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第11行：
第11行： − + + 第18行：
第19行： − 粒子在三维空间中的位置需要3个位置坐标。同样，粒子运动的方向和速度可以用三个速度分量来描述，每个分量都参照空间的三个维度。如果系统的时间演化是确定的，状态在某一时刻唯一地决定了它的过去和未来的位置和速度作为时间的函数，这样的系统有六个自由度。如果粒子的运动被限制在较低的维数上——例如，粒子必须沿着导线或在固定的表面上运动——那么系统的自由度就少于6度。另一方面，一个系统与一个扩展的对象，可以旋转或振动可以有六个以上的自由度。+ 第26行：
第27行： − 在经典力学中，一个点粒子在任意给定时间的状态通常用拉格朗日公式中的位置和速度坐标来描述，或者用哈密顿公式中的位置和动量坐标来描述。+ 第34行：
第35行： − 在统计力学中，一个自由度是描述系统微观状态的单个标量数。一个系统的所有微观状态的描述是系统相空间中的一个点。+ 第42行：
第43行： − 在化学三维理想链模型中，每个单体的取向需要用两个角度来描述。+ 第50行：
第51行： − 指定二次自由度常常是有用的。这些自由度以二次函数的形式，对系统的能量做出贡献。+ − + 第68行：
第69行： − 在三维空间中，3个自由度与一个粒子的运动有关。一个双原子气体分子有6个自由度。这个集合可以根据分子的平动、转动和振动来分解。整个分子的质心运动占3个自由度。此外，分子有两个转动度和一个振动模式。旋转发生在两个轴与两个原子之间的直线垂直的周围。原子-原子键的旋转不是物理旋转。这样，在一个双原子分子的时间里，产生了一个分解过程:+ − − <math>N = 6 = 3 + 2 + 1.</math> − − 6 = 3 + 2 + 1 + − For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition:+ − 对于一般的非线性分子，考虑了所有3个转动自由度，导致分解: − − <math>3N = 3 + 3 + (3N - 6)</math> − − 3N = 3 + 3 + (3N-6) </math > − 第96行：
第88行： − 这意味着一个原子分子的振动自由度。在特殊情况下，如吸附大分子，转动自由度可以限制为只有一个。+ 第104行：
第96行： − 根据上面的定义，还可以使用指定位置所需的最小坐标数来计算自由度。具体做法如下:+ + + 第110行：
第104行： − 对于单个粒子，我们需要二维平面中的两个坐标来指定它的位置，以及三维空间中的三个坐标。因此，它在三维空间中的自由度为3。+ + + 第116行：
第112行： − 对于由两个粒子组成的物体(例如。在一个三维空间中，两个双原子分子之间的距离不变(比如说 d) ，我们可以显示它的自由度为5。+ + + 第122行：
第120行： − 假设这个物体中的一个粒子具有坐标，而另一个粒子具有未知坐标。两坐标间距公式的应用+ + − <math>d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}</math> − − [ math > d = sqrt {(x _ 2-x _ 1) ^ 2 + (y _ 2-y _ 1) ^ 2 + (z _ 2-z _ 1) ^ 2][ math > − − results in one equation with one unknown, in which we can solve for {{math|''z''<sub>2</sub>}}. − − results in one equation with one unknown, in which we can solve for . − 结果是一个方程有一个未知数，在这个方程中我们可以解出。+ − + − One of , , , , , or can be unknown.+ − 其中之一，，，，，或者可能是未知的。
无编辑摘要
In physics and chemistry, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all states of a system is known as the system's phase space, and the degrees of freedom of the system are the dimensions of the phase space.
In physics and chemistry, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all states of a system is known as the system's phase space, and the degrees of freedom of the system are the dimensions of the phase space.
在物理和化学领域中，''自由度''是对物理系统状态的形式描述中的独立物理参数。
在物理和化学领域中，''自由度''是对物理系统状态形式描述中的独立物理参数。系统所有状态的集合称为系统的'''<font color="#ff8000"> 相空间</font>'''，系统的自由度是相空间的维数。
The location of a particle in three-dimensional space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. If the time evolution of the system is deterministic, where the state at one instant uniquely determines its past and future position and velocity as a function of time, such a system has six degrees of freedom. If the motion of the particle is constrained to a lower number of dimensions – for example, the particle must move along a wire or on a fixed surface – then the system has fewer than six degrees of freedom. On the other hand, a system with an extended object that can rotate or vibrate can have more than six degrees of freedom.
