotional patterns. Histrionic symptoms include "seeking reassurance, switching emotional, and feeling uncomfortable." Histrionic and Narcissistic Personality Disorders overlap because decisions are sporadic and unreliable. These individuals can experience these symptoms from failed attempts of depression like symptoms.

Narcissistic personality disorder is characterized as feelings of superiority, a sense of grandiosity, exhibitionism, charming but also exploitive behaviors in the interpersonal domain, success, beauty, feelings of entitlement and a lack of empathy. Those with this disorder often engage in assertive self enhancement and antagonistic self protection. All of these factors can lead an individual with narcissistic personality disorder to manipulate others.

Assessment tools

Further information: Psychological evaluation § Personality Assessment

Emotional manipulation scale

The emotional manipulation scale is a ten-item questionnaire developed in 2007 through factor analysis, primarily to measure the capability of manipulative behavior and the Machiavellianism personality trait. At the time of publication, emotional intelligence assessments did not specifically examine manipulative behavior or Machiavellianism and were instead predominantly focussed on Big Five personality trait assessment.

Managing the emotions of others scale

The Managing the emotions of others scale (MEOS) was developed in 2013 through factor analysis to measure the ability to change emotions of others. The survey questions measure six categories: mood (or emotional state) enhancement, mood worsening, concealing emotions, capacity for inauthenticity, poor emotion skills, and using diversion to enhance mood. The enhancement, worsening and diversion categories have been used to identify the ability and willingness of manipulative behavior. The MEOS has also been used for assessing emotional intelligence, and has been compared to the HEXACO model of personality structure, for which the capacity for inauthenticity category in the MEOS was found to correspond to low honesty-humility scores on the HEXACO.

In popular psychology

Harriet B. Braiker

Harriet B. Braiker identified the following ways that manipulators control their victims:

• Positive reinforcement: includes praise, superficial charm, superficial sympathy (crocodile tears), excessive apologizing, money, approval, gifts, attention, facial expressions such as a forced laugh or smile, and public recognition.
• Negative reinforcement: involves removing one from a negative situation as a reward.
• Gaslighting.
• Intermittent or partial reinforcement: Partial or intermittent negative reinforcement can create an effective climate of fear and doubt. Partial or intermittent positive reinforcement can encourage the victim to persist.
• Punishment: includes nagging, yelling, the silent treatment, intimidation, threats, swearing, emotional blackmail, guilt trips, sulking, crying, and playing the victim.
• Traumatic one-trial learning: using verbal abuse, explosive anger, or other intimidating behavior to establish dominance or superiority; even one incident of such behavior can condition or train victims to avoid upsetting, confronting or contradicting the manipulator.

According to Braiker, manipulators exploit the following vulnerabilities (buttons) that may exist in victims:

• the desire to please
• addiction to earning the approval and acceptance of others
• emotophobia (fear of negative emotion; i.e. a fear of expressing anger, frustration or disapproval)
• lack of assertiveness and ability to say no
• blurry sense of identity (with soft personal boundaries)
• low self-reliance
• external locus of control

Manipulators can have various possible motivations, including but not limited to:

• the need to advance their own purposes and personal gain at (virtually any) cost to others
• a strong need to attain feelings of power and superiority in relationships with others - compare megalomania (associated with, for example, narcissistic personality disorder)
• a want and need to feel in control
• a desire to gain a feeling of power over others in order to raise their perception of self-esteem
• furtherance of cult dynamics in recruiting or retaining followers
• boredom, or growing tired of one's surroundings; seeing manipulation as a game more than hurting others
• covert agendas, criminal or otherwise, including financial manipulation (often seen when intentionally targeting the elderly or unsuspecting, unprotected wealthy for the sole purpose of obtaining victims' financial assets)
• not identifying with underlying emotions (including experiencing commitment phobia), and subsequent rationalization (offenders do not manipulate consciously, but rather try to convince themselves of the invalidity of their own emotions)
• lack of self-control over impulsive and anti-social behaviour - leading to pre-emptive or reactionary manipulation to maintain image

George K. Simon

According to psychology author George K. Simon, successful psychological manipulation primarily involves the manipulator:

• Concealing aggressive intentions and behaviors and being affable.
• Knowing the psychological vulnerabilities of the victim to determine which tactics are likely to be the most effective.
• Having a sufficient level of ruthlessness to have no qualms about causing harm to the victim if necessary.

