The dynamics of a charged particle with velocity $\textbf{v}$ in electromagnetic field is dominated by the Lorentz force   $$\textbf{F} = q(\textbf{E}+{\textbf{v} \times \textbf{B}}), \tag{1}$$  and the corresponding Lagrangian is  $$L = \dfrac{1}{2}mv^2 - q\phi(\textbf{r},t)+q\textbf{v} \cdot \textbf{A}. \tag{2}$$      My question is: How to derive the equation $(1)$ just from $(2)$ ?
121578
We can achieve a simplified version of the Lorentz force by $$F=q\bigg[-\nabla(\phi-\mathbf{A}\cdot\mathbf{v})-\frac{d\mathbf{A}}{dt}\bigg],$$ where $\mathbf{A}$ is the magnetic vector potential and the scalar $\phi$ the electrostatic potential.  How is this derivable from a velocity-dependent potential  $$U=q\phi-q\mathbf{A}\cdot\mathbf{v}?$$  I fail to see how the total derivative of $\mathbf{A}$ can be disposed of and the signs partially reversed. I'm obviously missing something.
88220
As an exercise I sat down and derived the magnetic field produced by moving charges for a few contrived situations. I started out with Coulomb's Law and Special Relativity. For example, I derived the magnetic field produced by a current $I$ in an infinite wire. It's a relativistic effect; in the frame of a test charge, the electron density increases or decreases relative to the proton density in the wire due to relativistic length contraction, depending on the test charge's movement. The net effect is a frame-dependent Coulomb field whose effect on a test charge is exactly equivalent to that of a magnetic field according to the Biot–Savart Law.  My question is: Can  be derived using only  and Special Relativity?   If so, and the $B$-field is in all cases a purely relativistic effect, then Maxwell's equations can be re-written without reference to a $B$-field. Does this still leave room for magnetic monopoles?
106589
The electromagnetic force on a charge $ e $ is   $$ \vec F = e(\vec E + \vec v\times \vec B),$$   the Lorentz force. But, is this a separate assumption added to the full Maxwell's equations? (the result of some empirical evidence?) Or is it somewhere hidden in Maxwell's equations?
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My question and context with explanation are given below. Thank you in advance.   Faraday’s Law () states that a time-changing magnetic field vector induces a curl of the Electric field around that point. However it does not specify how quickly the necessary spatial gradient is developed in the Electric Field to satisfy that curl. Of course, the Electric Field vectors around that point cannot just jump to the required values instantaneously in accompaniment as soon as the magnetic field starts changing at a fixed rate, since it is required that the Electric Field has a sensible rate of change to in turn induce a curl of the magnetic Field and continue the propagation of the wave. It would seem the exact rates at which the Electric and Magnetic Fields are changing is critical in the mechanism of an EM wave’s propagation. How would I then calculate the rate of change of E somewhere along the wave given I know the rate of change of the B vector that began it, and given the wave is restricted to one dimension for simplicity? There should after all be some delay in how the EM wave moves from point to point in space, seeing as it moves at the finite speed of causality (3*10^8 m/s), as shown in the first image:    Some of my ideas so far have been that perhaps the rate of change of E is determined such that it exactly cancels the growth of the time-changing E vector that gave rise to a curl of B before, which developed at such a rate to induce a curl of E, which demands this rate of change of E which I am talking about now. In this way the wave might level out the growth of the fields in its wake and spread the change further outwards, in the same way smoothening out a crease spreads it elsewhere. However I am stuck on how to show this rigorously.  I am open to any difficulty of maths, so long as the answer provides the explanation. Below are some diagrams to help visualize what I am talking about:    Faraday's Law requires that a curl of E is developed, which under these restrictions requires a spatial gradient of E to appear across that point. Since the field across that point was previously in some other state, it is required that the Electric Field vectors that are infinitesimally close to that point undergo a rate of change through time to achieve the desired state of curl. The question is, how quickly is this desired state of curl achieved?
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In considering the (special) relativistic EM field, I understand that assuming a Lagrangian density of the form  $$\mathcal{L} =-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}  + \frac{1}{c}j_\mu A^\mu$$  and following the Euler-Lagrange equations recovers Maxwell's equations.   Does there exist a first-principles derivation of this Lagrangian? A reference or explanation would be greatly appreciated!
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Is there a proof from the first principle that for the Lagrangian $L$,    $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$  in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it?  On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the  answer, and if possible, the primary source (who published the answer first in the literature).      Notes added on Sept 22: - Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of. - We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider. - But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion. - Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future.