The location of a particle in three-dimensional space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. If the time evolution of the system is deterministic, where the state at one instant uniquely determines its past and future position and velocity as a function of time, such a system has six degrees of freedom. If the motion of the particle is constrained to a lower number of dimensions – for example, the particle must move along a wire or on a fixed surface – then the system has fewer than six degrees of freedom. On the other hand, a system with an extended object that can rotate or vibrate can have more than six degrees of freedom.
一个粒子在三维空间中的位置需要三个位置坐标进行标识。类似地，我们也可以根据三个速度分量来描述粒子运动的方向和速度，每个速度分量都参考该空间的三个维度。如果系统的时间演变是确定的，那么其中某个瞬间以及其过去和未来瞬间的位置和速度，均可以被该瞬间的状态确定为时间函数，相应地，这种系统具有六个自由度。如果将粒子的运动轨迹限制在更小的维度里面，例如，粒子必须沿着电线或在固定的表面上移动，那么系统的自由度则小于6。另一方面，如果一个系统带有可旋转或振动的扩展对象，那么它将具有六个以上的自由度。
In classical mechanics, the state of a point particle at any given time is often described with position and velocity coordinates in the Lagrangian formalism, or with position and momentum coordinates in the Hamiltonian formalism.
In classical mechanics, the state of a point particle at any given time is often described with position and velocity coordinates in the Lagrangian formalism, or with position and momentum coordinates in the Hamiltonian formalism.
在经典力学中，任何给定时间下'''<font color="#ff8000"> 质点</font>'''的状态，不同的力学形式会有不一样的描述，在拉格朗日力学中描述为位置和速度坐标，而在哈密顿力学中则描述为位置和动量坐标。
In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system. The specification of all microstates of a system is a point in the system's phase space.
In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system. The specification of all microstates of a system is a point in the system's phase space.
在统计力学中，自由度是描述系统微观状态的单个标量数。一个系统所有微观状态的规格参数都是基于该系统相空间中的一个点。
In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer.
In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer.
在化学的三维'''<font color="#ff8000"> 理想链</font>'''模型中，描述每个单元结构方向的必要参数是它们的两个角度。
It is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic function to the energy of the system.
It is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic function to the energy of the system.
指定二次自由度通常很重要。因为它们有助于建立系统能量的二次函数。
==Gas molecules==
== Gas molecules 气体分子 ==
[[Image:Degrees of freedom (diatomic molecule).png|thumb|right|Different ways of visualizing the 6 degrees of freedom of a diatomic molecule. (CM: [[center of mass]] of the system, T: [[translational motion]], R: [[rotation]]al motion, V: [[molecular vibration|vibrational motion]].)]]
[[Image:Degrees of freedom (diatomic molecule).png|thumb|right|Different ways of visualizing the 6 degrees of freedom of a diatomic molecule. (CM: [[center of mass]] of the system, T: [[translational motion]], R: [[rotation]]al motion, V: [[molecular vibration|vibrational motion]].)]]
In three-dimensional space, three degrees of freedom are associated with the movement of a particle. A diatomic gas molecule has 6 degrees of freedom. This set may be decomposed in terms of translations, rotations, and vibrations of the molecule. The center of mass motion of the entire molecule accounts for 3 degrees of freedom. In addition, the molecule has two rotational degrees of motion and one vibrational mode. The rotations occur around the two axes perpendicular to the line between the two atoms. The rotation around the atom–atom bond is not a physical rotation. This yields, for a diatomic molecule, a decomposition of:
In three-dimensional space, three degrees of freedom are associated with the movement of a particle. A diatomic gas molecule has 6 degrees of freedom. This set may be decomposed in terms of translations, rotations, and vibrations of the molecule. The center of mass motion of the entire molecule accounts for 3 degrees of freedom. In addition, the molecule has two rotational degrees of motion and one vibrational mode. The rotations occur around the two axes perpendicular to the line between the two atoms. The rotation around the atom–atom bond is not a physical rotation. This yields, for a diatomic molecule, a decomposition of:
在三维空间中，粒子的运动与它的三个自由度有关。双原子气体分子具有6个自由度（存在争议）。因此可以根据其分子的平移，旋转和振动来分解该运动集合。整个分子的质心运动具有3个自由度。除此之外，其具有两个旋转自由度和一个[存在争议]振动自由度。其中旋转运动是围绕垂直于两个原子之间的连线轴发生。值得注意的是这里围绕原子和原子键的旋转并不是物理旋转[存在争议]。因此对于双原子分子，可以将其分解为：
:<math>N = 6 = 3 + 2 + 1.</math>
:<math>N = 6 = 3 + 2 + 1.</math>
For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition:
For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition:
For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition:
对于一般的非线性分子，其所有的3个旋转自由度均需考虑，因此它的分解形式为：
:<math>3N = 3 + 3 + (3N - 6)</math>
:<math>3N = 3 + 3 + (3N - 6)</math>
which means that an -atom molecule has vibrational degrees of freedom for . In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited to only one.
which means that an -atom molecule has vibrational degrees of freedom for . In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited to only one.
