Spherical coordinate system

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https://en.wikipedia.org/wiki/Spherical_coordinate_system

Spherical coordinates (r, θ, φ) as commonly used in physics (ISO 80000-2:2019 convention): radial distance r (slant distance to origin), polar angle θ (theta) (angle with respect to positive polar axis), and azimuthal angle φ (phi) (angle of rotation from the initial meridian plane).

Spherical coordinates (r, θ, φ) as often used in mathematics: radial distance r, azimuthal angle θ, and polar angle φ. The meanings of θ and φ have been swapped compared to the physics convention. As in physics, ρ (rho) is often used instead of r, to avoid confusion with the value r in cylindrical and 2D polar coordinates.

A globe showing the radial distance, polar angle and azimuthal angle of a point P with respect to a unit sphere, in the mathematics convention. In this image, r equals 4/6, θ equals 90°, and φ equals 30°.

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin; its polar angle measured from a fixed polar axis or zenith direction; and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the fixed axis, measured from another fixed reference direction on that plane. When radius is fixed, the two angular coordinates make a coordinate system on the sphere sometimes called spherical polar coordinates.

The radial distance is also called the radius or radial coordinate. The polar angle may be called colatitude, zenith angle, normal angle, or inclination angle. The polar angle is often replaced by the elevation angle measured from the reference plane towards the positive Z axis; the depression angle is the negative of the elevation angle.

The use of symbols and the order of the coordinates differs among sources and disciplines. This article will use the ISO convention frequently encountered in physics: ${\displaystyle (r,\theta ,\phi )}$ gives the radial distance, polar angle, and azimuthal angle. By contrast, in many mathematics books, ${\displaystyle (\rho ,\theta ,\phi )}$ or ${\displaystyle (r,\theta ,\phi )}$ gives the radial distance, azimuthal angle, and polar angle, switching the meanings of θ and φ. Other conventions are also used, such as r for radius from the z-axis, so great care needs to be taken to check the meaning of the symbols.

According to the conventions of geographical coordinate systems, positions are measured by latitude, longitude, and height (altitude). There are a number of celestial coordinate systems based on different fundamental planes and with different terms for the various coordinates. The spherical coordinate systems used in mathematics normally use radians rather than degrees and measure the azimuthal angle counterclockwise from the x-axis to the y-axis rather than clockwise from north (0°) to east (+90°) like the horizontal coordinate system.

The spherical coordinate system can be seen as one possible generalization of the polar coordinate system in three-dimensional space. It can also be further extended to higher-dimensional spaces and is then referred to as a hyperspherical coordinate system.

Definition

To define a spherical coordinate system, one must choose two orthogonal directions, the zenith and the azimuth reference, and an origin point in space. These choices determine a reference plane that contains the origin and is perpendicular to the zenith. The spherical coordinates of a point P are then defined as follows:

• The radius or radial distance is the Euclidean distance from the origin O to P.
• The azimuth (or azimuthal angle) is the signed angle measured from the azimuth reference direction to the orthogonal projection of the line segment OP on the reference plane.
• The inclination (or polar angle) is the angle between the zenith direction and the line segment OP.

The sign of the azimuth is determined by choosing what is a positive sense of turning about the zenith. This choice is arbitrary, and is part of the coordinate system's definition.

The elevation angle is the signed angle between the reference plane and the line segment OP, where positive angles are oriented towards the zenith. Equivalently, it is 90 degrees (π/2 radians) minus the inclination angle.

If the inclination is zero or 180 degrees (π radians), the azimuth is arbitrary. If the radius is zero, both azimuth and inclination are arbitrary.

In linear algebra, the vector from the origin O to the point P is often called the position vector of P.

Conventions

Several different conventions exist for representing the three coordinates, and for the order in which they should be written. The use of ${\displaystyle (r,\theta ,\phi )}$ to denote radial distance, inclination (or elevation), and azimuth, respectively, is common practice in physics, and is specified by ISO standard 80000-2:2019, and earlier in ISO 31-11 (1992).

This article, as stated above, will use the ISO convention, ${\displaystyle (r,\theta ,\phi ),}$ gives the radial distance, polar angle, and azimuthal angle.

However, some authors (including mathematicians) use ρ for radial distance, φ for inclination (or elevation) and θ for azimuth, and r for radius from the z-axis, which "provides a logical extension of the usual polar coordinates notation". Some authors may also list the azimuth before the inclination (or elevation). Some combinations of these choices result in a left-handed coordinate system. The standard convention ${\displaystyle (r,\theta ,\phi )}$ conflicts with the usual notation for two-dimensional polar coordinates and three-dimensional cylindrical coordinates, where θ is often used for the azimuth.

The angles are typically measured in degrees (°) or radians (rad), where 360° = 2π rad. Degrees are most common in geography, astronomy, and engineering, whereas radians are commonly used in mathematics and theoretical physics. The unit for radial distance is usually determined by the context.

When the system is used for physical three-space, it is customary to use positive sign for azimuth angles that are measured in the counter-clockwise sense from the reference direction on the reference plane, as seen from the zenith side of the plane. This convention is used, in particular, for geographical coordinates, where the "zenith" direction is north and positive azimuth (longitude) angles are measured eastwards from some prime meridian.

Major conventions

coordinates	corresponding local geographical directions (Z, X, Y)	right/left-handed
(r, θinc, φaz,right)	(U, S, E)	right
(r, φaz,right, θel)	(U, E, N)	right
(r, θel, φaz,right)	(U, N, E)	left

Note: easting (E), northing (N), upwardness (U). Local azimuth angle would be measured, e.g., counterclockwise from S to E in the case of (U, S, E).

Unique coordinates

Any spherical coordinate triplet ${\displaystyle (r,\theta ,\phi )}$ specifies a single point of three-dimensional space. On the other hand, every point has infinitely many equivalent spherical coordinates. One can add or subtract any number of full turns to either angular measure without changing the angles themselves, and therefore without changing the point. It is also convenient, in many contexts, to allow negative radial distances, with the convention that ${\displaystyle (-r,-\theta ,\phi )}$ is equivalent to ${\displaystyle (r,\theta ,\phi )}$ for any r, θ, and φ. Moreover, ${\displaystyle (r,-\theta ,\phi )}$ is equivalent to ${\displaystyle (r,\theta ,\phi +{180}^{\circ })}$.

If it is necessary to define a unique set of spherical coordinates for each point, one must restrict their ranges. A common choice is

• radial distance: r ≥ 0,
• polar angle: 0° ≤ θ ≤ 180° (π rad),
• azimuth : 0° ≤ φ < 360° (2π rad).

However, the azimuth φ is often restricted to the interval (−180°, +180°], or (−π, +π ] in radians, instead of [0, 360°). This is the standard convention for geographic longitude.