103221
According to my text book:  The Lorentz Force F on a charged particle moving in an electromagnetic field is given by:  $$F= qE + q(v\times B)$$  then it states that "it is to be pointed out that only the electric force does work, while the no work is done by the magnetic force which is simply a deflecting force."  How can this by true won't the the magnetic force have a considerable magnitude, will it now impart kinetic energy to the charge? Any explanations will help a lot. Please keep the terminology simple, thanks.
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The Euler-Lagrange equation for particles is given by   $$ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q},\tag{1}$$  and for fields  it is   $$ \partial_{\mu} \frac{\partial \mathcal{L}}{\partial (\partial_{\mu}\phi)} = \frac{\partial \mathcal{L}}{\partial \phi}.\tag{2} $$  Comparing the two equations, the first one has a total time derivative $\frac{d}{dt}$ but the other one appears to have partial derivatives $\partial_{\mu}$. These derivatives come from integration by parts in the derivation of the EL equation. I was wondering why the field version has partial derivatives and the particle version has total derivatives?   I have also seen for the specific example (in Quantum Field Theory for the gifted amateur) of 1 dimensional waves on a string, the corresponding Euler-Lagrange equation is  $$ \frac{d}{dt} \frac{\partial \mathcal{L}}{\partial \left(\frac{d\phi}{dt}\right)} + \frac{d}{dx} \frac{\partial \mathcal{L}}{\partial \left(\frac{d\phi}{dx}\right)} = \frac{\partial \mathcal{L}}{\partial \phi}, \tag{3}$$  which uses total derivatives, so I am a bit confused.
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I'm currently working through the book Heisenberg's Quantum Mechanics (Razavy, 2010), and am reading the chapter on classical mechanics. I'm interested in part of their derivative of a generalized Lorentz force via a velocity-dependent potential.   I understand the generalized force   $$F_i = -\frac{\partial V}{\partial x_i} + \frac{d}{dt}\left(\frac{\partial V}{\partial v_i}\right)$$  that they derive from a Lagrangian of the form $L = \frac{1}{2}m|\vec v|^2 - V(\vec r,\vec v,t)$.  However, in the next (critical) step of the derivation, the author cites a theorem from Helmholtz saying      ...according to Helmholtz, for the existence of the Lagrangian, such a generalized force can be at most a linear function of acceleration, and it must satisfy the Helmholtz identities.   The three Helmholtz identities are then listed as:  $$\frac{\partial F_i}{\partial \dot{v_j}} = \frac{\partial F_j}{\partial \dot{v_i}},$$  $$\frac{\partial F_i}{\partial v_j} +  \frac{\partial F_j}{\partial v_i} = \frac{d}{dt}\left(\frac{\partial F_i}{\partial \dot{v_j}} +  \frac{\partial F_j}{\partial \dot{v_i}}\right),$$  $$\frac{\partial F_i}{\partial x_j} -  \frac{\partial F_j}{\partial x_i} = \frac{1}{2}\frac{d}{dt}\left(\frac{\partial F_i}{\partial v_j} - \frac{\partial F_j}{\partial v_i}\right).$$  I'm trying to understand where this theorem comes from. Razavy cited a 1887 paper by Helmholtz. I was able to find , but it is in German, so I could not verify whether or not it proved the theorem. Additionally, I could not find it in any recent literature. I searched online and in Goldstein's Classical Mechanics.   The only similar concept that I can find is in the  where we have three equations known as Helmholtz conditions. Are these two concepts one in the same? If so, how should I interpret the function $\Phi$ and the matrix $g_{ij}$ that appear in the Helmholtz conditions I found online?   If the cited theorem from Razavy does not relate from the inverse Lagrangian problem, could I have some help finding the right direction?
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Will the Lorentz force expression be valid for a magnetic field created by a ?  I haven't seen any derivation of Lorentz force expression yet and I don't know whether it was derived applying Lorentz transformations to field created by charge particles.  But if this is the case then magnetic field created by moving charges of course has zero divergence ,as we know (not same as magnetic field of a monopole) and a magnetic monopole can create magnetic field even when it is at rest. So should the Lorentz force expression be applied in the case of magnetic field by a monopole?  Edit:I was talking about the force  on a charge particle due to magnetic field created by a monopole.  $$F=q_e(v \times B)$$
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Given that:  1) virtual photons mediate the electric and magnetic force fields  2) the magnetic field is the curl of the magnetic vector potential  3) the electric field is the negative gradient of the electric scalar potential  How do we understand the vector and scalar potentials in terms of virtual photons?   Specifically, I mean curl-free magnetic vector and gradient-free electric scalar potential fields, where no electromagnetic waves or physical forces are present.   For instance, what's actually happening outside of the solenoid in the Aharonov-Bohm experiment that alters the phase of the passing electron? Are virtual particles involved there? This question has really stumped me because I only have an undergrad level of understanding in physics. Thanks for any help.
