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 velocity

In 3-D space, a particle with mass $m\,\!$, velocity $\mathbf{v}\,\!$ has kinetic energy $T\,\!$

$T =\frac{1}{2}m v^2 \,\!.$

Velocity is the derivative of position with respect to time. Use chain rule for several variables:

$\mathbf{v}=\frac{d\mathbf{r}}{dt}=\sum_i\ \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i+\frac{\partial \mathbf{r}}{\partial t}\,\!.$

Therefore,

$T =\frac{1}{2}m \left(\sum_i\ \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i+\frac{\partial \mathbf{r}}{\partial t}\right)^2\,\!.$

Rearranging the terms carefully,^[1]

$T =T_0+T_1+T_2\,\!:$

$T_0=\frac{1}{2}m\left(\frac{\partial \mathbf{r}}{\partial t}\right)^2\,\!,$

$T_1=\sum_i\ m\frac{\partial \mathbf{r}}{\partial t}\cdot \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i\,\!,$

$T_2=\sum_{i,j}\ \frac{1}{2}m\frac{\partial \mathbf{r}}{\partial q_i}\cdot \frac{\partial \mathbf{r}}{\partial q_j}\dot{q}_i\dot{q}_j\,\!,$

where $T_0\,\!$, $T_1\,\!$, $T_2\,\!$ 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.

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

$x_t=x_0\cos\omega t\,\!,$

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 rheonomous as it obeys constraint explicitly dependent on time

$\sqrt{(x - x_0\cos\omega t)^2+y^2} - L=0\,\!.$

See also

• Lagrangian mechanics
• Holonomic system
• Nonholonomic system
• Rheonomous

References

1. Goldstein, Herbert (1980). Classical Mechanics (3rd ed.). United States of America: Addison Wesley. p. 25. ISBN 0-201-65702-3.