For the polar angle θ, the range [0°, 180°] for inclination is equivalent to [−90°, +90°] for elevation. In geography, the latitude is the elevation.

Even with these restrictions, if polar angle is 0° or 180° (elevation is 90° or −90°) then the azimuth angle is arbitrary; and if r is zero, both azimuth and polar angle are arbitrary. To make the coordinates unique, one can use the convention that in these cases the arbitrary coordinates are set to zero.

Plotting

To plot a dot from its spherical coordinates (r, θ, φ), where θ is inclination, move r units from the origin in the zenith direction, rotate by θ about the origin towards the azimuth reference direction, and rotate by φ about the zenith in the proper direction.

Applications

Just as the two-dimensional Cartesian coordinate system is useful on the plane, a two-dimensional spherical coordinate system is useful on the surface of a sphere. In this system, the sphere is taken as a unit sphere, so the radius is unity and can generally be ignored. This simplification can also be very useful when dealing with objects such as rotational matrices.

Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as volume integrals inside a sphere, the potential energy field surrounding a concentrated mass or charge, or global weather simulation in a planet's atmosphere. A sphere that has the Cartesian equation x² + y² + z² = c² has the simple equation r = c in spherical coordinates.

Two important partial differential equations that arise in many physical problems, Laplace's equation and the Helmholtz equation, allow a separation of variables in spherical coordinates. The angular portions of the solutions to such equations take the form of spherical harmonics.

Another application is ergonomic design, where r is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out.

The output pattern of an industrial loudspeaker shown using spherical polar plots taken at six frequencies

Three dimensional modeling of loudspeaker output patterns can be used to predict their performance. A number of polar plots are required, taken at a wide selection of frequencies, as the pattern changes greatly with frequency. Polar plots help to show that many loudspeakers tend toward omnidirectionality at lower frequencies.

The spherical coordinate system is also commonly used in 3D game development to rotate the camera around the player's position

In geography

Main article: Geographic coordinate system

See also: ECEF

To a first approximation, the geographic coordinate system uses elevation angle (latitude) in degrees north of the equator plane, in the range −90° ≤ φ ≤ 90°, instead of inclination. Latitude is either geocentric latitude, measured at the Earth's center and designated variously by ψ, q, φ′, φ_c, φ_g or geodetic latitude, measured by the observer's local vertical, and commonly designated φ. The polar angle, which is 90° minus the latitude and ranges from 0 to 180°, is called colatitude in geography.

The azimuth angle (longitude), commonly denoted by λ, is measured in degrees east or west from some conventional reference meridian (most commonly the IERS Reference Meridian), so its domain is −180° ≤ λ ≤ 180°. For positions on the Earth or other solid celestial body, the reference plane is usually taken to be the plane perpendicular to the axis of rotation.

Instead of the radial distance, geographers commonly use altitude above or below some reference surface (vertical datum), which may be the mean sea level. The radial distance r can be computed from the altitude by adding the radius of Earth, which is approximately 6,360 ± 11 km (3,952 ± 7 miles).

However, modern geographical coordinate systems are quite complex, and the positions implied by these simple formulae may be wrong by several kilometers. The precise standard meanings of latitude, longitude and altitude are currently defined by the World Geodetic System (WGS), and take into account the flattening of the Earth at the poles (about 21 km or 13 miles) and many other details.

Planetary coordinate systems use formulations analogous to the geographic coordinate system.

In astronomy

A series of astronomical coordinate systems are used to measure the elevation angle from different fundamental planes. These reference planes are the observer's horizon, the celestial equator (defined by Earth's rotation), the plane of the ecliptic (defined by Earth's orbit around the Sun), the plane of the earth terminator (normal to the instantaneous direction to the Sun), and the galactic equator (defined by the rotation of the Milky Way).

Coordinate system conversions

See also: List of common coordinate transformations § To spherical coordinates

As the spherical coordinate system is only one of many three-dimensional coordinate systems, there exist equations for converting coordinates between the spherical coordinate system and others.

Cartesian coordinates

The spherical coordinates of a point in the ISO convention (i.e. for physics: radius r, inclination θ, azimuth φ) can be obtained from its Cartesian coordinates (x, y, z) by the formulae

$${\displaystyle \begin{array}{rl}r& =\sqrt{x^2+y^2+z^2}\\ \theta & =\arccos \frac{z}{\sqrt{x^2+y^2+z^2}}=\arccos \frac{z}{r}=\{\begin{array}{ll}\arctan \frac{\sqrt{x^2+y^2}}{z}& \text{if }z>0\\ \pi +\arctan \frac{\sqrt{x^2+y^2}}{z}& \text{if }z<0\\ +\frac{\pi }{2}& \text{if }z=0\text{ and }xy\ne 0\\ \text{undefined}& \text{if }x=y=z=0\end{array}\\ \phi & =\mathrm{sgn}(y)\arccos \frac{x}{\sqrt{x^2+y^2}}=\{\begin{array}{ll}\arctan (\frac{y}{x})& \text{if }x>0,\\ \arctan (\frac{y}{x})+\pi & \text{if }x<0\text{ and }y\ge 0,\\ \arctan (\frac{y}{x})-\pi & \text{if }x<0\text{ and }y<0,\\ +\frac{\pi }{2}& \text{if }x=0\text{ and }y>0,\\ -\frac{\pi }{2}& \text{if }x=0\text{ and }y<0,\\ \text{undefined}& \text{if }x=0\text{ and }y=0.\end{array}\end{array}}$$

The inverse tangent denoted in φ = arctan y/x must be suitably defined, taking into account the correct quadrant of (x, y). See the article on atan2.

Alternatively, the conversion can be considered as two sequential rectangular to polar conversions: the first in the Cartesian xy plane from (x, y) to (R, φ), where R is the projection of r onto the xy-plane, and the second in the Cartesian zR-plane from (z, R) to (r, θ). The correct quadrants for φ and θ are implied by the correctness of the planar rectangular to polar conversions.

These formulae assume that the two systems have the same origin, that the spherical reference plane is the Cartesian xy plane, that θ is inclination from the z direction, and that the azimuth angles are measured from the Cartesian x axis (so that the y axis has φ = +90°). If θ measures elevation from the reference plane instead of inclination from the zenith the arccos above becomes an arcsin, and the cos θ and sin θ below become switched.

Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (radius r, inclination θ, azimuth φ), where r ∈ [0, ∞), θ ∈ [0, π], φ ∈ [0, 2π), by

$${\displaystyle \begin{array}{rl}x& =r\sin \theta \cos \phi ,\\ y& =r\sin \theta \sin \phi ,\\ z& =r\cos \theta .\end{array}}$$

Cylindrical coordinates

Main article: Cylindrical coordinate system

Cylindrical coordinates (axial radius ρ, azimuth φ, elevation z) may be converted into spherical coordinates (central radius r, inclination θ, azimuth φ), by the formulas

$${\displaystyle \begin{array}{rl}r& =\sqrt{{\rho }^2+z^2},\\ \theta & =\arctan \frac{\rho }{z}=\arccos \frac{z}{\sqrt{{\rho }^2+z^2}},\\ \phi & =\phi .\end{array}}$$

Conversely, the spherical coordinates may be converted into cylindrical coordinates by the formulae

$${\displaystyle \begin{array}{rl}\rho & =r\sin \theta ,\\ \phi & =\phi ,\\ z& =r\cos \theta .\end{array}}$$

These formulae assume that the two systems have the same origin and same reference plane, measure the azimuth angle φ in the same senses from the same axis, and that the spherical angle θ is inclination from the cylindrical z axis.

Generalization

See also: Ellipsoidal coordinates

It is also possible to deal with ellipsoids in Cartesian coordinates by using a modified version of the spherical coordinates.

Let P be an ellipsoid specified by the level set

$${\displaystyle a{x}^2+b{y}^2+c{z}^2=d.}$$

The modified spherical coordinates of a point in P in the ISO convention (i.e. for physics: radius r, inclination θ, azimuth φ) can be obtained from its Cartesian coordinates (x, y, z) by the formulae

$${\displaystyle \begin{array}{rl}x& =\frac{1}{\sqrt{a}}r\sin \theta \cos \phi ,\\ y& =\frac{1}{\sqrt{b}}r\sin \theta \sin \phi ,\\ z& =\frac{1}{\sqrt{c}}r\cos \theta ,\\ r^2& =a{x}^2+b{y}^2+c{z}^2.\end{array}}$$

An infinitesimal volume element is given by

$${\displaystyle \mathrm{d}V=\left|\frac{\mathrm{\partial }(x,y,z)}{\mathrm{\partial }(r,\theta ,\phi )}\right|drd\theta d\phi =\frac{1}{\sqrt{abc}}{r}^2\sin \theta \mathrm{d}r\mathrm{d}\theta \mathrm{d}\phi =\frac{1}{\sqrt{abc}}{r}^2\mathrm{d}r\mathrm{d}\mathrm{\Omega }.}$$

The square-root factor comes from the property of the determinant that allows a constant to be pulled out from a column:

$${\displaystyle \left|\begin{array}{ccc}ka& b& c\\ kd& e& f\\ kg& h& i\end{array}\right|=k\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|.}$$

Integration and differentiation in spherical coordinates

Unit vectors in spherical coordinates

The following equations (Iyanaga 1977) assume that the colatitude θ is the inclination from the positive z axis, as in the physics convention discussed.

The line element for an infinitesimal displacement from (r, θ, φ) to (r + dr, θ + dθ, φ + dφ) is

$${\displaystyle \mathrm{d}\mathbf{r}=\mathrm{d}r\hat{\mathbf{r}}+r\mathrm{d}\theta \hat{\mathit{\theta }}+r\sin \theta \mathrm{d}\phi \hat{\mathit{\phi }},}$$

where

$${\displaystyle \begin{array}{rl}\hat{\mathbf{r}}& =\sin \theta \cos \phi \hat{\mathbf{x}}+\sin \theta \sin \phi \hat{\mathbf{y}}+\cos \theta \hat{\mathbf{z}},\\ \hat{\mathit{\theta }}& =\cos \theta \cos \phi \hat{\mathbf{x}}+\cos \theta \sin \phi \hat{\mathbf{y}}-\sin \theta \hat{\mathbf{z}},\\ \hat{\mathit{\phi }}& =-\sin \phi \hat{\mathbf{x}}+\cos \phi \hat{\mathbf{y}}\end{array}}$$

are the local orthogonal unit vectors in the directions of increasing r, θ, and φ, respectively, and x̂, ŷ, and ẑ are the unit vectors in Cartesian coordinates. The linear transformation to this right-handed coordinate triplet is a rotation matrix,

$${\displaystyle R=\left(\begin{array}{ccc}\sin \theta \cos \phi & \sin \theta \sin \phi & \phantom{}\cos \theta \\ \cos \theta \cos \phi & \cos \theta \sin \phi & -\sin \theta \\ -\sin \phi & \cos \phi & \phantom{}0\end{array}\right).}$$

This gives the transformation from the spherical to the cartesian, the other way around is given by its inverse. Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose.

The Cartesian unit vectors are thus related to the spherical unit vectors by:

$${\displaystyle \left[\begin{array}{c}\hat{\mathbf{x}}\\ \hat{\mathbf{y}}\\ \hat{\mathbf{z}}\end{array}\right]=\left[\begin{array}{ccc}\sin \theta \cos \phi & \cos \theta \cos \phi & -\sin \phi \\ \sin \theta \sin \phi & \cos \theta \sin \phi & \phantom{}\cos \phi \\ \cos \theta & -\sin \theta & \phantom{}0\end{array}\right]\left[\begin{array}{c}\hat{\mathit{r}}\\ \hat{\mathit{\theta }}\\ \hat{\mathit{\phi }}\end{array}\right]}$$

The general form of the formula to prove the differential line element, is

$${\displaystyle \mathrm{d}\mathbf{r}=\sum _i\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{x}_i}\mathrm{d}{x}_i=\sum _i\left|\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{x}_i}\right|\frac{\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{x}_i}}{\left|\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{x}_i}\right|}\mathrm{d}{x}_i=\sum _i\left|\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{x}_i}\right|\mathrm{d}{x}_i{\hat{\mathit{x}}}_i,}$$

that is, the change in ${\displaystyle \mathbf{r}}$ is decomposed into individual changes corresponding to changes in the individual coordinates.

