Scleronomous

Amechanical systemisscleronomousif the equations ofconstraintsdo not contain the time as an explicitvariableand the equation of constraints can be described by generalized coordinates. Such constraints are calledscleronomicconstraints. The opposite of scleronomous isrheonomous.

Application

Main article:Generalized velocity

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

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

Velocity is the derivative of position$r\,\!$with respect to time$t\,\!$.Usechain rule for several variables:

$\mathbf {v} ={\frac {d\mathbf {r} }{dt}}=\sum _{i}\ {\frac {\partial \mathbf {r} }{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial \mathbf {r} }{\partial t}}\,\!.$

where$q_{i}\,\!$aregeneralized coordinates.

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,

$T=T_{0}+T_{1}+T_{2}\,\!:$

$T_{0}={\frac {1}{2}}m\left({\frac {\partial \mathbf {r} }{\partial t}}\right)^{2}\,\!,$

$T_{1}=\sum _{i}\ m{\frac {\partial \mathbf {r} }{\partial t}}\cdot {\frac {\partial \mathbf {r} }{\partial q_{i}}}{\dot {q}}_{i}\,\!,$

$T_{2}=\sum _{i,j}\ {\frac {1}{2}}m{\frac {\partial \mathbf {r} }{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} }{\partial q_{j}}}{\dot {q}}_{i}{\dot {q}}_{j}\,\!,$

where$T_{0}\,\!$,$T_{1}\,\!$,$T_{2}\,\!$are respectivelyhomogeneous functionsof 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

A simple pendulum

As shown at right, a simplependulumis 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.

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 asimple 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
• Mass matrix