Scleronomous

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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 m\,\!, velocity {\mathbf {v}}\,\! has kinetic energy

T={\frac {1}{2}}mv^{2}\,\!.

Velocity is the derivative of position with respect 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; it obeys rheonomic constraint

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

See also

References

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

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