To apply this to the present case, one needs to calculate how ${\displaystyle \mathbf{r}}$ changes with each of the coordinates. In the conventions used,

$${\displaystyle \mathbf{r}=\left[\begin{array}{c}r\sin \theta \cos \phi \\ r\sin \theta \sin \phi \\ r\cos \theta \end{array}\right].}$$

Thus,

$${\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }r}=\left[\begin{array}{c}\sin \theta \cos \phi \\ \sin \theta \sin \phi \\ \cos \theta \end{array}\right]=\hat{\mathbf{r}},\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }\theta }=\left[\begin{array}{c}r\cos \theta \cos \phi \\ r\cos \theta \sin \phi \\ -r\sin \theta \end{array}\right]=r\hat{\mathit{\theta }},\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }\phi }=\left[\begin{array}{c}-r\sin \theta \sin \phi \\ \phantom{}r\sin \theta \cos \phi \\ 0\end{array}\right]=r\sin \theta \hat{\mathit{\phi }}.}$$

The desired coefficients are the magnitudes of these vectors:

$${\displaystyle \left|\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }r}\right|=1,\left|\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }\theta }\right|=r,\left|\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }\phi }\right|=r\sin \theta .}$$

The surface element spanning from θ to θ + dθ and φ to φ + dφ on a spherical surface at (constant) radius r is then

$${\displaystyle \mathrm{d}{S}_r=\Vert \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }\theta }\times \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }\phi }\Vert \mathrm{d}\theta \mathrm{d}\phi =\left|r\hat{\mathit{\theta }}\times r\sin \theta \hat{\mathit{\phi }}\right|=r^2\sin \theta \mathrm{d}\theta \mathrm{d}\phi .}$$

Thus the differential solid angle is

$${\displaystyle \mathrm{d}\mathrm{\Omega }=\frac{\mathrm{d}{S}_r}{r^2}=\sin \theta \mathrm{d}\theta \mathrm{d}\phi .}$$

The surface element in a surface of polar angle θ constant (a cone with vertex the origin) is

$${\displaystyle \mathrm{d}{S}_{\theta }=r\sin \theta \mathrm{d}\phi \mathrm{d}r.}$$

The surface element in a surface of azimuth φ constant (a vertical half-plane) is

$${\displaystyle \mathrm{d}{S}_{\phi }=r\mathrm{d}r\mathrm{d}\theta .}$$

The volume element spanning from r to r + dr, θ to θ + dθ, and φ to φ + dφ is specified by the determinant of the Jacobian matrix of partial derivatives,

$${\displaystyle J=\frac{\mathrm{\partial }(x,y,z)}{\mathrm{\partial }(r,\theta ,\phi )}=\left(\begin{array}{ccc}\sin \theta \cos \phi & r\cos \theta \cos \phi & -r\sin \theta \sin \phi \\ \sin \theta \sin \phi & r\cos \theta \sin \phi & \phantom{}r\sin \theta \cos \phi \\ \cos \theta & -r\sin \theta & \phantom{}0\end{array}\right),}$$

namely

$${\displaystyle \mathrm{d}V=\left|\frac{\mathrm{\partial }(x,y,z)}{\mathrm{\partial }(r,\theta ,\phi )}\right|\mathrm{d}r\mathrm{d}\theta \mathrm{d}\phi =r^2\sin \theta \mathrm{d}r\mathrm{d}\theta \mathrm{d}\phi =r^2\mathrm{d}r\mathrm{d}\mathrm{\Omega }.}$$

Thus, for example, a function f(r, θ, φ) can be integrated over every point in R³ by the triple integral

$${\displaystyle \underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}f(r,\theta ,\phi )r^2\sin \theta \mathrm{d}r\mathrm{d}\theta \mathrm{d}\phi .}$$

The del operator in this system leads to the following expressions for the gradient, divergence, curl and (scalar) Laplacian,

$${\displaystyle \begin{array}{rl}\mathrm{\nabla }f=& \frac{\mathrm{\partial }f}{\mathrm{\partial }r}\hat{\mathbf{r}}+\frac{1}{r}\frac{\mathrm{\partial }f}{\mathrm{\partial }\theta }\hat{\mathit{\theta }}+\frac{1}{r\sin \theta }\frac{\mathrm{\partial }f}{\mathrm{\partial }\phi }\hat{\mathit{\phi }},\\ \mathrm{\nabla }\cdot \mathbf{A}=& \frac{1}{r^2}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r^2{A}_r\right)+\frac{1}{r\sin \theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(\sin \theta A_{\theta }\right)+\frac{1}{r\sin \theta }\frac{\mathrm{\partial }{A}_{\phi }}{\mathrm{\partial }\phi },\\ \mathrm{\nabla }\times \mathbf{A}=& \frac{1}{r\sin \theta }\left(\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(A_{\phi }\sin \theta \right)-\frac{\mathrm{\partial }{A}_{\theta }}{\mathrm{\partial }\phi }\right)\hat{\mathbf{r}}\\ & +\frac{1}{r}\left(\frac{1}{\sin \theta }\frac{\mathrm{\partial }{A}_r}{\mathrm{\partial }\phi }-\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r{A}_{\phi }\right)\right)\hat{\mathit{\theta }}\\ & +\frac{1}{r}\left(\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r{A}_{\theta }\right)-\frac{\mathrm{\partial }{A}_r}{\mathrm{\partial }\theta }\right)\hat{\mathit{\phi }},\\ {\mathrm{\nabla }}^{2}f=& \frac{1}{r^2}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r^2\frac{\mathrm{\partial }f}{\mathrm{\partial }r}\right)+\frac{1}{r^2\sin \theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(\sin \theta \frac{\mathrm{\partial }f}{\mathrm{\partial }\theta }\right)+\frac{1}{r^2{\sin }^2\theta }\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{\phi }^2}\\ =& \left(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{r}^2}+\frac{2}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\right)f+\frac{1}{r^2\sin \theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(\sin \theta \frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\right)f+\frac{1}{r^2{\sin }^2\theta }\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\phi }^2}f.\end{array}}$$

Further, the inverse Jacobian in Cartesian coordinates is

$${\displaystyle J^{-1}=\left(\begin{array}{ccc}{\displaystyle \frac{x}{r}}& {\displaystyle \frac{y}{r}}& {\displaystyle \frac{z}{r}}\\ \\ {\displaystyle \frac{xz}{r^2\sqrt{x^2+y^2}}}& {\displaystyle \frac{yz}{r^2\sqrt{x^2+y^2}}}& {\displaystyle \frac{-(x^2+y^2)}{r^2\sqrt{x^2+y^2}}}\\ \\ {\displaystyle \frac{-y}{x^2+y^2}}& {\displaystyle \frac{x}{x^2+y^2}}& 0\end{array}\right).}$$

The metric tensor in the spherical coordinate system is ${\displaystyle g=J^{T}J}$.

