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Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles. [1] There is also a third system, based on two poles ( biangular coordinates ). The term "bipolar" is further used on occasion to describe other curves having two singular points (foci), such as ellipses , hyperbolas , and Cassini ovals .
However, only a handful of forces result in formulae for in terms of known functions. The solution for φ {\displaystyle \varphi } can be expressed as an integral over u {\displaystyle u} φ = φ 0 + L 2 m ∫ u d u E t o t − V ( 1 / u ) − L 2 u 2 2 m {\displaystyle \varphi =\varphi _{0}+{\frac {L}{\sqrt {2m}}}\int ^{u}{\frac {du}{\sqrt {E ...
In classical mechanics, the central-force problem is to determine the motion of a particle in a single central potential field.A central force is a force (possibly negative) that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center.
The factor of two in the Coriolis force arises from two equal contributions: (i) the apparent change of an inertially constant velocity with time because rotation makes the direction of the velocity seem to change (a dv B /dt term) and (ii) an apparent change in the velocity of an object when its position changes, putting it nearer to or ...
A single force acting at any point O′ of a rigid body can be replaced by an equal and parallel force F acting at any given point O and a couple with forces parallel to F whose moment is M = Fd, d being the separation of O and O′. Conversely, a couple and a force in the plane of the couple can be replaced by a single force, appropriately ...
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are the radial distance r along the line connecting the point to a fixed point called the origin; the polar angle θ between this radial line and a given polar axis; [a] and
Using the spherical coordinates (r, θ, φ) as commonly used in physics (ISO 80000-2:2019 convention), where r is the radial distance to origin, θ is polar angle (also known as colatitude, zenith angle, normal angle, or inclination angle), and φ is the azimuthal angle, the Lagrangian for a central potential is = (˙ + ˙ + ˙) ().
In the cylindrical coordinate system, a z-coordinate with the same meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple (r, θ, z). [8] Spherical coordinates take this a step further by converting the pair of cylindrical coordinates ( r , z ) to polar coordinates ( ρ , φ ) giving a triple ( ρ , θ ...