Understanding the Equation: ( frac{d}{dt} left( frac{partial r}{partial q} right) frac{partial dot{r}}{partial q} )

To understand the mathematical equation [ frac{d}{dt} left( frac{partial r}{partial q} right) frac{partial dot{r}}{partial q}, ] we need to delve into the concepts of the chain rule and total derivatives. In this article, we will break down each component and explain the steps required to grasp this equation.

Understanding the Variables

In the equation, $r$ is a function of two variables: $q$ and $t$. Additionally, $q$ itself is a function of time $t$. We can express this as:

$r r(q(t), t)$. $q q(t)$.

Total Derivative

The total derivative of $r$ with respect to $t$ takes into account both the direct dependence on $t$ and the indirect dependence through $q$. Mathematically, this is expressed as:

[ frac{dr}{dt} frac{partial r}{partial t} frac{partial r}{partial q} frac{dq}{dt}. ]

differentiating $frac{partial r}{partial q}$

To find the derivative $frac{d}{dt} left( frac{partial r}{partial q} right)$, we apply the chain rule:

[ frac{d}{dt} left( frac{partial r}{partial q} right) frac{partial}{partial t} left( frac{partial r}{partial q} right) frac{partial}{partial q} left( frac{partial r}{partial q} right) frac{dq}{dt}. ]

Lets break this down further:

The term $frac{partial}{partial t} left( frac{partial r}{partial q} right)$ represents how $frac{partial r}{partial q}$ changes directly with time. The second term $frac{partial}{partial q} left( frac{partial r}{partial q} right) frac{dq}{dt}$ captures the change in $frac{partial r}{partial q}$ due to changes in $q$ over time.

Linking to $dot{r}$

The notation $dot{r}$ typically denotes $frac{dr}{dt}$. Thus, $frac{partial dot{r}}{partial q}$ indicates how the rate of change of $r$ with respect to time varies with changes in $q$.

Conclusion

The equality [ frac{d}{dt} left( frac{partial r}{partial q} right) frac{partial dot{r}}{partial q} ] holds because both sides express how the partial derivative of $r$ with respect to $q$ evolves over time, accounting for both direct changes and changes due to the dependence of $q$ on $t$.

This equation is pivotal in understanding how functions of multiple variables evolve over time. By breaking down the components, we can see the intricate interplay between direct and indirect time dependencies, which is vital in fields such as physics, engineering, and economics where time-dependent systems are prevalent.