Distance in spherical coordinates

In spherical coordinates, given two points with φ being the azimuthal coordinate

$${\displaystyle \begin{array}{rl}\mathbf{r}& =(r,\theta ,\phi ),\\ {\mathbf{r}}^{\prime }& =(r^{\prime },{\theta }^{\prime },{\phi }^{\prime })\end{array}}$$

The distance between the two points can be expressed as

$${\displaystyle \begin{array}{rl}\mathbf{D}& =\sqrt{r^2+r^{\prime 2}-2r{r}^{\prime }(\sin \theta \sin {\theta }^{\prime }\cos (\phi -{\phi }^{\prime })+\cos \theta \cos {\theta }^{\prime })}\end{array}}$$

Kinematics

In spherical coordinates, the position of a point or particle (although better written as a triple${\displaystyle (r,\theta ,\phi )}$) can be written as

$${\displaystyle \mathbf{r}=r\hat{\mathbf{r}}.}$$

Its velocity is then

$${\displaystyle \mathbf{v}=\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}=\dot{r}\hat{\mathbf{r}}+r\dot{\theta }\hat{\mathit{\theta }}+r\dot{\phi }\sin \theta \hat{\mathit{\phi }}}$$

and its acceleration is

$${\displaystyle \begin{array}{rl}\mathbf{a}=\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}=& \left(\ddot{r}-r{\dot{\theta }}^2-r{\dot{\phi }}^2{\sin }^2\theta \right)\hat{\mathbf{r}}\\ & +\left(r\ddot{\theta }+2\dot{r}\dot{\theta }-r{\dot{\phi }}^2\sin \theta \cos \theta \right)\hat{\mathit{\theta }}\\ & +\left(r\ddot{\phi }\sin \theta +2\dot{r}\dot{\phi }\sin \theta +2r\dot{\theta }\dot{\phi }\cos \theta \right)\hat{\mathit{\phi }}\end{array}}$$

The angular momentum is

$${\displaystyle \mathbf{L}=\mathbf{r}\times \mathbf{p}=\mathbf{r}\times m\mathbf{v}=m{r}^2(-\dot{\phi }\sin \theta \hat{\mathit{\theta }}+\dot{\theta }\hat{\mathit{\phi }})}$$

Where ${\displaystyle m}$ is mass. In the case of a constant φ or else θ = π/2, this reduces to vector calculus in polar coordinates.

The corresponding angular momentum operator then follows from the phase-space reformulation of the above,

$${\displaystyle \mathbf{L}=-i\hslash \mathbf{r}\times \mathrm{\nabla }=i\hslash \left(\frac{\hat{\mathit{\theta }}}{\sin (\theta )}\frac{\mathrm{\partial }}{\mathrm{\partial }\varphi }-\hat{\mathit{\varphi }}\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\right).}$$

The torque is given as

$${\displaystyle \tau =\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t}=\mathbf{r}\times \mathbf{F}=-m\left(2r\dot{r}\dot{\phi }\sin \theta +r^2\ddot{\phi }\sin \theta +2r^2\dot{\theta }\dot{\phi }\cos \theta \right)\hat{\mathit{\theta }}+m\left(r^2\ddot{\theta }+2r\dot{r}\dot{\theta }-r^2{\dot{\phi }}^2\sin \theta \cos \theta \right)\hat{\mathit{\phi }}}$$

The kinetic energy is given as

$${\displaystyle E_k=\frac{1}{2}m\left[\left({\dot{r}}^2\right)+{\left(r\dot{\theta }\right)}^2+{\left(r\dot{\phi }\sin \theta \right)}^2\right]}$$

Exercise

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Exercise

Cycling is a popular form of exercise.

Weight training

Exercise is a body activity that enhances or maintains physical fitness and overall health and wellness.

It is performed for various reasons, including weight loss or maintenance, to aid growth and improve strength, develop muscles and the cardiovascular system, hone athletic skills, improve health, or simply for enjoyment. Many individuals choose to exercise outdoors where they can congregate in groups, socialize, and improve well-being as well as mental health.

In terms of health benefits, usually, 2.5 hours of moderate-intensity exercise per week is recommended for reducing the risk of health issues. At the same time, even doing a small amount of exercise is healthier than doing none. Already doing an hour and a quarter (11 minutes/day) of exercise could reduce the risk of early death, cardiovascular disease, stroke, and cancer.

Classification

Physical exercises are generally grouped into three types, depending on the overall effect they have on the human body:

• Aerobic exercise is any physical activity that uses large muscle groups and causes the body to use more oxygen than it would while resting. The goal of aerobic exercise is to increase cardiovascular endurance. Examples of aerobic exercise include running, cycling, swimming, brisk walking, skipping rope, rowing, hiking, dancing, playing tennis, continuous training, and long distance running.
• Anaerobic exercise, which includes strength and resistance training, can firm, strengthen, and increase muscle mass, as well as improve bone density, balance, and coordination. Examples of strength exercises are push-ups, pull-ups, lunges, squats, bench press. Anaerobic exercise also includes weight training, functional training, Eccentric Training, interval training, sprinting, and high-intensity interval training which increase short-term muscle strength.
• Flexibility exercises stretch and lengthen muscles. Activities such as stretching help to improve joint flexibility and keep muscles limber. The goal is to improve the range of motion which can reduce the chance of injury.

Physical exercise can also include training that focuses on accuracy, agility, power, and speed.

Types of exercise can also be classified as dynamic or static. 'Dynamic' exercises such as steady running, tend to produce a lowering of the diastolic blood pressure during exercise, due to the improved blood flow. Conversely, static exercise (such as weight-lifting) can cause the systolic pressure to rise significantly, albeit transiently, during the performance of the exercise.

Health effects

Main article: Exercise physiology

Exercise in a gym.

Physical exercise is important for maintaining physical fitness and can contribute to maintaining a healthy weight, regulating the digestive system, building and maintaining healthy bone density, muscle strength, and joint mobility, promoting physiological well-being, reducing surgical risks, and strengthening the immune system. Some studies indicate that exercise may increase life expectancy and the overall quality of life. People who participate in moderate to high levels of physical exercise have a lower mortality rate compared to individuals who by comparison are not physically active. Moderate levels of exercise have been correlated with preventing aging by reducing inflammatory potential. The majority of the benefits from exercise are achieved with around 3500 metabolic equivalent (MET) minutes per week, with diminishing returns at higher levels of activity. For example, climbing stairs 10 minutes, vacuuming 15 minutes, gardening 20 minutes, running 20 minutes, and walking or bicycling for transportation 25 minutes on a daily basis would together achieve about 3000 MET minutes a week. A lack of physical activity causes approximately 6% of the burden of disease from coronary heart disease, 7% of type 2 diabetes, 10% of breast cancer and 10% of colon cancer worldwide. Overall, physical inactivity causes 9% of premature mortality worldwide.

Fitness

Main article: Physical fitness

Most people can increase fitness by increasing physical activity levels. Increases in muscle size from resistance training are primarily determined by diet and testosterone. This genetic variation in improvement from training is one of the key physiological differences between elite athletes and the larger population. There is evidence that exercising in middle age may lead to better physical ability later in life.

Early motor skills and development is also related to physical activity and performance later in life. Children who are more proficient with motor skills early on are more inclined to be physically active, and thus tend to perform well in sports and have better fitness levels. Early motor proficiency has a positive correlation to childhood physical activity and fitness levels, while less proficiency in motor skills results in a more sedentary lifestyle.

The type and intensity of physical activity performed may have an effect on a person's fitness level. There is some weak evidence that high-intensity interval training may improve a person's VO2 max slightly more than lower intensity endurance training. However, unscientific fitness methods could lead to sports injuries.