Techniques of manipulators may include:

Techniques	Description
Lying (by commission)	It is hard to tell if somebody is lying at the time they do it, although often the truth may be apparent later when it is too late. One way to minimize the chances of being lied to is to understand that some personality types (particularly psychopaths) are experts at lying and cheating, doing it frequently, and often in subtle ways.
Lying by omission	This is a subtle form of lying by withholding a significant amount of the truth. This technique is also used in propaganda.
Denial	Manipulator refuses to admit that they have done something wrong.
Rationalization	An excuse made by the manipulator for inappropriate behavior. Rationalization is closely related to spin.
Minimization	This is a type of denial coupled with rationalization. The manipulator asserts that their behavior is not as harmful or irresponsible as someone else was suggesting.
Selective inattention or selective attention	Manipulator refuses to pay attention to anything that may distract from their agenda.
Diversion	Manipulator not giving a straight answer to a straight question and instead being diversionary, steering the conversation onto another topic.
Evasion	Similar to diversion but giving irrelevant, rambling, or vague responses
Covert intimidation	Manipulator putting the victim onto the defensive by using veiled (subtle, indirect or implied) threats.
Guilt trip	A special kind of intimidation tactic. A manipulator suggests to the conscientious victim that they do not care enough, are too selfish or have it too easy. This can result in the victim feeling bad, keeping them in a self-doubting, anxious and submissive position.
Shaming	Manipulator uses sarcasm and put-downs to increase fear and self-doubt in the victim. Manipulators use this tactic to make others feel unworthy and therefore defer to them. Manipulators can make one feel ashamed for even daring to challenge them. It is an effective way to foster a sense of inadequacy in the victim.
Vilifying the victim	This tactic is a powerful means of putting the victim on the defensive while simultaneously masking the aggressive intent of the manipulator, while the manipulator falsely accuses the victim as being an abuser in response when the victim stands up for or defends themselves or their position.
Playing the victim role	Manipulator portrays themself as a victim of circumstance or of someone else's behavior in order to gain pity, sympathy or evoke compassion and thereby get something from another. Caring and conscientious people often cannot stand to see anyone suffering and the manipulator often finds it easy to play on sympathy to get cooperation.
Playing the servant role	Cloaking a self-serving agenda in the guise of a service to a more noble cause.
Seduction	Manipulator uses charm, praise, flattery or overtly supporting others in order to get them to lower their defenses and give their trust and loyalty to the manipulator. They will also offer help with the intent to gain trust and access to an unsuspecting victim they have charmed.
Projecting the blame (blaming others)	Manipulating scapegoats in often subtle, hard-to-detect ways. Often, the manipulator will project their own thinking onto the victim, making the victim look like they have done something wrong. Manipulators will also claim that the victim is the one who is at fault for believing lies that they were conned into believing, as if the victim forced the manipulator to be deceitful. All blame, except for the part that is used by the manipulator to accept false guilt, is done in order to make the victim feel guilty about making healthy choices, correct thinking and good behaviors. It is frequently used as a means of psychological and emotional manipulation and control. Manipulators lie about lying, only to re-manipulate the original, less believable story into a "more acceptable" truth that the victim will believe. Projecting lies as being the truth is another common method of control and manipulation. Manipulators may falsely accuse the victim of "deserving to be treated that way". They often claim that the victim is crazy or abusive, especially when there is evidence against the manipulator.
Feigning innocence	Manipulator tries to suggest that any harm done was unintentional or that they did not do something that they were accused of. Manipulator may put on a look of surprise or indignation. This tactic makes the victim question their own judgment and possibly their own sanity.
Feigning confusion	Manipulator tries to play dumb by pretending they do not know what the victim is talking about or is confused about an important issue brought to their attention. The manipulator intentionally confuses the victim in order for the victim to doubt their own accuracy of perception, often pointing out key elements that the manipulator intentionally included in case there is room for doubt. Sometimes manipulators will have used cohorts in advance to help back up their story.
Brandishing anger	Manipulator uses anger to brandish sufficient emotional intensity and rage to shock the victim into submission. The manipulator is not actually angry, they just put on an act. They just want what they want and get "angry" when denied. Controlled anger is often used as a manipulation tactic to avoid confrontation, avoid telling the truth or to further hide intent. There are often threats used by the manipulator of going to the police, or falsely reporting abuses that the manipulator intentionally contrived to scare or intimidate the victim into submission. Blackmail and other threats of exposure are other forms of controlled anger and manipulation, especially when the victim refuses initial requests or suggestions by the manipulator. Anger is also used as a defense so the manipulator can avoid telling truths at inconvenient times or circumstances. Anger is often used as a tool or defense to ward off inquiries or suspicion. The victim becomes more focused on the anger instead of the manipulation tactic.
Bandwagon effect	Manipulator comforts the victim into submission by claiming (whether true or false) that many people already have done something, and the victim should as well. Such manipulation can be seen in peer pressure situations, often occurring in scenarios where the manipulator attempts to influence the victim into trying drugs or other substances.

