Χρήστης:Ιων/αμμοδοχείο2

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Σημεία στο πολικό σύστημα συντεταγμένων με τον πόλο O και τον πολικό άξονα L. Σε πράσινο χρώμα, το σημείο με ακτινική συντεταγμένη 3 και γωνιακή συντεταγμένη 60 μοιρών, ή (3,60°). Σε μπλε χρώμα, το σημείο (4,210°).

Στα μαθηματικά, το πολικό σύστημα συντεταγμένων είναι ένα διδιάστατο σύστημα συντεταγμένων στο οποίο κάθε σημείο πάνω σε ένα επίπεδο καθορίζεται από μια απόσταση από ένα καθορισμένο σημείο και μια γωνία από μια καθορισμένη κατεύθυνση.

Το καθορισμένο σημείο (ανάλογο με την αρχή ενός Καρτεσιανού συστήματος) ονομάζεται πόλος, και η ακτινική απόσταση από τον πόλο στην καθορισμένη κατεύθυνση είναι ο πολικός άξονας. Η απόσταση από τον πόλο ονομάζεται ακτινική συντεταγμένη ή ακτίνα, και η γωνία είναι η γωνιακή συντεταγμένη, η πολική γωνία ή αζιμούθιο.

Ιστορία

Βλέπε επίσης:ιστορία των τριγωνομετρικών συναρτήσεων Οι έννοιες της γωνίας και της ακτίνας χρησιμοποιούντο ήδη από τους αρχαίους της πρώτης χιλιετίας π.Χ. Ο Έλληνας αστρονόμος και αστρολόγος Ίππαρχος (190–120 π.Χ.) δημιούργησε έναν πίνακα συναρτήσεων χορδών δίνοντας το μήκος της χορδής για κάθε γωνία, και υπάρχουν αναφορές ότι χρησιμοποιούσε πολικές συντεταγμένες για να καθορίσει τις αστρικές θέσεις. In On Spirals, Archimedes describes the Archimedean spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.

From the 8th century CE onward, astronomers developed methods for approximating and calculating the direction to Makkah (qibla)—and its distance—from any location on the Earth. From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. The calculation is essentially the conversion of the equatorial polar coordinates of Mecca (i.e. its longitude and latitude) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the great circle through the given location and the Earth's poles, and whose polar axis is the line through the location and its antipodal point.

There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidge's Origin of Polar Coordinates. Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs.

In Method of Fluxions (written 1671, published 1736), Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems. In the journal Acta Eruditorum (1691), Jacob Bernoulli used a system with a point on a line, called the pole and polar axis respectively. Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates.

The actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacock's 1816 translation of Lacroix's Differential and Integral Calculus. Alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them.

Conventions

A polar grid with several angles labeled in degrees

The radial coordinate is often denoted by r, and the angular coordinate by θ or t.

Angles in polar notation are generally expressed in either degrees or radians (2π rad being equal to 360°). Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics.

In many contexts, a positive angular coordinate means that the angle θ is measured counterclockwise from the axis.

In mathematical literature, the polar axis is often drawn horizontal and pointing to the right.

Uniqueness of polar coordinates

Adding any number of full turns (360°) to the angular coordinate does not change the corresponding direction. Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the same point can be expressed with an infinite number of different polar coordinates (r, θ ± n×360°) or (−r, θ ± (2n + 1)180°), where n is any integer. Moreover, the pole itself can be expressed as (0, θ) for any angle θ.

Where a unique representation is needed for any point, it is usual to limit r to non-negative numbers (r ≥ 0) and θ to the interval [0, 360°) or (−180°, 180°] (in radians, [0, 2π) or (−π, π]). One must also choose a unique azimuth for the pole, e.g., θ = 0.

Εναλλσγή μεταξύ πολικών και Καρτεσιανών συντεταγμένων

Διάγραμμα που εικονογραφεί την σχέση μεταξύ πολικών και Καρτεσιανών συντεταγμένων.

Μια καμπύλη στο Καρτεσιανό επίπεδο μπορεί να σχεδιασθεί σε πολικές συντεταγμένες. Σε αυτή την κινούμενη εικόνα, η $y=\sin(6x)+2$ σχεδιάζεται πάνω σε $r=\sin(6\theta )+2$. Κάντε κλικ στην εικόνα για λεπτομέρειες.