Cardiovascular system

Main article: Cardiovascular fitness

The beneficial effect of exercise on the cardiovascular system is well documented. There is a direct correlation between physical inactivity and cardiovascular disease, and physical inactivity is an independent risk factor for the development of coronary artery disease. Low levels of physical exercise increase the risk of cardiovascular diseases mortality.

Children who participate in physical exercise experience greater loss of body fat and increased cardiovascular fitness. Studies have shown that academic stress in youth increases the risk of cardiovascular disease in later years; however, these risks can be greatly decreased with regular physical exercise.

There is a dose-response relationship between the amount of exercise performed from approximately 700–2000 kcal of energy expenditure per week and all-cause mortality and cardiovascular disease mortality in middle-aged and elderly men. The greatest potential for reduced mortality is seen in sedentary individuals who become moderately active.

Studies have shown that since heart disease is the leading cause of death in women, regular exercise in aging women leads to healthier cardiovascular profiles.

Most beneficial effects of physical activity on cardiovascular disease mortality can be attained through moderate-intensity activity (40–60% of maximal oxygen uptake, depending on age). Persons who modify their behavior after myocardial infarction to include regular exercise have improved rates of survival. Persons who remain sedentary have the highest risk for all-cause and cardiovascular disease mortality. According to the American Heart Association, exercise reduces the risk of cardiovascular diseases, including heart attack and stroke.

Some have suggested that increases in physical exercise might decrease healthcare costs, increase the rate of job attendance, as well as increase the amount of effort women put into their jobs.

Immune system

Although there have been hundreds of studies on physical exercise and the immune system, there is little direct evidence on its connection to illness. Epidemiological evidence suggests that moderate exercise has a beneficial effect on the human immune system; an effect which is modeled in a J curve. Moderate exercise has been associated with a 29% decreased incidence of upper respiratory tract infections (URTI), but studies of marathon runners found that their prolonged high-intensity exercise was associated with an increased risk of infection occurrence. However, another study did not find the effect. Immune cell functions are impaired following acute sessions of prolonged, high-intensity exercise, and some studies have found that athletes are at a higher risk for infections. Studies have shown that strenuous stress for long durations, such as training for a marathon, can suppress the immune system by decreasing the concentration of lymphocytes. The immune systems of athletes and nonathletes are generally similar. Athletes may have a slightly elevated natural killer cell count and cytolytic action, but these are unlikely to be clinically significant.

Vitamin C supplementation has been associated with a lower incidence of upper respiratory tract infections in marathon runners.

Biomarkers of inflammation such as C-reactive protein, which are associated with chronic diseases, are reduced in active individuals relative to sedentary individuals, and the positive effects of exercise may be due to its anti-inflammatory effects. In individuals with heart disease, exercise interventions lower blood levels of fibrinogen and C-reactive protein, an important cardiovascular risk marker. The depression in the immune system following acute bouts of exercise may be one of the mechanisms for this anti-inflammatory effect.

Cancer

A systematic review evaluated 45 studies that examined the relationship between physical activity and cancer survival rates. According to the review, "[there] was consistent evidence from 27 observational studies that physical activity is associated with reduced all-cause, breast cancer–specific, and colon cancer–specific mortality. There is currently insufficient evidence regarding the association between physical activity and mortality for survivors of other cancers." Evidence suggests that exercise may positively affect the quality of life in cancer survivors, including factors such as anxiety, self-esteem and emotional well-being. For people with cancer undergoing active treatment, exercise may also have positive effects on health-related quality of life, such as fatigue and physical functioning. This is likely to be more pronounced with higher intensity exercise.

Exercise may contribute to a reduction of cancer-related fatigue in survivors of breast cancer. Although there is only limited scientific evidence on the subject, people with cancer cachexia are encouraged to engage in physical exercise. Due to various factors, some individuals with cancer cachexia have a limited capacity for physical exercise. Compliance with prescribed exercise is low in individuals with cachexia and clinical trials of exercise in this population often have high drop-out rates.

There is low-quality evidence for an effect of aerobic physical exercises on anxiety and serious adverse events in adults with hematological malignancies. Aerobic physical exercise may result in little to no difference in the mortality, quality of life, or physical functioning. These exercises may result in a slight reduction in depression and reduction in fatigue.

Neurobiological

A woman engaging in aerobic exercise (jogging)

The neurobiological effects of physical exercise are numerous and involve a wide range of interrelated effects on brain structure, brain function, and cognition. A large body of research in humans has demonstrated that consistent aerobic exercise (e.g., 30 minutes every day) induces persistent improvements in certain cognitive functions, healthy alterations in gene expression in the brain, and beneficial forms of neuroplasticity and behavioral plasticity; some of these long-term effects include: increased neuron growth, increased neurological activity (e.g., c-Fos and BDNF signaling), improved stress coping, enhanced cognitive control of behavior, improved declarative, spatial, and working memory, and structural and functional improvements in brain structures and pathways associated with cognitive control and memory. The effects of exercise on cognition have important implications for improving academic performance in children and college students, improving adult productivity, preserving cognitive function in old age, preventing or treating certain neurological disorders, and improving overall quality of life.

In healthy adults, aerobic exercise has been shown to induce transient effects on cognition after a single exercise session and persistent effects on cognition following regular exercise over the course of several months. People who regularly perform an aerobic exercise (e.g., running, jogging, brisk walking, swimming, and cycling) have greater scores on neuropsychological function and performance tests that measure certain cognitive functions, such as attentional control, inhibitory control, cognitive flexibility, working memory updating and capacity, declarative memory, spatial memory, and information processing speed. The transient effects of exercise on cognition include improvements in most executive functions (e.g., attention, working memory, cognitive flexibility, inhibitory control, problem solving, and decision making) and information processing speed for a period of up to 2 hours after exercising.

Aerobic exercise induces short- and long-term effects on mood and emotional states by promoting positive affect, inhibiting negative affect, and decreasing the biological response to acute psychological stress. Over the short-term, aerobic exercise functions as both an antidepressant and euphoriant, whereas consistent exercise produces general improvements in mood and self-esteem.

Regular aerobic exercise improves symptoms associated with a variety of central nervous system disorders and may be used as adjunct therapy for these disorders. There is clear evidence of exercise treatment efficacy for major depressive disorder and attention deficit hyperactivity disorder. The American Academy of Neurology's clinical practice guideline for mild cognitive impairment indicates that clinicians should recommend regular exercise (two times per week) to individuals who have been diagnosed with this condition. Reviews of clinical evidence also support the use of exercise as an adjunct therapy for certain neurodegenerative disorders, particularly Alzheimer's disease and Parkinson's disease. Regular exercise is also associated with a lower risk of developing neurodegenerative disorders. A large body of preclinical evidence and emerging clinical evidence supports the use of exercise as an adjunct therapy for the treatment and prevention of drug addictions. Regular exercise has also been proposed as an adjunct therapy for brain cancers.

