I'm trying to derive the conservation of energy in 3D from the equation $\vec{F}=m\vec{a}$.

David Morin, in his book "Introduction to Classical Mechanics With Problems and Solutions" p. 138-139, proves the conservation of energy in 1D in the following way:

enter image description here

I wanted to prove the 3D version in the same way, so I got the term

$$\int_C m\vec{v} \cdot d\vec{v}$$

or, if parametrized,

$$\int_{t_0}^t m\vec{v}(t) \cdot \frac{d\vec{v}(t)}{dt} \ dt$$

This should obviously yield $$\frac{1}{2}m|\vec{v(t)}|^2 - \frac{1}{2}m|\vec{v(t_0)}|^2.$$

But what I'm wondering is, how do I deal with the dot product? Please see the diagram below.

enter image description here

The angle between $d\vec{v}$ and $\vec{v}$ looks too complicated to be taken into account at infinitesimal level. (and note that even $d\theta$ is not the angle between these two)

  • $\begingroup$ Notice \begin{align} \frac{1}{2}\frac{d}{dt} (v(t)\cdot v(t)) = 2v'(t)\cdot v(t) \end{align} $\endgroup$ – Jacky Chong Jun 25 at 2:41
  • $\begingroup$ @JackyChong Thanks. I see how I can apply integration by parts to obtain the desired result. Can you also comment on the direction of $d\vec{v}$ please? $\endgroup$ – curious Jun 25 at 3:46
  • $\begingroup$ Technically conservation of energy is a more fundamental principle than even Newton's laws of motion. It's a bit strange to derive the former from the latter. But that is more of a Physics concern than a mathematical one. $\endgroup$ – Deepak Jun 25 at 3:51
  • $\begingroup$ @Deepak Why is it? Isn't $F=ma$ the governing equation of Newtonian Mechanics? $\endgroup$ – curious Jun 25 at 4:01
  • 1
    $\begingroup$ @curious It is a key equation (I wouldn't use the word "governing") in Newtonian mechanics. But conservation of energy is a far more general principle - it applies not just to mechanics but practically everything - thermodynamics, electromagnetism etc. With consideration of mass-energy equivalence, it also applies to Special and General Relativity. That's why I said conservation of energy is a much more fundamental principle. Newtonian mechanics is not a perfectly accurate description of the natural world (as we understand it) but that doesn't affect the validity of conservation. $\endgroup$ – Deepak Jun 25 at 7:58

You can write the $d\vec v$ in terms of the components along $\vec v$ and perpendicular to it. $$d\vec v=d|\vec v| \hat v+v d\theta\hat\theta$$ When you multiply with $\vec v=|\vec v|\hat v$, the second term will wanish. So all you need to consider is the radial component (1 dimension)

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  • $\begingroup$ Wow! Perfect! What a genius mathematician! Thanks a bunch. $\endgroup$ – curious Jun 25 at 3:53

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