Coordinates

Basis  A basis is a set of elements that generate a group or field. Groups have a minimum set that generates the group.Groups have a minimum set that generates the group. Cyclic groups have a single element basis.Cyclic groups have a single element basis.  Vector spaces use the scalars and basis vectors to generate the space. Example  A basis B i for M 2 ( R ) is  The vector equation has only one solution so they are linearly independent.

Cartesian Coordinates  Three coordinates x 1, x 2, x 3x 1, x 2, x 3 Replace x, y, zReplace x, y, z Usual right-handed systemUsual right-handed system  A vector can be expressed in coordinates, or from a basis. Unit vectors form a basisUnit vectors form a basis x1x1 x2x2 x3x3 Summation convention used

Cartesian Algebra  Vector algebra requires vector multiplication. Wedge productWedge product Usual 3D cross productUsual 3D cross product  The dot product is also defined for Cartesian vectors. Kronecker delta:  ij = 1, i = j  ij = 0, i ≠ j Permutation epsilon:  ijk = 0, any i, j, k the same  ijk = 1, if i, j, k an even permutation of 1, 2, 3  ijk = -1, if i, j, k an odd permutation of 1, 2, 3

Coordinate Transformation  A vector can be described by many Cartesian coordinate systems. Transform from one system to another Transformation matrix L x1x1 x2x2 x3x3 A physical property that transforms like this is a Cartesian vector.

General Transformation  Transformation and inverse q m = q m (x 1, x 2, x 3, t)q m = q m (x 1, x 2, x 3, t) x i = x i (q 1, q 2, q 3, t)x i = x i (q 1, q 2, q 3, t)  Generalized coordinates need not be distances.  For a small displacement a non-zero determinant of the transformation matrix guarantees an inverse transformation.  For a small displacement  If  Then the inverse exists

Other Coordinates  Polar-cylindrical coordinates r: q 1 = (x 1 + x 2 ) 1/2 r: q 1 = (x 1 + x 2 ) 1/2  : q 2 = tan -1 (x 2 /x 1 )  : q 2 = tan -1 (x 2 /x 1 ) z: q 3 = x 3 z: q 3 = x 3  Spherical coordinates r: q 1 = (x 1 + x 2 + x 3 ) 1/2 r: q 1 = (x 1 + x 2 + x 3 ) 1/2  : q 2 = cot -1 (x 3 / (x 1 + x 2 ) 1/2 )  : q 2 = cot -1 (x 3 / (x 1 + x 2 ) 1/2 )  : q 3 = tan -1 (x 2 /x 1 )  : q 3 = tan -1 (x 2 /x 1 )  Translation with constant velocity q 1 = x 1 – vt q 2 = x 2 q 3 = x 3  Translation with constant acceleration q 1 = x 1 – gt 2 q 2 = x 2 q 3 = x 3

Constraints  Coordinates may be constrained to a manifold Surface of a sphereSurface of a sphere Spiral wireSpiral wire  A function of the coordinates and time: holonomic  (x 1, x 2, x 3, t) = 0  (x 1, x 2, x 3, t) = 0 If time appears the constraint is moving.If time appears the constraint is moving. If time does not appear the constraint is fixed.If time does not appear the constraint is fixed.  Non-holonomic constraints include terms like velocity or acceleration.

System of Points  Coordinates with two indices: x  i  represents the point  represents the point i represents the coordinate index (use i through n ) i represents the coordinate index (use i through n )  A rigid body has k holonomic constraints.  j (x  i, t) = 0  j (x  i, t) = 0 System has f = 3 N – k degrees of freedomSystem has f = 3 N – k degrees of freedom next