# Velocity-dependent potentials and the dissipation function

From this previous question Charge, velocity-dependent potentials and Lagrangian where the citation is shown at the page 22, §1.5 of the book Classical Mechanics of Goldstein, we read that

"an electric charge $$q$$ of mass $$m$$ moving at a velocity $$\mathbf{v}$$ in a region containing both electric field $$\mathbf E(t,x,y,z)$$ and magnetic field $$\mathbf B(t,x,y,z)$$ ($$\mathbf B$$ and $$\mathbf E$$ are derivable from a scalar potential $$\phi(t, x, y, z)$$ and a vector potential $$\mathbf A(t,x,y,z)$$), nowing that $$\mathbf E=- \nabla \phi - \frac{\partial {\mathbf A}} {\partial t}, \quad \mathbf B= \nabla \times {\mathbf A}. "\tag{1.61}$$

Why must be

$$\color{red}{\large \mathcal U=q \phi - q {\mathbf A} \cdot{\mathbf v} \quad ?}\tag{1.62}$$

Exist a physics proof of this equality?

Because that combination, after use of the Euler-Lagrange equation, gives you the Lorentz force.

Lagrangians can never really be "proved". Once you know the equation of motion (be it classical mechanics, quantum or electrodynamics), you can "come up" with a Lagrangian that gives the correct answer. Lagrangian are better for they are scalars, manifestly symmetric, and more compact.

• In the meantime, I would like to thank you very much for your reply. Could you do me a kindness, if it is possible, to have a simple proof...because that combination, after use of the Euler-Lagrange equation, gives you the Lorentz force. Thank you very much. Sep 6 '19 at 21:15
• phys.ufl.edu/~pjh/teaching/phy4605/notes/… Sep 6 '19 at 21:16
• Is it also the answer of Claudius' user? I have understood 50% the method of Claudius and 50% your notes. I'm not gonna make a mixture and just get confused, am I? Sep 6 '19 at 21:18
• ... I beg your pardon? Who's Claudius? Sep 6 '19 at 21:19
• Ah I see, on your other question. I think he does the beginning, whereas the link goes all the way. I'm happy to discuss steps of the proof in the chat you'd like. Sep 6 '19 at 21:20

If you use that definition for a potential then the generalised force Qi( see Goldstein equation 1.53) can be replaced by a velocity dependent potential( equation 1.58) that leads to a beautiful equation called Lagrange equation( equation 1.57)

• While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review Apr 7 '20 at 10:56
• I'm just a newbie here Apr 7 '20 at 11:07
• As it was already said in the review, it is good practice to be more explanatory (write corresponding formulas in the answer), and if the latter seems to make too much effort instead of posting an answer make rather a comment. The comment section offers quite a lot editing space (:-)). Apr 7 '20 at 12:22
• I cannot comment as it says that one should have at least 50 reputation. Apr 7 '20 at 13:35
• @YasirSadiq There is a reason for that. You shouldn't try to get around the site design by posting comments as answers. You should post answers as answers, and then when you get enough reputation it shows that you know what you are talking about and are a little more familiar with the site, and hence you will be able to leave comments Apr 7 '20 at 16:57