Martin Kantor

Kantor advises in his 2006 book The Psychopathology of Everyday Life: How Antisocial Personality Disorder Affects All of Us that vulnerability to psychopathic manipulators involves being too:

• Dependent – dependent people need to be loved and are therefore gullible and liable to say yes to something to which they should say no.
• Immature – has impaired judgment and so tends to believe exaggerated advertising claims.
• Naïve – cannot believe there are dishonest people in the world, or takes it for granted that if there are any, they will not be allowed to prey on others.
• Impressionable – overly seduced by charmers.
• Trusting – people who are honest often assume that everyone else is honest. They are more likely to commit themselves to people they hardly know without checking credentials, etc., and less likely to question so-called experts.
• Carelessness – not giving sufficient amount of thought or attention to harm or errors.
• Lonely – lonely people may accept any offer of human contact. A psychopathic stranger may offer human companionship for a price.
• Narcissistic – narcissists are prone to falling for unmerited flattery.
• Impulsive – make snap decisions.
• Altruistic – the opposite of psychopathic: too honest, too fair, too empathetic.
  • Not being listed in the book, implication carries that self-unaware psychopaths will be manipulated easily because of their lack of regret, meanness, boldness and disinhibition.
• Frugal – cannot say no to a bargain even if they know the reason it is so cheap.
• Materialistic – easy prey for loan sharks or get-rich-quick schemes.
• Greedy – the greedy and dishonest may fall prey to a psychopath who can easily entice them to act in an immoral way.
• Masochistic – lack self-respect and so unconsciously let psychopaths take advantage of them. They think they deserve it out of a sense of guilt.
• The elderly – the elderly can become fatigued and less capable of multi-tasking. When hearing a sales pitch they are less likely to consider that it could be a con. They are prone to giving money to someone with a hard-luck story. See elder abuse.

Supercontinent cycle

From Wikipedia, the free encyclopedia

Map of Pangaea with modern continental outlines

The supercontinent cycle is the quasi-periodic aggregation and dispersal of Earth's continental crust. There are varying opinions as to whether the amount of continental crust is increasing, decreasing, or staying about the same, but it is agreed that the Earth's crust is constantly being reconfigured. One complete supercontinent cycle is said to take 300 to 500 million years. Continental collision makes fewer and larger continents while rifting makes more and smaller continents.

Description

Simplified representation of the proposed series of supercontinents to the modern day

The most recent supercontinent, Pangaea, formed about 300 million years ago (0.3 Ga). There are two different views on the history of earlier supercontinents. The first proposes a series of supercontinents: Vaalbara (c. 3.6 to c. 2.8 billion years ago); Ur (c. 3 billion years ago); Kenorland (c. 2.7 to 2.1 billion years ago); Columbia (c. 1.8 to 1.5 billion years ago); Rodinia (c. 1.25 billion to 750 million years ago); and Pannotia (c. 600 million years ago), whose dispersal produced the fragments that ultimately collided to form Pangaea.