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What was the improvement that Maxwell did to the  and why?  I understand that he combined the main equations so that you could get a wave equation for the vectors of $\bf E$ and $\bf B$ but is there anything else?
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The current density is defined as: $$\textbf{J} (\textbf{r},t) := \rho(\textbf{r},t) \cdot \textbf{v}(\textbf{r},t)$$ where $\rho (\textbf{r},t)$ and $\textbf{v}(\textbf{r},t)$ is the charge density and the particle velocity at point $\textbf{r}$ and at time $t$. Based on that definition, how could one derive the following relation? $$I = \iint_{\mathcal{A}} \textbf{J}(\textbf{r},t)\cdot d\textbf{S}$$ where $\mathcal{A}$ is a surface. I am asking this because authors tend to define the current density $\textbf{J}$ through the latter equation (having the current $I$ predefined). I wish to understand how the two are equivalent.
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Taking the curl of the electric field must be possible, because Faraday's law involves it: $$\nabla \times \mathbf{E} = - \partial \mathbf{B} / \partial t$$ But I've just , where it says  The curl of the gradient of any twice-differentiable scalar field $\phi$ is always the zero vector: $$\nabla \times (\nabla \phi)=\mathbf{0}$$  Seeing as $\mathbf{E} = - \nabla V$, where $V$ is the electric potential, this would suggest $\nabla \times \mathbf{E} = \mathbf{0}$. What presumably monumentally obvious thing am I missing?
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How can I derive the ? I don't want to use hyperbolic functions and the fact that the light waves travel by forming spherical wavefronts. Is there a way to derive the Lorentz transformations applying the conditions I have mentioned. I was unable to understand the method given in Landau and lifshitz deeply. That's why I want a method other than the one using hyperbolic functions
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For a point charge that moves with the trajectory $ \vec r(t)$, we know that it has the singular charge and current densities  $$ \rho (\vec x, t) = q \delta^3(\vec x - \vec r(t)) $$ $$ \vec J(\vec x, t) = q \frac{d\vec r}{dt} \delta ^3 (\vec x - \vec r (t))$$  How do you prove that these densities satisfy the continuity equation?  $$ \frac{\partial \rho}{\partial t} + \nabla \cdot \vec J = 0 $$  I considered using the property of the dirac delta  $$ \delta (f(x)) = \sum_i^N  \frac{\delta (x - x_i)}{|\frac{df}{dx}|_{x = x_i}}$$  where $x_i$s are the roots of $f(x)$. In the problem at hand, we see that the dummy variable x is t, and $f(x) = \vec x - \vec r(t)$ so that the only root is $t_0$ which makes $\vec r(t_0) = \vec x$.  However I just don't know how to apply the derivative to the dirac delta. I know that   $$ \delta '[f] = f'(0)$$  Any help please? Thank you very much!
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We consider in Lorentz spacetime, $(x^0,x^1,x^2,x^3)=(t,x,y,z)$, choose the unit of time such that $c=1$.   Given a four vector $A_\mu$, and let the Lagrangian  $$L(x^i,\dot x^i,t)=-m\sqrt{1-\dot x_i\dot x^i}+qA_0+qA_i\dot x^i,\tag{1}$$ where we use Einstein's convention for summation. (See  with electron charge $q=-e$.)    By the Euler-Lagrange equation $$\frac{d}{dt}\frac{\partial L}{\partial\dot x^i}-\frac{\partial L}{\partial x^i}=0\tag{2}$$ we can get three equations, and if we use the fact that  $$\frac{d}{d\tau}=\frac{1}{\sqrt{1-\dot x_i\dot x^i}}\frac{d}{dt}\tag{3}$$  then these equations are, for $i=1,2,3$ $$m\frac{d^2x^i}{d\tau^2}=q(\frac{\partial A_\mu}{\partial x^i}-\frac{\partial A_i}{\partial x^\mu})\frac{dx^\mu}{d\tau}\tag{4}$$  My question follows, can we obtain a fourth equation which is just letting $i=0$?   The  says that this fourth equation is from the property that the action is invariant under Lorentz transformation. But I think this is not very convincing. I tried to view the first three equations as two four vectors having three equation space components, but I do not think this fact can lead to the equality for the last time component.  References:   L. Susskind, .