Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

If we consider the the relativistic Lorentz force law:

$$\frac{d}{dt} (m\gamma \vec{u})=e(\vec{E}+\vec{u} \times \vec{B})$$

How can we deduce:

$$\frac{d}{dt} (m\gamma c^2)=e \vec{E} \cdot \vec{u}$$

Clearly dotting with $\vec{u}$ will give us the RHS. Which leaves us:

$$\vec{u} \cdot \frac{d}{dt} (m\gamma \vec{u})=e \vec{u} \cdot \vec{E}$$

Could anyone help explain how to proceed and if this is the correct method?

EDIT: If it helps: with reference to these notes i'm working through: http://www.maths.ox.ac.uk/system/files/coursematerial/2012/2393/8/WoodhouseLectures.pdf Page 86, eq (178), the paragraph underneath states 'The first equation (which follows from the second)', this is what i'm trying to prove (a warning, the notes are riddled with errors..).

share|improve this question
    
Have you tried integrating by parts? –  Jerry Schirmer Apr 2 '13 at 18:17
    
@JerrySchirmer: So saying: '$\vec{u} \cdot \frac{d}{dt} (m\gamma \vec{u})= \frac{d}{dt}(m \gamma \frac12 \vec{u}\cdot\vec{u})-m \frac12 \vec{u}\cdot\vec{u} \frac{d}{dt}\gamma$'? –  Freeman Apr 2 '13 at 18:21
add comment

2 Answers

up vote 2 down vote accepted

Let's set $c=1$ for simplicity.

Using your observations, it suffices to show that (just combine the second and third equations you write down) $$ \dot \gamma = \vec u \cdot \frac{d}{dt}(\gamma \vec u). $$ To prove this, the following facts are useful: $$ \dot \gamma = \gamma^3\vec u \cdot\dot{\vec u}, \qquad \gamma^2\vec u^2 +1 = \gamma^2. $$ Now just compute \begin{align} \vec u \cdot \frac{d}{dt}(\gamma \vec u) &=\vec u \cdot (\dot \gamma \vec u + \gamma \dot{\vec u}) \\ &=\vec u \cdot (\gamma^3(\vec u \cdot \dot{\vec u})\vec u + \gamma \dot{\vec u}) \\ &= \gamma \vec u \cdot \dot{\vec u}(\gamma^2 \vec u^2 + 1) \\ &= \gamma^3\vec u \cdot\dot {\vec u} \\ &= \dot \gamma \end{align}

share|improve this answer
    
That's brilliant, thank you very much! I should have got that... You know how it is when your brain is tired and the coffee has worn off! –  Freeman Apr 2 '13 at 18:57
    
@Freeman Sure thing! –  joshphysics Apr 2 '13 at 19:02
add comment

The LHS of the equation you're trying to get to is none other than the time rate of change of the (relativistic) kinetic energy once you add back in the derivative of the constant term - mc^2.

The relativistic analog of the Newtonian relation Force-dot-velocity = rate of change of K.E. holds as well. From here you just substitute the force law and the desired formula results as expected.

share|improve this answer
    
Thank you for this, as a mathematician it's great to see a more physical interpretation of it. Thank you so much for your time! –  Freeman Apr 2 '13 at 19:00
    
Funny, as I'm always fretting if I should be making my physics more rigorous. The grass is always greener I suppose. =p –  David H Apr 2 '13 at 19:05
    
Haha.. very true indeed! –  Freeman Apr 2 '13 at 19:07
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.