|
|
| Line 1: |
Line 1: |
| In [[classical mechanics]], '''holonomic constraints''' are relations between the coordinates (and possibly time) which can be expressed in the following form:
| | The author's name is Christy Brookins. It's not a common thing but what I like performing is to climb but I don't have the time lately. Some time ago he selected to reside in North Carolina and he doesn't strategy on altering it. My day job is a travel agent.<br><br>Take a look at my site: psychic readers ([http://koreanyelp.com/index.php?document_srl=1798&mid=SchoolNews visit the up coming internet page]) |
| <math>f(q_1, q_2, q_3,\ldots, q_{n}, t) = 0</math>, where <math>\{ q_1, q_2, q_3, \ldots, q_{n} \}</math> are the n coordinates which describe the system. For example, the motion of a particle constrained to lie on the surface of a [[sphere]] is subject to a holonomic constraint, but if the particle is able to fall off the sphere under the influence of gravity, the constraint becomes non-holonomic.
| |
| | |
| Velocity-dependent constraints such as
| |
| <math>f(q_{1},q_{2},...,q_{n},\dot{q}_{1},\dot{q}_{2},...,\dot{q}_{n},t)=0</math>
| |
| are not usually holonomic.
| |
| | |
| ==Holonomic system (physics)==
| |
| In [[classical mechanics]] a system may be defined as '''holonomic''' if all constraints of the system are holonomic. For a constraint to be holonomic it must be expressible as a [[function (mathematics)|function]]:
| |
| :<math> f(x_1,\ x_2,\ x_3,\ \dots,\ x_N,\ t)=0, \, </math>
| |
| i.e. a '''holonomic constraint''' depends only on the coordinates <math>x_j\,\!</math> and time <math>t\,\!</math> . It does not depend on the velocities. A constraint that cannot be expressed in the form shown above is a [[nonholonomic system|nonholonomic constraint]].
| |
| | |
| ===Transformation to general coordinates===
| |
| The holonomic constraint equations can help us easily remove some of the dependent variables in our system. For example, if we want to remove <math>x_d\,\!</math> which is a parameter in the constraint equation <math>f_i\,\!</math>, we can rearrange the equation into the following form, assuming it can be done,
| |
| :<math>x_d=g_i(x_1,\ x_2,\ x_3,\ \dots,\ x_{d-1},\ x_{d+1},\ \dots,\ x_N,\ t), \, </math>
| |
| | |
| and replace the <math>x_d\,\!</math> in every equation of the system using the above function. This can always be done for general physical system, provided that <math>f_i\,\!</math> is <math>C^1\,\!</math>, then by [[implicit function theorem]], the solution <math>g_i\,</math> is guaranteed in some open set. Thus, it is possible to remove all occurrences of the dependent variable <math>x_d\,\!</math>. | |
| | |
| Suppose that a physical system has <math>N\,\!</math> degrees of freedom. Now, <math>h\,\!</math> holonomic constraints are imposed on the system. Then, the number of degrees of freedom is reduced to <math>m=N - h\,\!</math>. We can use <math>m\,\!</math> independent generalized coordinates (<math>q_j\,\!</math>) to completely describe the motion of the system. The transformation equation can be expressed as follows:
| |
| :<math>x_i=x_i(q_1,\ q_2,\ \dots,\ q_m,\ t)\ ,\qquad\qquad\qquad i=1,\ 2,\ \dots N. \, </math>
| |
| | |
| ===Differential form===
| |
| Consider the following differential form of a constraint equation:
| |
| :<math>\sum_j\ c_{ij} dq_j+c_i dt=0; \, </math>
| |
| where ''c''<sub>''ij''</sub>, ''c''<sub>''i''</sub> are the coefficients of the differentials ''dq''<sub>''j''</sub> and ''dt'' for the ''i''th constraint.
| |
| | |
| If the differential form is integrable, i.e., if there is a function <math>f_i(q_1,\ q_2,\ q_3,\ \dots,\ q_N,\ t)=0\,\!</math> satisfying the equality
| |
| :<math>df_i=\sum_j\ c_{ij} dq_j+c_i dt=0,\,</math>
| |
| then this constraint is a holonomic constraint; otherwise, nonholonomic. Therefore, all holonomic and some nonholonomic constraints can be expressed using the differential form. Not all nonholonomic constraints can be expressed this way. Examples of nonholonomic constraints which can not be expressed this way are those that are dependent on generalized velocities. With a constraint equation in differential form, whether the constraint is holonomic or nonholonomic depends on the integrability of the differential form.
| |
| | |
| ===Classification of physical systems===
| |
| In order to study classical physics rigorously and methodically, we need to classify systems. Based on previous discussion, we can classify physical systems into holonomic systems and [[non-holonomic system]]s. One of the conditions for the applicability of many theorems and equations is that the system must be a holonomic system. For example, if a physical system is a holonomic system and a [[monogenic system]], then [[Hamilton's principle]] is the necessary and sufficient condition for the correctness of [[Lagrange's equation]].<ref name="Herb1980">{{cite book |last=Goldstein |first=Herbert |authorlink=Herbert Goldstein |title=Classical Mechanics |year=1980 |location=United States of America |publisher=Addison Wesley |edition=3rd |isbn=0-201-65702-3 |page=45}}</ref>
| |
| | |
| ===Examples===
| |
| [[File:SimplePendulum01.JPG|frame|right|A simple pendulum]]
| |
| As shown at right, a simple [[pendulum]] is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string’s length is a constant. Therefore, this system is holonomic; it obeys the holonomic constraint
| |
| :'''<math> {x^2+y^2} - L^2=0, </math>''' | |
| where <math>(x,\ y)\,\!</math> is the position of the weight and <math>L\,\!</math> is length of the string.
| |
| | |
| The particles of a [[rigid body]] obey the holonomic constraint
| |
| :<math>(\mathbf{r}_i - \mathbf{r}_j)^2 - L_{ij}^2=0, \, </math> | |
| where <math>\mathbf{r}_i\,\!</math>, <math>\mathbf{r}_j\,\!</math> are respectively the positions of particles <math>P_i\,\!</math> and <math>P_j\,\!</math>, and <math>L_{ij}\,\!</math> is the distance between them.
| |
| | |
| ==References== | |
| <references />
| |
| | |
| [[Category:Classical mechanics]]
| |
The author's name is Christy Brookins. It's not a common thing but what I like performing is to climb but I don't have the time lately. Some time ago he selected to reside in North Carolina and he doesn't strategy on altering it. My day job is a travel agent.
Take a look at my site: psychic readers (visit the up coming internet page)