Scleronomous

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 velocityIn 3-D space, a particle with mass $m,!$ , velocity $mathbf\{v\},!$ has kinetic energy:$T\; =frac\{1\}\{2\}m\; v^2\; ,!$ .

Velocity is the derivative of position with respect time. Use chain rule for several variables::$mathbf\{v\}=frac\{dmathbf\{r\{dt\}=sum\_i\; frac\{partial\; mathbf\{r\{dq\_i\}dot\{q\}\_i+frac\{partial\; mathbf\{r\{dt\},!$ .

Therefore,:$T\; =frac\{1\}\{2\}msum\_i\; left(frac\{partial\; mathbf\{r\{partial\; q\_i\}dot\{q\}\_i+frac\{partial\; mathbf\{r\{partial\; t\}\; ight)^2,!$ .

Rearranging the terms carefullycite book |last=Goldstein|first=Herbert|title=Classical Mechanics|year=1980| location=United States of America | publisher=Addison Wesley| edition= 3rd| isbn=0201657023 | language=English| pages=pp. 25] ,

:$T\; =T\_0+T\_1+T\_2,!$ ::$T\_0=frac\{1\}\{2\}mleft(frac\{partial\; mathbf\{r\{partial\; t\}\; ight)^2,!$ ,:$T\_1=sum\_i\; mfrac\{partial\; mathbf\{r\{partial\; t\}cdot\; frac\{partial\; mathbf\{r\{partial\; q\_i\}dot\{q\}\_i,!$ ,:$T\_2=sum\_\{i,j\}\; frac\{1\}\{2\}mfrac\{partial\; mathbf\{r\{partial\; q\_i\}cdot\; frac\{partial\; mathbf\{r\{partial\; q\_j\}dot\{q\}\_idot\{q\}\_j,!$ .

$T\_0,!$ , $T\_1,!$ , $T\_2,!$ 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:

:$frac\{partial\; mathbf\{r\{partial\; t\}=0,!$ .Therefore, only term $T\_2,!$ does not vanish::$T\; =T\_2,!$ .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: $sqrt\{x^2+y^2\}\; -\; L=0,!$ ,

where $(x,\; y),!$ is the position of the weight and $L,!$ is length of the string.

Refer to figure at right, Assume the top end of the string is attached to a pivot point undergoing a simple harmonic motion:$x\_t=x\_0cosomega\; t,!$ ,

where $x\_0,!$ is amplitude, $omega,!$ is angular frequency, and $t,!$ 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 a rheonomous; it obeys rheonomic constraint:$sqrt\{(x\; -\; x\_0cosomega\; t)^2+y^2\}\; -\; L=0,!$ .

ee also

*Pendulum
*Lagrangian mechanics
*Holonomic
*Rheonomous

References