Οι πολικές συντεταγμένες r και θ μπορούν να μετατραπούν στις Καρτεσιανές συντεταγμένες x και y χρησιμοποιώντας τις τριγωνομετρικές συναρτήσεις του ημιτόνου και του συνημιτόνου:

$x=r\cos \theta \,$

$y=r\sin \theta \,$

Οι Καρτεσιανές συντεταγμένες x και y μπορούν να μετατραπούν στις πολικές συντεταγμένες r και θ με r ≥ 0 και θ στο διάστημα (−π, π] με:

$r={\sqrt {x^{2}+y^{2}}}\quad$ (όπως στο Πυθαγόρειο θεώρημα του Ευκλείδιου κανόνα), και

$\theta =\operatorname {atan2} (y,x)\quad$,

όπου atan2 είναι μια κοινή παραλλαγή της συνάρτησης αντίστροφης εφαπτομένης που ορίζεται ως

$\operatorname {atan2} (y,x)={\begin{cases}\arctan({\frac {y}{x}})&{\mbox{if }}x>0\\\arctan({\frac {y}{x}})+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 0\\\arctan({\frac {y}{x}})-\pi &{\mbox{if }}x<0{\mbox{ and }}y<0\\{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y>0\\-{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y<0\\{\text{undefined}}&{\mbox{if }}x=0{\mbox{ and }}y=0\end{cases}}$

Η τιμή του θ ανωτέρω είναι η κύρια τιμή of the complex number function arg applied to x+iy. An angle in the range [0, 2π) may be obtained by adding 2π to the value in case it is negative.

Polar equation of a curve

The equation defining an algebraic curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of θ. The resulting curve then consists of points of the form (r(θ), θ) and can be regarded as the graph of the polar function r.

Different forms of symmetry can be deduced from the equation of a polar function r. If r(−θ) = r(θ) the curve will be symmetrical about the horizontal (0°/180°) ray, if r(π − θ) = r(θ) it will be symmetric about the vertical (90°/270°) ray, and if r(θ − α) = r(θ) it will be rotationally symmetric α counterclockwise about the pole.

Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, and cardioid.

For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.

Κύκλος

Ένας κύκλος με εξίσωση r(θ) = 1

Η γενική εξίσωση για έναν κύκλο με κέντρο στο (r₀, $\varphi$) και ακτίνα a είναι

$r^{2}-2rr_{0}\cos(\theta -\varphi )+r_{0}^{2}=a^{2}.\,$

This can be simplified in various ways, to conform to more specific cases, such as the equation

$r(\theta )=a\,$

for a circle with a center at the pole and radius a.

When r₀ = a, or when the origin lies on the circle, the equation becomes

$r=2a\cos(\theta -\varphi )$.

In the general case, the equation can be solved for r, giving

$r=r_{0}\cos(\theta -\varphi )+{\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\theta -\varphi )}}$,

the solution with a minus sign in front of the square root gives the same curve.

Line

A polar rose with equation r(θ) = 2 sin 4θ

Radial lines (those running through the pole) are represented by the equation

$\theta =\varphi \,$,

where φ is the angle of elevation of the line; that is, φ = arctan m where m is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line θ = φ perpendicularly at the point (r₀, φ) has the equation

$r(\theta )={r_{0}}\sec(\theta -\varphi ).\,$

Otherwise stated (r₀, φ) is the point in which the tangent intersects the imaginary circle of radius r₀.

Polar rose

A polar rose is a famous mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,

$r(\theta )=a\cos(k\theta +\varphi _{0})\,$

for any constant φ₀ (including 0). If k is an integer, these equations will produce a k-petaled rose if k is odd, or a 2k-petaled rose if k is even. If k is rational but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The variable a represents the length of the petals of the rose.

Archimedean spiral

One arm of an Archimedean spiral with equation r(θ) = θ / 2π for 0 < θ < 6π

The Archimedean spiral is a famous spiral that was discovered by Archimedes, which also can be expressed as a simple polar equation. It is represented by the equation

$r(\theta )=a+b\theta .\,$

Changing the parameter a will turn the spiral, while b controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one for θ > 0 and one for θ < 0. The two arms are smoothly connected at the pole. Taking the mirror image of one arm across the 90°/270° line will yield the other arm. This curve is notable as one of the first curves, after the conic sections, to be described in a mathematical treatise, and as being a prime example of a curve that is best defined by a polar equation.