这意味着当N>2时，N原子分子具有3N-6个振动自由度。不过在特殊情况下，例如一个吸附的大分子，旋转自由度只能限制为一个。
As defined above one can also count degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows:
As defined above one can also count degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows:
如上所述，还可以使用指定维度空间所需的最少坐标数来确定自由度。比如：
# For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space. Thus its degree of freedom in a 3-D space is 3.
# For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space. Thus its degree of freedom in a 3-D space is 3.
For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space. Thus its degree of freedom in a 3-D space is 3.
For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space. Thus its degree of freedom in a 3-D space is 3.
对于单个粒子，我们需要在二维平面中指定2个坐标，在三维空间中指定3个坐标。因此，它在三维空间中的自由度为3。
# For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D space with constant distance between them (let's say d) we can show (below) its degrees of freedom to be 5.
# For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D space with constant distance between them (let's say d) we can show (below) its degrees of freedom to be 5.
For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D space with constant distance between them (let's say d) we can show (below) its degrees of freedom to be 5.
For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D space with constant distance between them (let's say d) we can show (below) its degrees of freedom to be 5.
在三维空间中，由2个粒子（例如双原子分子）组成且彼此之间具有恒定距离（假设d）的物体，我们可以（在下面）显示其自由度为5。
Let's say one particle in this body has coordinate {{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>, ''z''<sub>1</sub>)}} and the other has coordinate {{math|(''x''<sub>2</sub>, ''y''<sub>2</sub>, ''z''<sub>2</sub>)}} with {{math| ''z''<sub>2</sub>}} unknown. Application of the formula for distance between two coordinates
Let's say one particle in this body has coordinate {{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>, ''z''<sub>1</sub>)}} and the other has coordinate {{math|(''x''<sub>2</sub>, ''y''<sub>2</sub>, ''z''<sub>2</sub>)}} with {{math| ''z''<sub>2</sub>}} unknown. Application of the formula for distance between two coordinates
Let's say one particle in this body has coordinate and the other has coordinate with unknown. Application of the formula for distance between two coordinates
Let's say one particle in this body has coordinate and the other has coordinate with unknown. Application of the formula for distance between two coordinates
假设这个物体中的一个粒子的坐标为{{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>, ''z''<sub>1</sub>)}}，另一个粒子的坐标为{{math|(''x''<sub>2</sub>, ''y''<sub>2</sub>, ''z''<sub>2</sub>)}}，其中{{math| ''z''<sub>2</sub>}}未知。那么两个坐标之间距离的公式可以描述为：
:<math>d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}</math>
:<math>d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}</math>
results in one equation with one unknown, in which we can solve for {{math|''z''<sub>2</sub>}}. One of {{math|''x''<sub>1</sub>}}, {{math|''x''<sub>2</sub>}}, {{math|''y''<sub>1</sub>}}, {{math|''y''<sub>2</sub>}}, {{math|''z''<sub>1</sub>}}, or {{math|''z''<sub>2</sub>}} can be unknown.
One of {{math|''x''<sub>1</sub>}}, {{math|''x''<sub>2</sub>}}, {{math|''y''<sub>1</sub>}}, {{math|''y''<sub>2</sub>}}, {{math|''z''<sub>1</sub>}}, or {{math|''z''<sub>2</sub>}} can be unknown.
results in one equation with one unknown, in which we can solve for . One of , , , , , or can be unknown.
其等式含有一个未知数{{math|''z''<sub>2</sub>}}，不过我们可以对其求解。因此实际上是允许{{math|''x''<sub>1</sub>}}, {{math|''x''<sub>2</sub>}}, {{math|''y''<sub>1</sub>}}, {{math|''y''<sub>2</sub>}}, {{math|''z''<sub>1</sub>}}, 或者 {{math|''z''<sub>2</sub>}}其中之一是未知的。