9.6 The Hamilton Principles and Their Applications
551
The space position of a free particle requires three independent coordinates to be
determined. The free system composed of the n particles can only be completely
determined when it requires n independent coordinates.
The independent variables which can completely determine the position of a
system are called the generalized coordinates of the system, or it is called the
coordinates for short. The denotations q 1 , q 2 , … are often used to express the generalized coordinates. The characteristics of generalized coordinates is no dependencies
between them, therefore they are called the independent coordinates. The generalized coordinates are not necessarily to choose rectangular coordinates, also can
choose other variables, such as the distance, arc length, angle degree and area etc.
A variable as long as that can locate the system can be used as the generalized
coordinates.
A system composed of the n particle is limited by m holonomic constraints, the
configuration of the system can be completely determined by k = 3n −m generalized
coordinates. The motion of the system with k generalized coordinates namely with k
degrees of freedom can be regarded as the motion of a representative point in the kdimensional space, a dimension of the space corresponds to a generalized coordinate
in q i , this k-dimensional space is called the configuration space. In brief, the set
composed of all the configurations of a system is called the configuration space of
the system.
The virtual displacement of generalized coordinates is called the variation of
coordinate. The number of independent variation of the generalized coordinates of
a system is called the number of degrees of freedom of the system, it is called the
degrees of freedom for short. The particle system composed of n particles has m
holonomic constraints, the number of generalized coordinates is
k = 3n − m
(9.6.36)
If the above motion is constrained by other m constraint equations
ϕ j (t, x 1 , x 2 , . . . , x n , y 1 , y 2 , . . . , y n , z 1 , z 2 , . . . , z n ) = 0 ( j = 1, 2, . . . , m, m < 3n)
(9.6.37)
then the number of independent variables has only 3n − m. The generalized coordinates q 1 , q 2 , …, q 3n−m can be use to express these independent variables. If using the
3n − m independent variables to express the original coordinates x i , y i , z i , namely
⎧
⎪ ⎨
⎪ ⎩
x i = x i (q 1 , q 2 , . . . , q 3n−m , t)
y i = y i (q 1 , q 2 , . . . , q 3n−m , t)
z i = z i (q 1 , q 2 , . . . , q 3n−m , t)
(i = 1, 2, . . . , n)
(9.6.38)
then T and V are also the function of the generalized coordinates q 1 , q 2 , …, q 3n−m ,
namely
551
The space position of a free particle requires three independent coordinates to be
determined. The free system composed of the n particles can only be completely
determined when it requires n independent coordinates.
The independent variables which can completely determine the position of a
system are called the generalized coordinates of the system, or it is called the
coordinates for short. The denotations q 1 , q 2 , … are often used to express the generalized coordinates. The characteristics of generalized coordinates is no dependencies
between them, therefore they are called the independent coordinates. The generalized coordinates are not necessarily to choose rectangular coordinates, also can
choose other variables, such as the distance, arc length, angle degree and area etc.
A variable as long as that can locate the system can be used as the generalized
coordinates.
A system composed of the n particle is limited by m holonomic constraints, the
configuration of the system can be completely determined by k = 3n −m generalized
coordinates. The motion of the system with k generalized coordinates namely with k
degrees of freedom can be regarded as the motion of a representative point in the kdimensional space, a dimension of the space corresponds to a generalized coordinate
in q i , this k-dimensional space is called the configuration space. In brief, the set
composed of all the configurations of a system is called the configuration space of
the system.
The virtual displacement of generalized coordinates is called the variation of
coordinate. The number of independent variation of the generalized coordinates of
a system is called the number of degrees of freedom of the system, it is called the
degrees of freedom for short. The particle system composed of n particles has m
holonomic constraints, the number of generalized coordinates is
k = 3n − m
(9.6.36)
If the above motion is constrained by other m constraint equations
ϕ j (t, x 1 , x 2 , . . . , x n , y 1 , y 2 , . . . , y n , z 1 , z 2 , . . . , z n ) = 0 ( j = 1, 2, . . . , m, m < 3n)
(9.6.37)
then the number of independent variables has only 3n − m. The generalized coordinates q 1 , q 2 , …, q 3n−m can be use to express these independent variables. If using the
3n − m independent variables to express the original coordinates x i , y i , z i , namely
⎧
⎪ ⎨
⎪ ⎩
x i = x i (q 1 , q 2 , . . . , q 3n−m , t)
y i = y i (q 1 , q 2 , . . . , q 3n−m , t)
z i = z i (q 1 , q 2 , . . . , q 3n−m , t)
(i = 1, 2, . . . , n)
(9.6.38)
then T and V are also the function of the generalized coordinates q 1 , q 2 , …, q 3n−m ,
namely