Conic sections

Ellipse, showing semi-latus rectum

A conic section with one focus on the pole and the other somewhere on the 0° ray (so that the conic's major axis lies along the polar axis) is given by:

$r={\ell \over {1+e\cos \theta }}$

where e is the eccentricity and $\ell$ is the semi-latus rectum (the perpendicular distance at a focus from the major axis to the curve). If e > 1, this equation defines a hyperbola; if e = 1, it defines a parabola; and if e < 1, it defines an ellipse. The special case e = 0 of the latter results in a circle of radius $\ell$.

Μιγαδικοί αριθμοί

Εικονογράφηση ενός μιγαδικού αριθμού z σχεδιασμένου στο μιγαδικό επίπεδο

An illustration of a complex number plotted on the complex plane using Euler's formula

Κάθε μιγαδικός αριθμός μπορεί να παρασταθεί ως ένα σημείο στο μιγαδικό επίπεδο, and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form). The complex number z can be represented in rectangular form as

$z=x+iy\,$

where i is the imaginary unit, or can alternatively be written in polar form (via the conversion formulae given above) as

$z=r\cdot (\cos \theta +i\sin \theta )$

and from there as

$z=re^{i\theta }\,$

where e is Euler's number, which are equivalent as shown by Euler's formula. (Note that this formula, like all those involving exponentials of angles, assumes that the angle θ is expressed in radians.) To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used.

For the operations of multiplication, division, and exponentiation of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form. From the laws of exponentiation:

• Multiplication:

$r_{0}e^{i\theta _{0}}\cdot r_{1}e^{i\theta _{1}}=r_{0}r_{1}e^{i(\theta _{0}+\theta _{1})}\,$

• Division:

${\frac {r_{0}e^{i\theta _{0}}}{r_{1}e^{i\theta _{1}}}}={\frac {r_{0}}{r_{1}}}e^{i(\theta _{0}-\theta _{1})}\,$

• Exponentiation (De Moivre's formula):

$(re^{i\theta })^{n}=r^{n}e^{in\theta }\,$

Calculus

Calculus can be applied to equations expressed in polar coordinates.

The angular coordinate θ is expressed in radians throughout this section, which is the conventional choice when doing calculus.

Διαφορικός λογισμός

Χρησιμοποιώντας το x = r cos θ και το y = r sin θ , μπορεί να εξαχθεί μια σχέση μεταξύ παραγώγων στις Καρτεσιανές και τις πολικές συντεταγμένες. Για μια δεδομένη συνάρτηση, 'το 'u(x,y), ικανοποιεί

$r{\frac {\partial u}{\partial r}}=r{\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial r}}+r{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial r}},$

${\frac {\partial u}{\partial \theta }}={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial \theta }}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial \theta }},$

ή

$r{\frac {\partial u}{\partial r}}=r{\frac {\partial u}{\partial x}}\cos \theta +r{\frac {\partial u}{\partial y}}\sin \theta =x{\frac {\partial u}{\partial x}}+y{\frac {\partial u}{\partial y}},$

${\frac {\partial u}{\partial \theta }}=-{\frac {\partial u}{\partial x}}r\sin \theta +{\frac {\partial u}{\partial y}}r\cos \theta =-y{\frac {\partial u}{\partial x}}+x{\frac {\partial u}{\partial y}}.$

Ούτως, έχουμε τους ακόλουθους τύπους:

$r{\frac {\partial }{\partial r}}=x{\frac {\partial }{\partial x}}+y{\frac {\partial }{\partial y}}\,$

${\frac {\partial }{\partial \theta }}=-y{\frac {\partial }{\partial x}}+x{\frac {\partial }{\partial y}}.$

Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function u(r,θ), it follows that

${\frac {\partial u}{\partial x}}={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial x}}+{\frac {\partial u}{\partial \theta }}{\frac {\partial \theta }{\partial x}},$

${\frac {\partial u}{\partial y}}={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial y}}+{\frac {\partial u}{\partial \theta }}{\frac {\partial \theta }{\partial y}},$

or

${\frac {\partial u}{\partial x}}={\frac {\partial u}{\partial r}}{\frac {x}{\sqrt {x^{2}+y^{2}}}}-{\frac {\partial u}{\partial \theta }}{\frac {y}{x^{2}+y^{2}}}=\cos \theta {\frac {\partial u}{\partial r}}-{\frac {1}{r}}\sin \theta {\frac {\partial u}{\partial \theta }},$

