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I'm a teaching assistant in calculus and my students who are studying mechanical engineering asked me to explain the multivariable chain rule. So I thought it could be fun if I could give an example of its use in classical mechanics; by considering some classical system. It should be simple classical mechanics since they're only in the first semester and haven't heard about lagrangian and other stuff yet.

Do you have any suggestions? Specifically, I'm looking for some function that represents a physical system, which I can use the chain rule on.

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    $\begingroup$ Yeah unfortunately there is no objective answer to this so it should remain closed. Like you have the rocket equation $v = -v_e\ln(m/m_0)$ so you get $m^{-1}~dm/dt$, you have even just $[R \cos(\omega t), R\sin(\omega t)] $ giving you $v=\omega r,a=\omega^2 r$, you have implicit differentiation of arctangent, $\tan f = x$ so you have $f'(x) \sec^2(f) = 1$ and thus $f'(x)=1/(1+x^2)$... Lots of possibilities, I would have a hard time choosing just one. $\endgroup$
    – CR Drost
    Commented Sep 25, 2020 at 6:25

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A simple example of using the chain rule is to determine the rate of change of kinetic energy for an object with mass $m$ acted on by a constant force $F$:

$\displaystyle \frac d {dt} \left( \frac 1 2 m v^2 \right) = \frac d {dv} \left( \frac 1 2 m v^2 \right) \frac {dv}{dt} = mv \frac {dv}{dt} = vF$

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In kinematics, the multivariate chain rule is very useful. Especially in deriving accelerations from velocities.

Consider the position of a particle attached to a pendulum riding on a cart. The cart position $x$ and the pendulum angle $\theta$ (as well as the constant $\ell$) describe the position of the particle as follows:

$$ \vec{\rm pos}(x,\theta) = \pmatrix{x+\ell \sin \theta \\ -\ell\cos \theta \\ 0} $$

To find the velocity at any time using the chain rule

$$ \begin{aligned} \vec{\rm vel}(x,\theta,\dot{x},\dot{\theta}) & = \left( \frac{\partial}{\partial x} \vec{\rm pos}(x,\theta) \right) \dot{x} + \left( \frac{\partial}{\partial \theta} \vec{\rm pos}(x,\theta) \right) \dot{\theta} \\ & = \pmatrix{1\\0\\0} \dot{x} + \pmatrix{\ell \cos \theta \\ \ell \sin \theta \\ 0} \dot{\theta} = \pmatrix{\dot{x} + \dot{\theta} (\ell \cos \theta) \\ \dot{\theta} (\ell \sin \theta) \\ 0} \end{aligned} $$

and again the find the accelleration

$$ \begin{aligned} \vec{\rm acc}(x,\theta,\dot{x},\dot{\theta},\ddot{x},\ddot{\theta}) & = \left( \frac{\partial}{\partial x} \vec{\rm vel}(x,\theta,\dot{x},\dot{\theta}) \right) \dot{x} + \left( \frac{\partial}{\partial \theta} \vec{\rm vel}(x,\theta,\dot{x},\dot{\theta}) \right) \dot{\theta} + \\ & + \left( \frac{\partial}{\partial \dot{x}} \vec{\rm vel}(x,\theta,\dot{x},\dot{\theta}) \right) \ddot{x} + \left( \frac{\partial}{\partial \dot{\theta}} \vec{\rm vel}(x,\theta,\dot{x},\dot{\theta}) \right) \ddot{\theta} \\ & = \pmatrix{0\\0\\0} \dot{x} + \pmatrix{-\dot{\theta} \ell \sin \theta \\ \dot{\theta} \ell \cos \theta \\ 0} \dot{\theta} + \pmatrix{1\\0\\0} \ddot{x} + \pmatrix{\ell \cos \theta \\ \ell \sin \theta \\ 0} \ddot{\theta} \\ & = \pmatrix{\ddot{x} + \ddot{\theta} \ell \cos \theta - \dot{\theta}^2 \ell \sin \theta \\ \ddot{\theta} \ell \sin \theta + \dot{\theta}^2 \ell \cos \theta \\ 0 } \end{aligned} $$

And you have introduced the students to the wonderful world of dynamics (kinematics actually here) by application of calculus.

Any particle constrained to some motion with one or more parameters will do here, even as simple as circular motion as the acceleration will always feature a partial derivative to position and a partial to velocity.

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