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}}mv^{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\,\!.

References

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

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Scleronomous

Application

Example: pendulum

See also

References