McCarthy Formalism: Difference between revisions
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A mechanical system is '''scleronomous''' if the equations of constraints do not contain the time as an explicit variable. Such constraints are called '''scleronomic''' constraints. | |||
==Application== | |||
:Main article:[[Generalized velocity]] | |||
In 3-D space, a particle with mass <math>m\,\!</math>, velocity <math>\mathbf{v}\,\!</math> has [[kinetic energy]] | |||
:<math>T =\frac{1}{2}m v^2 \,\!.</math> | |||
Velocity is the derivative of position with respect time. Use [[chain rule#Chain rule for several variables|chain rule for several variables]]: | |||
:<math>\mathbf{v}=\frac{d\mathbf{r}}{dt}=\sum_i\ \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i+\frac{\partial \mathbf{r}}{\partial t}\,\!.</math> | |||
Therefore, | |||
:<math>T =\frac{1}{2}m \left(\sum_i\ \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i+\frac{\partial \mathbf{r}}{\partial t}\right)^2\,\!.</math> | |||
Rearranging the terms carefully,<ref name="Herb1980">{{cite book |last=Goldstein|first=Herbert|title=Classical Mechanics|year=1980| location=United States of America | publisher=Addison Wesley| edition= 3rd| isbn=0-201-65702-3 | page=25}}</ref> | |||
:<math>T =T_0+T_1+T_2\,\!:</math> | |||
:<math>T_0=\frac{1}{2}m\left(\frac{\partial \mathbf{r}}{\partial t}\right)^2\,\!,</math> | |||
:<math>T_1=\sum_i\ m\frac{\partial \mathbf{r}}{\partial t}\cdot \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i\,\!,</math> | |||
:<math>T_2=\sum_{i,j}\ \frac{1}{2}m\frac{\partial \mathbf{r}}{\partial q_i}\cdot \frac{\partial \mathbf{r}}{\partial q_j}\dot{q}_i\dot{q}_j\,\!,</math> | |||
where <math>T_0\,\!</math>, <math>T_1\,\!</math>, <math>T_2\,\!</math> are respectively [[homogeneous function]]s of degree 0, 1, and 2 in generalized velocities. If this system is scleronomous, then the position does not depend explicitly with time: | |||
:<math>\frac{\partial \mathbf{r}}{\partial t}=0\,\!.</math> | |||
Therefore, only term <math>T_2\,\!</math> does not vanish: | |||
:<math>T = T_2\,\!.</math> | |||
Kinetic energy is a homogeneous function of degree 2 in generalized velocities . | |||
==Example: pendulum== | |||
[[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 scleronomous; it obeys scleronomic constraint | |||
: <math> \sqrt{x^2+y^2} - L=0\,\!,</math> | |||
where <math>(x,y)\,\!</math> is the position of the weight and <math>L\,\!</math> is length of the string. | |||
[[File:Pendulum02.JPG|frame|right|A simple pendulum with oscillating pivot point]] | |||
Take a more complicated example. Refer to the next figure at right, Assume the top end of the string is attached to a pivot point undergoing a [[simple harmonic motion]] | |||
:<math>x_t=x_0\cos\omega t\,\!,</math> | |||
where <math>x_0\,\!</math> is amplitude, <math>\omega\,\!</math> is angular frequency, and <math>t\,\!</math> is time. | |||
Although the top end of the string is not fixed, the length of this inextensible string is still a constant. The distance between the top end and the weight must stay the same. Therefore, this system is rheonomous; it obeys rheonomic constraint | |||
:<math> \sqrt{(x - x_0\cos\omega t)^2+y^2} - L=0\,\!.</math> | |||
==See also== | |||
*[[Lagrangian mechanics]] | |||
*[[Holonomic system]] | |||
*[[Nonholonomic system]] | |||
*[[Rheonomous]] | |||
== References == | |||
<references /> | |||
[[Category:Mechanics]] | |||
[[Category:Classical mechanics]] | |||
[[Category:Lagrangian mechanics]] | |||
[[de:Skleronom]] |
Latest revision as of 06:14, 12 October 2013
A mechanical system is scleronomous if the equations of constraints do not contain the time as an explicit variable. Such constraints are called scleronomic constraints.
Application
- Main article:Generalized velocity
In 3-D space, a particle with mass , velocity has kinetic energy
Velocity is the derivative of position with respect time. Use chain rule for several variables:
Therefore,
Rearranging the terms carefully,[1]
where , , are respectively homogeneous functions of degree 0, 1, and 2 in generalized velocities. If this system is scleronomous, then the position does not depend explicitly with time:
Therefore, only term does not vanish:
Kinetic energy is a homogeneous function of degree 2 in generalized velocities .
Example: 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 scleronomous; it obeys scleronomic constraint
where is the position of the weight and is length of the string.
Take a more complicated example. Refer to the next figure at right, Assume the top end of the string is attached to a pivot point undergoing a simple harmonic motion
where is amplitude, is angular frequency, and is time.
Although the top end of the string is not fixed, the length of this inextensible string is still a constant. The distance between the top end and the weight must stay the same. Therefore, this system is rheonomous; it obeys rheonomic constraint
See also
References
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