McCarthy Formalism

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,

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

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

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

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

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

A simple 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.

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 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

• Lagrangian mechanics
• Holonomic system
• Nonholonomic system
• Rheonomous