scleronomous

{{main|Generalized velocity}}

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

$$T\; =\backslash frac\{1\}\{2\}m\; v^2\; .$$

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

$$\backslash mathbf\{v\}\; =\; \backslash frac\{d\backslash mathbf\{r\}\}\{dt\}\; =\; \backslash sum\_i\backslash \; \backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; q\_i\}\; \backslash dot\{q\}\_i\; +\; \backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; t\}\; .$$

where $q\_i$ are generalized coordinates.

Therefore,

$$T\; =\; \backslash frac\{1\}\{2\}\; m\; \backslash left(\backslash sum\_i\backslash \; \backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; q\_i\}\backslash dot\{q\}\_i+\backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; t\}\backslash right)^2\; .$$

Rearranging the terms carefully,{{cite book |last=Goldstein|first=Herbert|title=Classical Mechanics|url=https://archive.org/details/classicalmechani00gold_919|url-access=limited|year=1980| location=United States of America | publisher=Addison Wesley| edition= 3rd| isbn=0-201-65702-3 | page=[https://archive.org/details/classicalmechani00gold_919/page/n34 25]}}

$$\backslash begin\{align\}$$

T &= T_0 + T_1 + T_2 : \\[1ex]

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,

\end{align}

where $T\_0\backslash ,\backslash !$, $T\_1\backslash ,\backslash !$, $T\_2$ 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:

$$\backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; t\}=0.$$

Therefore, only term $T\_2$ does not vanish:

$$T\; =\; T\_2.$$

Kinetic energy is a homogeneous function of degree 2 in generalized velocities.

File:SimplePendulum01.svg

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

$$\backslash sqrt\{x^2+y^2\}\; -\; L\; =\; 0,$$

where $(x,y)$ is the position of the weight and $L$ is length of the string.

File:Pendulum02.JPG

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

$$x\_t=x\_0\backslash cos\backslash omega\; t\; ,$$

where $x\_0$ is amplitude, $\backslash omega$ 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

$$\backslash sqrt\{(x\; -\; x\_0\backslash cos\backslash omega\; t)^2+y^2\}\; -\; L\; =\; 0.$$

• Lagrangian mechanics
• Holonomic system
• Nonholonomic system
• Rheonomous
• Mass matrix

Category:Mechanics

Category:Classical mechanics

Category:Lagrangian mechanics

de:Skleronom