Reference frame and coordinate system are another set of basic elements to describe the motion of objects. The frame of reference is the basic condition to describe the motion of an object, and the coordinate system is the general means to describe the motion of an object.
1. frame of reference
When studying the relative position changes between objects, one object must be selected in advance in order to determine the position changes of other objects relative to this object. Generally speaking, this preselected object is called a reference object, and the spatial relationship established with the reference object as the center is called a frame of reference.
Choosing a suitable frame of reference can not only simplify the mechanical model of object motion, but also help to explore the law of motion. On the contrary, choosing an unreasonable or wrong frame of reference will only lead to the complexity of mechanical model and confusion of understanding. The "geocentric theory" in human history is a typical wrong choice of frame of reference, which causes irreparable sorrow and regret. In modern society, some people sometimes make similar mistakes. For example, some people always want to find the driving force of crustal tectonic movement in the earth itself, that is, the choice of reference system is wrong.
Frame of reference can be divided into inertial frame of reference and non-inertial frame of reference. All inertial reference systems are equivalent and can be converted to each other.
Sometimes, for the motion analysis of the same object under different conditions, different reference frames can be selected.
When analyzing the motion law of spherical particles on the earth, this book adopts different reference systems with the earth, the sun and the silver core as the reference systems.
2. Coordinate system
In order to study the change of the object's motion position and the change of the target's position, we must first locate the target (often abstracted as a point). Usually, a set of ordered numbers to determine the position of a point is called the coordinates of this point, and the reference system used to determine the corresponding relationship between coordinates and points is called the coordinate system.
Coordinate system is the basis of the combination of shape and number. The method of discussing problems by using coordinate system is coordinate method.
After the reference system is selected, the reference body is generally selected as the origin of the coordinate system to establish the coordinate system, and the object to be studied is placed in the coordinate system to discuss the change of its relative position.
Several commonly used coordinate systems are: affine coordinate system, rectangular coordinate system and polar coordinate system.
The affine coordinate system in space consists of a point in space and vectors of three non-* * planes. Affine coordinate system can be planar or spatial, and can also be extended to high-dimensional situations.
Give the affine coordinate system a certain constraint condition-make the included angle between the coordinate axes a right angle, then the affine coordinate system will be transformed into a rectangular coordinate system.
The conclusion of the formation of affine coordinate system is also true in rectangular coordinate system; On the contrary, the conclusion that holds in Cartesian coordinate system may not hold in affine coordinate system.
Polar coordinate system is also a very important coordinate system. If we can analyze some movements around a fixed point in polar coordinate system, we will often find an unusually simple mathematical model, which will make the analysis conclusion clear at a glance.
Polar coordinates are a kind of coordinates: in Euclidean plane, a reference object is taken as a fixed point O (sometimes F), generally the right direction is given as a positive direction, the unit vector is taken as a ray ox, the point O is called a pole, and Ox is called a polar axis, thus forming a polar coordinate system. For any point S in the polar coordinate system, its polar coordinates are generally expressed as S(r, θ), and R and θ are called the polar diameter and polar angle of point S respectively. There is no one-to-one correspondence between the midpoint and the coordinates in polar coordinates. Only in the principal value interval, the points correspond to the coordinates.
The choice of coordinate system is artificial, and choosing the appropriate coordinate system according to different situations will bring convenience to solving problems.
Because of the nature of coordinates and the relative position relationship between coordinate systems, coordinate transformation equation can be established, so as to complete coordinate transformation.
If two sets of curves are given on the plane, so that any point on the plane is the intersection of one of the two sets of curves, the codes of the two sets of curves can be used as the coordinates of all the intersections by numbering the two sets of curves respectively. This kind of coordinate is usually called curve coordinate or curve coordinate.
Affine coordinates, rectangular coordinates and polar coordinates are all special cases of curve coordinates. Geographical coordinates on the earth are curvilinear coordinates composed of longitude and latitude.