${\frac {\partial u}{\partial y}}={\frac {\partial u}{\partial r}}{\frac {y}{\sqrt {x^{2}+y^{2}}}}+{\frac {\partial u}{\partial \theta }}{\frac {x}{x^{2}+y^{2}}}=\sin \theta {\frac {\partial u}{\partial r}}+{\frac {1}{r}}\cos \theta {\frac {\partial u}{\partial \theta }}.$

Hence, we have the following formulae:

${\frac {\partial }{\partial x}}=\cos \theta {\frac {\partial }{\partial r}}-{\frac {1}{r}}\sin \theta {\frac {\partial }{\partial \theta }}\,$

${\frac {\partial }{\partial y}}=\sin \theta {\frac {\partial }{\partial r}}+{\frac {1}{r}}\cos \theta {\frac {\partial }{\partial \theta }}.$

To find the Cartesian slope of the tangent line to a polar curve r(θ) at any given point, the curve is first expressed as a system of parametric equations.

$x=r(\theta )\cos \theta \,$

$y=r(\theta )\sin \theta \,$

Differentiating both equations with respect to θ yields

${\frac {dx}{d\theta }}=r'(\theta )\cos \theta -r(\theta )\sin \theta \,$

${\frac {dy}{d\theta }}=r'(\theta )\sin \theta +r(\theta )\cos \theta .\,$

Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point (r(θ), θ):

${\frac {dy}{dx}}={\frac {r'(\theta )\sin \theta +r(\theta )\cos \theta }{r'(\theta )\cos \theta -r(\theta )\sin \theta }}.$

For other useful formulas including divergence, gradient, and Laplacian in polar coordinates, see curvilinear coordinates.

Ολοκληρωτικός λογισμός (μήκος τόξου)

Το μήκος τόξου (μήκος ενός τμήματος γραμμής) που ορίζεται από μια πολική συνάρτηση βρίσκεται από την ολοκλήρωση πάνω στην καμπύλη r(θ). Έστω ότι L είναι το μήκος του τμήματος της καμπύλης που διατρέχει από το σημείο A μέχρι το σημείο B, όπου αυτά τα σημεία αντιστοιχούν σε θ = a και θ = b τέτοια ώστε 0 < b − a < 2π. Το μήκος του L δίνεται από το ακόλουθο ολοκλήρωμα

$L=\int _{a}^{b}{\sqrt {\left[r(\theta )\right]^{2}+\left[{{dr(\theta )} \over {d\theta }}\right]^{2}}}d\theta$

Ολοκληρωτικός λογισμός (χωρίο)

Η περιοχή ολοκλήρωσης R περικλείεται από την καμπύλη r(θ) και τις ακτίνες θ = a και θ = b.

Έστω ότι το R δηλώνει ένα χωρίο περικλειόμενο από μια καμπύλη r(θ) και τις ακτίνες θ = a και θ = b, όπου 0 < b − a ≤ 2π. Τότε, το χωρίο R είναι

${\frac {1}{2}}\int _{a}^{b}\left[r(\theta )\right]^{2}\,d\theta .$

Το χωρίο R προσεγγίζεται με n τομείς (εδώ n = 5).

Ένα επιπεδόμετρο, το οποίο υπολογίζει μηχανικά πολικά ολοκληρώματα

Αυτό το αποτέλεσμα μπορεί να προκύψει ως εξής. Πρώτον, το διάστημα [a, b] διαιρείται σε n υποδιαστήματα, όπου n is an arbitrary positive integer. Thus Δθ, the length of each subinterval, is equal to b − a (the total length of the interval), divided by n, the number of subintervals. For each subinterval i = 1, 2, …, n, let θ_i be the midpoint of the subinterval, and construct a sector with the center at the pole, radius r(θ_i), central angle Δθ and arc length r(θ_i)Δθ. The area of each constructed sector is therefore equal to

$\left[r(\theta _{i})\right]^{2}\pi \cdot {\frac {\Delta \theta }{2\pi }}={\frac {1}{2}}\left[r(\theta _{i})\right]^{2}\Delta \theta .$

Hence, the total area of all of the sectors is

$\sum _{i=1}^{n}{\tfrac {1}{2}}r(\theta _{i})^{2}\,\Delta \theta .$

As the number of subintervals n is increased, the approximation of the area continues to improve. In the limit as n → ∞, the sum becomes the Riemann sum for the above integral.