Depression

A number of medical reviews have indicated that exercise has a marked and persistent antidepressant effect in humans, an effect believed to be mediated through enhanced BDNF signaling in the brain. Several systematic reviews have analyzed the potential for physical exercise in the treatment of depressive disorders. The 2013 Cochrane Collaboration review on physical exercise for depression noted that, based upon limited evidence, it is more effective than a control intervention and comparable to psychological or antidepressant drug therapies. Three subsequent 2014 systematic reviews that included the Cochrane review in their analysis concluded with similar findings: one indicated that physical exercise is effective as an adjunct treatment (i.e., treatments that are used together) with antidepressant medication; the other two indicated that physical exercise has marked antidepressant effects and recommended the inclusion of physical activity as an adjunct treatment for mild–moderate depression and mental illness in general. One systematic review noted that yoga may be effective in alleviating symptoms of prenatal depression. Another review asserted that evidence from clinical trials supports the efficacy of physical exercise as a treatment for depression over a 2–4 month period. These benefits have also been noted in old age, with a review conducted in 2019 finding that exercise is an effective treatment for clinically diagnosed depression in older adults.

A meta-analysis from July 2016 concluded that physical exercise improves overall quality of life in individuals with depression relative to controls.

Continuous aerobic exercise can induce a transient state of euphoria, colloquially known as a "runner's high" in distance running or a "rower's high" in crew, through the increased biosynthesis of at least three euphoriant neurochemicals: anandamide (an endocannabinoid), β-endorphin (an endogenous opioid), and phenethylamine (a trace amine and amphetamine analog).

Sleep

Preliminary evidence from a 2012 review indicated that physical training for up to four months may increase sleep quality in adults over 40 years of age. A 2010 review suggested that exercise generally improved sleep for most people, and may help with insomnia, but there is insufficient evidence to draw detailed conclusions about the relationship between exercise and sleep. A 2018 systematic review and meta-analysis suggested that exercise can improve sleep quality in people with insomnia.

Libido

One 2013 study found that exercising improved sexual arousal problems related to antidepressant use.

Respiratory system

People who participate in physical exercise experience increased cardiovascular fitness. There is some level of concern about additional exposure to air pollution when exercising outdoors, especially near traffic.

Mechanism of effects

Skeletal muscle

Main article: Skeletal muscle

Resistance training and subsequent consumption of a protein-rich meal promotes muscle hypertrophy and gains in muscle strength by stimulating myofibrillar muscle protein synthesis (MPS) and inhibiting muscle protein breakdown (MPB). The stimulation of muscle protein synthesis by resistance training occurs via phosphorylation of the mechanistic target of rapamycin (mTOR) and subsequent activation of mTORC1, which leads to protein biosynthesis in cellular ribosomes via phosphorylation of mTORC1's immediate targets (the p70S6 kinase and the translation repressor protein 4EBP1). The suppression of muscle protein breakdown following food consumption occurs primarily via increases in plasma insulin. Similarly, increased muscle protein synthesis (via activation of mTORC1) and suppressed muscle protein breakdown (via insulin-independent mechanisms) has also been shown to occur following ingestion of β-hydroxy β-methylbutyric acid.

Aerobic exercise induces mitochondrial biogenesis and an increased capacity for oxidative phosphorylation in the mitochondria of skeletal muscle, which is one mechanism by which aerobic exercise enhances submaximal endurance performance. These effects occur via an exercise-induced increase in the intracellular AMP:ATP ratio, thereby triggering the activation of AMP-activated protein kinase (AMPK) which subsequently phosphorylates peroxisome proliferator-activated receptor gamma coactivator-1α (PGC-1α), the master regulator of mitochondrial biogenesis.

Diagram of the molecular signaling cascades that are involved in myofibrillar muscle protein synthesis and mitochondrial biogenesis in response to physical exercise and specific amino acids or their derivatives (primarily L-leucine and HMB). Many amino acids derived from food protein promote the activation of mTORC1 and increase protein synthesis by signaling through Rag GTPases.

Abbreviations and representations

Resistance training stimulates muscle protein synthesis (MPS) for a period of up to 48 hours following exercise (shown by dotted line). Ingestion of a protein-rich meal at any point during this period will augment the exercise-induced increase in muscle protein synthesis (shown by solid lines).

Other peripheral organs

Summary of long-term adaptations to regular aerobic and anaerobic exercise. Aerobic exercise can cause several central cardiovascular adaptations, including an increase in stroke volume (SV) and maximal aerobic capacity (VO₂ max), as well as a decrease in resting heart rate (RHR). Long-term adaptations to resistance training, the most common form of anaerobic exercise, include muscular hypertrophy, an increase in the physiological cross-sectional area (PCSA) of muscle(s), and an increase in neural drive, both of which lead to increased muscular strength. Neural adaptations begin more quickly and plateau prior to the hypertrophic response.

Developing research has demonstrated that many of the benefits of exercise are mediated through the role of skeletal muscle as an endocrine organ. That is, contracting muscles release multiple substances known as myokines which promote the growth of new tissue, tissue repair, and multiple anti-inflammatory functions, which in turn reduce the risk of developing various inflammatory diseases. Exercise reduces levels of cortisol, which causes many health problems, both physical and mental. Endurance exercise before meals lowers blood glucose more than the same exercise after meals. There is evidence that vigorous exercise (90–95% of VO₂ max) induces a greater degree of physiological cardiac hypertrophy than moderate exercise (40 to 70% of VO₂ max), but it is unknown whether this has any effects on overall morbidity and/or mortality.^[126] Both aerobic and anaerobic exercise work to increase the mechanical efficiency of the heart by increasing cardiac volume (aerobic exercise), or myocardial thickness (strength training). Ventricular hypertrophy, the thickening of the ventricular walls, is generally beneficial and healthy if it occurs in response to exercise.

Central nervous system

Further information: Neurobiological effects of physical exercise § Neuroplasticity

The effects of physical exercise on the central nervous system are mediated in part by specific neurotrophic factor hormones that are released into the blood stream by muscles, including BDNF, IGF-1, and VEGF.

Public health measures

Community-wide and school campaigns are often used in an attempt to increase a population's level of physical activity. Studies to determine the effectiveness of these types of programs need to be interpreted cautiously as the results vary. There is some evidence that certain types of exercise programmes for older adults, such as those involving gait, balance, co-ordination and functional tasks, can improve balance. Following progressive resistance training, older adults also respond with improved physical function. Brief interventions promoting physical activity may be cost-effective, however this evidence is weak and there are variations between studies.