The second view (Protopangea-Paleopangea), based on both palaeomagnetic and geological evidence, is that supercontinent cycles did not occur before about 0.6 Ga (during the Ediacaran Period). Instead, the continental crust comprised a single supercontinent from about 2.7 Ga (gigaannums, or billion years ago) until it broke up for the first time, somewhere around 0.6 Ga. This reconstruction is based on the observation that if only small peripheral modifications are made to the primary reconstruction, the data show that the palaeomagnetic poles converged to quasi-static positions for long intervals between about 2.7–2.2, 1.5–1.25 and 0.75–0.6 Ga. During the intervening periods, the poles appear to have conformed to a unified apparent polar wander path. Thus the paleomagnetic data are adequately explained by the existence of a single Protopangea–Paleopangea supercontinent with prolonged quasi-integrity. The prolonged duration of this supercontinent could be explained by the operation of lid tectonics (comparable to the tectonics operating on Mars and Venus) during Precambrian times, as opposed to the plate tectonics seen on the contemporary Earth. However, this approach was widely criticized as it is based on incorrect application of paleomagnetic data.

The kinds of minerals found inside ancient diamonds suggest that the cycle of supercontinental formation and breakup began roughly 3.0 billion years ago (3.0 Ga). Before 3.2 billion years ago only diamonds with peridotitic compositions (commonly found in the Earth's mantle) formed, whereas after 3.0 billion years ago eclogitic diamonds (rocks from the Earth's surface crust) became prevalent. This change is thought to have come about as subduction and continental collision introduced eclogite into subcontinental diamond-forming fluids.

The supercontinent cycle and the Wilson cycle produced the supercontinents Rodinia and Pangaea

The hypothesized supercontinent cycle is overlaid by the Wilson Cycle named after plate tectonics pioneer John Tuzo Wilson, which describes the periodic opening and closing of oceanic basins from a single plate rift. The oldest seafloor material found today dates to only 170 million years old, whereas the oldest continental crust material found today dates to 4 billion years, showing the relative brevity of the regional Wilson cycles compared to the planetary pulse seen in the arrangement of the continents.

Effects on sea level

It is known that sea level is generally low when the continents are together and high when they are apart. For example, sea level was low at the time of formation of Pangaea (Permian) and Pannotia (latest Neoproterozoic), and rose rapidly to maxima during Ordovician and Cretaceous times, when the continents were dispersed. Major influences on sea level during the break up of supercontinents include: oceanic crust age, lost back-arc basins, marine sediment depths, emplacement of large igneous provinces, and the effect of passive margin extension. Of these, oceanic crust age, and marine sediment depths seem to play some of the largest roles in creating a sea level model. The addition of the other controlling parameters help stabilize models when data is sparse. As previously mentioned, the age of the oceanic lithosphere provides a first order control on the depth of the ocean basins, and therefore on global sea level. Oceanic lithosphere forms at mid-ocean ridges and moves outwards, conductively cooling and shrinking, which decreases the thickness and increases the density of the oceanic lithosphere, and lowers the seafloor away from mid-ocean ridges. For oceanic lithosphere that is less than about 75 million years old, a simple cooling half-space model of conductive cooling works, in which the depth of the ocean basins d in areas in which there is no nearby subduction is a function of the age of the oceanic lithosphere t. In general,

$${\displaystyle d(t)=\frac{2}{\sqrt{\pi }}{a}_{\mathrm{e}\mathrm{f}\mathrm{f}}{T}_1\sqrt{\kappa t}+d_{\mathrm{r}}}$$

where κ is the thermal diffusivity of the mantle lithosphere (c. 8×10⁻⁷ m²/s), a_eff is the effective thermal expansion coefficient for rock (c. 5.7×10⁻⁵ °C⁻¹), T₁ is the temperature of ascending magma compared to the temperature at the upper boundary (c. 1220 °C for the Atlantic and Indian Oceans, c. 1120 °C for the eastern Pacific) and d_r is the depth of the ridge below the ocean surface. After plugging in rough numbers for the sea floor, the equation becomes:

for the eastern Pacific Ocean:

$${\displaystyle d(t)=350\sqrt{t}+2500}$$

and for the Atlantic and Indian Oceans:

$${\displaystyle d(t)=390\sqrt{t}+2500}$$

where d is in meters and t is in millions of years, so that just-formed crust at the mid-ocean ridges lies at about 2,500 m depth, whereas 50-million-year-old seafloor lies at a depth of about 5,000 m.