A mechanical device that computes area integrals is the planimeter, which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element linkage effects Green's theorem, converting the quadratic polar integral to a linear integral.

Generalization

Using Cartesian coordinates, an infinitesimal area element can be calculated as dA = dx dy. The substitution rule for multiple integrals states that, when using other coordinates, the Jacobian determinant of the coordinate conversion formula has to be considered:

$J=\det {\frac {\partial (x,y)}{\partial (r,\theta )}}={\begin{vmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \theta }}\\[8pt]{\frac {\partial y}{\partial r}}&{\frac {\partial y}{\partial \theta }}\end{vmatrix}}={\begin{vmatrix}\cos \theta &-r\sin \theta \\\sin \theta &r\cos \theta \end{vmatrix}}=r\cos ^{2}\theta +r\sin ^{2}\theta =r.$

Hence, an area element in polar coordinates can be written as

$dA=dx\,dy\ =J\,dr\,d\theta =r\,dr\,d\theta .$

Now, a function that is given in polar coordinates can be integrated as follows:

$\iint _{R}f(x,y)\,dA=\int _{a}^{b}\int _{0}^{r(\theta )}f(r,\theta )\,r\,dr\,d\theta .$

Here, R is the same region as above, namely, the region enclosed by a curve r(θ) and the rays θ = a and θ = b.

The formula for the area of R mentioned above is retrieved by taking f identically equal to 1. A more surprising application of this result yields the Gaussian integral

$\int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}.$

Vector calculus

Vector calculus can also be applied to polar coordinates. For a planar motion, let $\mathbf {r}$ be the position vector (rcos(θ), rsin(θ)), with r and θ depending on time t.

We define the unit vectors

${\hat {\mathbf {r} }}=(\cos(\theta ),\sin(\theta ))$

in the direction of r and

${\hat {\boldsymbol {\theta }}}=(-\sin(\theta ),\cos(\theta ))={\hat {\mathbf {k} }}\times {\hat {\mathbf {r} }}\ ,$

in the plane of the motion perpendicular to the radial direction, where ${\hat {\mathbf {k} }}$ is a unit vector normal to the plane of the motion.

Then

$\mathbf {r} =(x,\ y)=r(\cos \theta ,\ \sin \theta )=r{\hat {\mathbf {r} }}\ ,$

${\dot {\mathbf {r} }}=({\dot {x}},\ {\dot {y}})={\dot {r}}(\cos \theta ,\ \sin \theta )+r{\dot {\theta }}(-\sin \theta ,\ \cos \theta )={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\theta }}{\hat {\boldsymbol {\theta }}}\ ,$

${\ddot {\mathbf {r} }}=({\ddot {x}},\ {\ddot {y}})={\ddot {r}}(\cos \theta ,\ \sin \theta )+2{\dot {r}}{\dot {\theta }}(-\sin \theta ,\ \cos \theta )+r{\ddot {\theta }}(-\sin \theta ,\ \cos \theta )-r{\dot {\theta }}^{2}(\cos \theta ,\ \sin \theta )\ =$

$\left({\ddot {r}}-r{\dot {\theta }}^{2}\right){\hat {\mathbf {r} }}+\left(2{\dot {r}}{\dot {\theta }}+r{\ddot {\theta }}\right){\hat {\boldsymbol {\theta }}}\ =({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+{\frac {1}{r}}\;{\frac {d}{dt}}\left(r^{2}{\dot {\theta }}\right){\hat {\boldsymbol {\theta }}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}$

Centrifugal and Coriolis terms

Δείτε επίσης: Mechanics of planar particle motion και Centrifugal force (rotating reference frame)

The term $r{\dot {\theta }}^{2}$ is sometimes referred to as the centrifugal term, and the term $2{\dot {r}}{\dot {\theta }}$ as the Coriolis term. For example, see Shankar. Although these equations bear some resemblance in form to the centrifugal and Coriolis effects found in rotating reference frames, nonetheless these are not the same things. For example, the physical centrifugal and Coriolis forces appear only in non-inertial frames of reference. In contrast, these terms that appear when acceleration is expressed in polar coordinates are a mathematical consequence of differentiation; these terms appear wherever polar coordinates are used. In particular, these terms appear even when polar coordinates are used in inertial frames of reference, where the physical centrifugal and Coriolis forces never appear.