Environmental approaches appear promising: signs that encourage the use of stairs, as well as community campaigns, may increase exercise levels. The city of Bogotá, Colombia, for example, blocks off 113 kilometers (70 mi) of roads on Sundays and holidays to make it easier for its citizens to get exercise. Such pedestrian zones are part of an effort to combat chronic diseases and to maintain a healthy BMI.

Parents can promote physical activity by modelling healthy levels of physical activity or by encouraging physical activity. According to the Centers for Disease Control and Prevention in the United States, children and adolescents should do 60 minutes or more of physical activity each day. Implementing physical exercise in the school system and ensuring an environment in which children can reduce barriers to maintain a healthy lifestyle is essential.

The European Commission's Directorate-General for Education and Culture (DG EAC) has dedicated programs and funds for Health Enhancing Physical Activity (HEPA) projects within its Horizon 2020 and Erasmus+ program, as research showed that too many Europeans are not physically active enough. Financing is available for increased collaboration between players active in this field across the EU and around the world, the promotion of HEPA in the EU and its partner countries, and the European Sports Week. The DG EAC regularly publishes a Eurobarometer on sport and physical activity.

Exercise trends

Main article: Exercise trends

Worldwide there has been a large shift toward less physically demanding work. This has been accompanied by increasing use of mechanized transportation, a greater prevalence of labor-saving technology in the home, and fewer active recreational pursuits. Personal lifestyle changes, however, can correct the lack of physical exercise.

Research published in 2015 suggests that incorporating mindfulness into physical exercise interventions increases exercise adherence and self-efficacy, and also has positive effects both psychologically and physiologically.

• Sports activities for exercising
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  Running helps in achieving physical fitness.

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  Skateboarding is good for cardiovascular health.

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  Swimming as an exercise tones muscles and builds strength.

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  Athletics (ex. pole vault) as a form of exercise

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  Football as an exercise

Social and cultural variation

Exercising looks different in every country, as do the motivations behind exercising. In some countries, people exercise primarily indoors (such as at home or health clubs), while in others, people primarily exercise outdoors. People may exercise for personal enjoyment, health and well-being, social interactions, competition or training, etc. These differences could potentially be attributed to a variety of reasons including geographic location and social tendencies.

In Colombia, for example, citizens value and celebrate the outdoor environments of their country. In many instances, they use outdoor activities as social gatherings to enjoy nature and their communities. In Bogotá, Colombia, a 70-mile stretch of road known as the Ciclovía is shut down each Sunday for bicyclists, runners, rollerbladers, skateboarders and other exercisers to work out and enjoy their surroundings.

Similarly to Colombia, citizens of Cambodia tend to exercise socially outside. In this country, public gyms have become quite popular. People will congregate at these outdoor gyms not only to use the public facilities, but also to organize aerobics and dance sessions, which are open to the public.

Sweden has also begun developing outdoor gyms, called utegym. These gyms are free to the public and are often placed in beautiful, picturesque environments. People will swim in rivers, use boats, and run through forests to stay healthy and enjoy the natural world around them. This works particularly well in Sweden due to its geographical location.

Exercise in some areas of China, particularly among those who are retired, seems to be socially grounded. In the mornings, square dances are held in public parks; these gatherings may include Latin dancing, ballroom dancing, tango, or even the jitterbug. Dancing in public allows people to interact with those with whom they would not normally interact, allowing for both health and social benefits.

These sociocultural variations in physical exercise show how people in different geographic locations and social climates have varying motivations and methods of exercising. Physical exercise can improve health and well-being, as well as enhance community ties and appreciation of natural beauty.

Nutrition and recovery

Proper nutrition is as important to health as exercise. When exercising, it becomes even more important to have a good diet to ensure that the body has the correct ratio of macronutrients while providing ample micronutrients, to aid the body with the recovery process following strenuous exercise.

Active recovery is recommended after participating in physical exercise because it removes lactate from the blood more quickly than inactive recovery. Removing lactate from circulation allows for an easy decline in body temperature, which can also benefit the immune system, as an individual may be vulnerable to minor illnesses if the body temperature drops too abruptly after physical exercise.

Exercise has an effect on appetite, but whether it increases or decreases appetite varies from individual to individual, and is affected by the intensity and duration of the exercise.

Excessive exercise

Main article: Overtraining

Excessive exercise or overtraining occurs when a person exceeds their body's ability to recover from strenuous exercise.

History

See also: Aerobic exercise § History, Fitness culture, and History of physical training and fitness

Roper's gymnasium, Philadelphia, US, c. 1831

The benefits of exercise have been known since antiquity. Dating back to 65 BCE, it was Marcus Cicero, Roman politician and lawyer, who stated: "It is exercise alone that supports the spirits, and keeps the mind in vigor." Exercise was also seen to be valued later in history during the Early Middle Ages as a means of survival by the Germanic peoples of Northern Europe.

More recently, exercise was regarded as a beneficial force in the 19th century. In 1858 Archibald MacLaren opened a gymnasium at the University of Oxford and instituted a training regimen for Major Frederick Hammersley and 12 non-commissioned officers. This regimen was assimilated into the training of the British Army, which formed the Army Gymnastic Staff in 1860 and made sport an important part of military life. Several mass exercise movements were started in the early twentieth century as well. The first and most significant of these in the UK was the Women's League of Health and Beauty, founded in 1930 by Mary Bagot Stack, that had 166,000 members in 1937.

The link between physical health and exercise (or lack of it) was further established in 1949 and reported in 1953 by a team led by Jerry Morris. Morris noted that men of similar social class and occupation (bus conductors versus bus drivers) had markedly different rates of heart attacks, depending on the level of exercise they got: bus drivers had a sedentary occupation and a higher incidence of heart disease, while bus conductors were forced to move continually and had a lower incidence of heart disease.

Other animals

Studies of animals indicate that physical activity may be more adaptable than changes in food intake to regulate energy balance.

Mice having access to activity wheels engaged in voluntary exercise and increased their propensity to run as adults. Artificial selection of mice exhibited significant heritability in voluntary exercise levels, with "high-runner" breeds having enhanced aerobic capacity, hippocampal neurogenesis, and skeletal muscle morphology.

The effects of exercise training appear to be heterogeneous across non-mammalian species. As examples, exercise training of salmon showed minor improvements of endurance, and a forced swimming regimen of yellowtail amberjack and rainbow trout accelerated their growth rates and altered muscle morphology favorable for sustained swimming. Crocodiles, alligators, and ducks showed elevated aerobic capacity following exercise training. No effect of endurance training was found in most studies of lizards, although one study did report a training effect. In lizards, sprint training had no effect on maximal exercise capacity, and muscular damage from over-training occurred following weeks of forced treadmill exercise.