As the mean level of the sea floor decreases, the volume of the ocean basins increases, and if other factors that can control sea level remain constant, sea level falls. The converse is also true: younger oceanic lithosphere leads to shallower oceans and higher sea levels if other factors remain constant.

The surface area of the oceans can change when continents rift (stretching the continents decreases ocean area and raises sea level) or as a result of continental collision (compressing the continents increases ocean area and lowers sea level). Increasing sea level will flood the continents, while decreasing sea level will expose continental shelves.

Because the continental shelf has a very low slope, a small increase in sea level will result in a large change in the percent of continents flooded.

If the world ocean on average is young, the seafloor will be relatively shallow, and sea level will be high: more of the continents are flooded. If the world ocean is on average old, seafloor will be relatively deep, and sea level will be low: more of the continents will be exposed.

There is thus a relatively simple relationship between the supercontinent cycle and the mean age of the seafloor.

• Supercontinent = much old seafloor = low sea level
• Dispersed continents = much young seafloor = high sea level

There will also be a climatic effect of the supercontinent cycle that will amplify this further:

• Supercontinent = continental climate dominant = continental glaciation likely = still lower sea level
• Dispersed continents = maritime climate dominant = continental glaciation unlikely = sea level is not lowered by this mechanism

Relation to global tectonics

There is a progression of tectonic regimes that accompanies the supercontinent cycle:

During break-up of the supercontinent, rifting environments dominate. This is followed by passive margin environments, while seafloor spreading continues and the oceans grow. This in turn is followed by the development of collisional environments that become increasingly important with time. First collisions are between continents and island arcs, but lead ultimately to continent-continent collisions. This was the situation during the Paleozoic supercontinent cycle; it is being observed for the Mesozoic–Cenozoic supercontinent cycle, still in progress.

Relation to climate

Main article: Greenhouse and icehouse Earth

There are two types of global earth climates: icehouse and greenhouse. Icehouse is characterized by frequent continental glaciations and severe desert environments. Greenhouse is characterized by warm climates. Both reflect the supercontinent cycle. It is now a short greenhouse phase of an icehouse world.

• Icehouse climate
  • Continents moving together
  • Sea level low due to lack of seafloor production
  • Climate cooler, arid
  • Associated with aragonite seas
  • Formation of supercontinents
• Greenhouse climate
  • Continents dispersed
  • Sea level high
  • High level of seafloor spreading
  • Relatively large amounts of CO₂ production at oceanic rifting zones
  • Climate warm and humid
  • Associated with calcite seas

Periods of icehouse climate: much of Neoproterozoic, late Paleozoic, late Cenozoic.

Periods of greenhouse climate: Early Paleozoic, Mesozoic–early Cenozoic.

Relation to evolution

The principal mechanism for evolution is natural selection among diverse populations. As genetic drift occurs more frequently in small populations, diversity is an observed consequence of isolation. Less isolation, and thus less diversification, occurs when the continents are all together, producing both one continent and one ocean with one coast. In Latest Neoproterozoic to Early Paleozoic times, when the tremendous proliferation of diverse metazoa occurred, isolation of marine environments resulted from the breakup of Pannotia.

A north–south arrangement of continents and oceans leads to much more diversity and isolation than east–west arrangements. North-to-south arrangements give climatically different zones along the communication routes to the north and south, which are separated by water or land from other continental or oceanic zones of similar climate. Formation of similar tracts of continents and ocean basins oriented east–west would lead to much less isolation, diversification, and slower evolution, since each continent or ocean is in fewer climatic zones. Through the Cenozoic, isolation has been maximized by a north–south arrangement.

Diversity, as measured by the number of families, follows the supercontinent cycle very well.