Inertial frame of reference S and instantaneous non-inertial co-rotating frame of reference S′. The co-rotating frame rotates at angular rate Ω equal to the rate of rotation of the particle about the origin of S′ at the particular moment t. Particle is located at vector position r(t) and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angle θ normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distance r need not be related to the radius of curvature of the path.

Co-rotating frame

For a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneous co-rotating frame of reference. To define a co-rotating frame, first an origin is selected from which the distance r(t) to the particle is defined. An axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin. Then, at the selected moment t, the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, dθ/dt. Next, the terms in the acceleration in the inertial frame are related to those in the co-rotating frame. Let the location of the particle in the inertial frame be (r(t), θ(t)), and in the co-rotating frame be (r(t), θ′(t)). Because the co-rotating frame rotates at the same rate as the particle, dθ′/dt = 0. The fictitious centrifugal force in the co-rotating frame is mrΩ², radially outward. The velocity of the particle in the co-rotating frame also is radially outward, because dθ′/dt = 0. The fictitious Coriolis force therefore has a value −2m(dr/dt)Ω, pointed in the direction of increasing θ only. Thus, using these forces in Newton's second law we find:

${\boldsymbol {F}}+{\boldsymbol {F_{cf}}}+{\boldsymbol {F_{Cor}}}=m{\ddot {\boldsymbol {r}}}\ ,$

where over dots represent time differentiations, and F is the net real force (as opposed to the fictitious forces). In terms of components, this vector equation becomes:

$F_{r}+mr\Omega ^{2}=m{\ddot {r}}$

$F_{\theta }-2m{\dot {r}}\Omega =mr{\ddot {\theta }}\ ,$

which can be compared to the equations for the inertial frame:

$F_{r}=m{\ddot {r}}-mr{\dot {\theta }}^{2}\$

$F_{\theta }=mr{\ddot {\theta }}+2m{\dot {r}}{\dot {\theta }}\ .$

This comparison, plus the recognition that by the definition of the co-rotating frame at time t it has a rate of rotation Ω = dθ/dt, shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame.

For general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous osculating circle of its motion, not to a fixed center of polar coordinates. For more detail, see centripetal force.

Connection to spherical and cylindrical coordinates

The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate system.

Applications

Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. They are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. For instance, the examples above show how elementary polar equations suffice to define curves—such as the Archimedean spiral—whose equation in the Cartesian coordinate system would be much more intricate. Moreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates. The initial motivation for the introduction of the polar system was the study of circular and orbital motion.

Position and navigation

Polar coordinates are used often in navigation, as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance, aircraft use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a clockwise direction, rather than counterclockwise, as in the mathematical system. Heading 360 corresponds to magnetic north, while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively. Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read zero-niner-zero by air traffic control).

Modeling

Systems displaying radial symmetry provide natural settings for the polar coordinate system, with the central point acting as the pole. A prime example of this usage is the groundwater flow equation when applied to radially symmetric wells. Systems with a radial force are also good candidates for the use of the polar coordinate system. These systems include gravitational fields, which obey the inverse-square law, as well as systems with point sources, such as radio antennas.

Radially asymmetric systems may also be modeled with polar coordinates. For example, a microphone's pickup pattern illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves. The curve for a standard cardioid microphone, the most common unidirectional microphone, can be represented as r = 0.5 + 0.5sin(θ) at its target design frequency. The pattern shifts toward omnidirectionality at lower frequencies.

Βλέπε επίσης

• List of canonical coordinate transformations
• Curvilinear coordinates

Αναφορές

General

• Anton, Howard (2002). Calculus (Seventh έκδοση). Anton Textbooks, Inc. ISBN 0-471-38157-8. 
• Finney, Ross (1994). Calculus: Graphical, Numerical, Algebraic (Single Variable Version έκδοση). Addison-Wesley Publishing Co. ISBN 0-201-55478-X. 

Specific

Εξωτερικοί σύνδεσμοι

• Hazewinkel, Michiel, επιμ.. (2001), «Polar coordinates», Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=p/p073410 
• Graphing Software στο Curlie
• Coordinate Converter — converts between polar, Cartesian and spherical coordinates
• Polar Coordinate System Dynamic Demo

{{Orthogonal coordinate systems}}

Category:Coordinate systems