Difference between cartesian coordinates and generalised coordinates
Answers
Usually, you start with Cartesian coordinates. These are the (x,y,z) coordinates that you learn about in high school.
Generalized (or curvilinear) coordinates are other triplets of numbers which describe the same space, such as spherical or cylindrical coordinates. Basically, take the idea behind spherical or cylindrical (or polar) coordinates as taking your “straight grid lines” and placing them in some curved way in space. Then, take arbitrarily squiggly lines instead, and those are generalized coordinates.
The key to understanding how they differ is that the dot product (or metric) of two vectors is not simply the sum of squares of their components [or with a difference, for Minkowski geometry] anymore; it is instead something more complicated.
The advantage is that sometimes a question is more naturally phrased in those coordinates, or some mathematical expression is more simply written that way. You usually trade some degree of nonlinearity for fewer variables.
Answer:
Usually, you start with Cartesian coordinates. These are the (x,y,z) coordinates that you learn about in high school.
Generalized (or curvilinear) coordinates are other triplets of numbers which describe the same space, such as spherical or cylindrical coordinates. Basically, take the idea behind spherical or cylindrical (or polar) coordinates as taking your “straight grid lines” and placing them in some curved way in space. Then, take arbitrarily squiggly lines instead, and those are generalized coordinates.
The key to understanding how they differ is that the dot product (or metric) of two vectors is not simply the sum of squares of their components [or with a difference, for Minkowski geometry] anymore; it is instead something more complicated.
The advantage is that sometimes a question is more naturally phrased in those coordinates, or some mathematical expression is more simply written that way. You usually trade some degree of nonlinearity for fewer variables.
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