LOG#204. Mechanics and light.

Gravitational or electrical forces are inverse squared laws:

    \[F_N=G_N\dfrac{Mm}{R^2}\]

    \[F_C=K_C\dfrac{Qq}{R^2}\]

Strikingly similar, they are both also conservative forces. For gravity:

    \[U_g=-G_N\dfrac{Mm}{R}\]

and

    \[U_e=K_C\dfrac{Qq}{R}\]

for the electrical force. Defining the potentials V_g=U_g/m, and V_e=U_e/q, you get the gravitational and electrical potentials

    \[V_g(r)=-G_N\dfrac{M}{R}\]

    \[V_e(r)=K_C\dfrac{Q}{R}\]

Conservative fields are defined from these potentials

    \[E=-\nabla V_e\]

    \[g=-\nabla V_g\]

In general, for any field \Psi, if conservative, \Psi=-\nabla V. The gravitational field reads, from newtonian gravity (module a sign)

    \[g=\dfrac{F}{m}=G_N\dfrac{M}{R^2}\]

and you would get E=-K_CQ/R^2 in the coulombian field case! Focusing on the gravitational case (a similar field could be done with the electrical field)…The momentum

    \[p=mv=m\dfrac{dx}{dt}\]

is conserved under any vertical (radial) gravitational field. Imagine you do a traslation

    \[x'=x-\alpha\]

The momentum in the x component reads p_x=m\dfrac{dx}{dt}=m\dfrac{dx'}{dt}! Note the momentum in the y or vertical component would not be conserved due to F_g! Thus, symmetry is important. Imagine a spring holding from the upper horizontal surface. Then

    \[x(t)=A\sin(\omega t)\]

where A=constant and

    \[\dot{x}=A\omega \cos (\omega t)\]

with m\omega^2=k, then

    \[m\ddot{x}=-kx\]

and

    \[E=T+U=\dfrac{m}{2}\dot{x}^2+\dfrac{k}{2}x^2=A^2\dfrac{k}{2}=\dfrac{m\omega^2 A^2}{2}=constant\]

Since the spring force is conservative, F=-kx=-kd(x^2/2)/dx, the total energy is conserved. Note the symmetry that says E does not depend on the time and it is constant!

Going 3D is important here. We will use components to avoid vector arrows for convenience. Newton’s second law is

    \[\sum_i F_i=ma_i\]

    \[v_i=\dfrac{d}{dt}x_i\]

    \[a_i=\dfrac{d}{dt}v_i=\dot{v}_i=\dfrac{d^2}{dt^2}x_i=\ddot{x}_i\]

    \[F_i=\dfrac{d}{dt}p_i\]

If F_i=0, then p_i=constant!

Kinetic energy for non-relativistic particles read

    \[T=\dfrac{1}{2}mv^2=\dfrac{1}{2}m\dot{q_i}^2=\dfrac{p_i^2}{2m}\]

If p_i is conserved, then the kinetic energy is also conserved. This is valid for the free particle. In the case of conservative forces, the potential energy reads

    \[a_i=\dfrac{d^2}{dt^2}x_i=\dfrac{f_i}{m}\]

and it yields a uniform motion with solution

    \[v_i(t)=v_{0i}+\dfrac{f_i}{m}t\]

    \[x_i(t)=x_{0i}+v_{0i}t+\dfrac{f_i}{2m}t^2\]

and

    \[f(x_i-x_{0i})=\dfrac{1}{2}m(v^2_i-v_{0i})\]

The first term is precisely:

    \[W(0\rightarrow f)=\Delta T=\dfrac{1}{2}\Delta v^2\]

And thus,

    \[W(i\rightarrow f)=-\Delta E_p=-\Delta U\]

with

    \[U=-F\cdot \Delta x\]

or

    \[f_i=-\dfrac{dU}{dx^i}\]

i. e., f=-\nabla U, Q.E.D. for any conservative force. E=T+U holds for conservative forces with certain properties in the potential energy (depending on coordinates in a homogeneus way). For the harmonic oscillator:

    \[a=\ddot{x}\]

    \[\ddot{x}+\omega^2x=0\]

and the solution

    \[x(t)=A\sin (\omega t)+B\cos (\omega t)\]

with x(0)=x_0 at t=t_0. t=t_0=0 in general, so

    \[x(t)=x_0\cos (\omega t)\]

or any other sinusoidal waveform as well. The velocity

    \[v(t)=\dot{x}=-x_0 \omega \sin (\omega t)\]

    \[a(t)=-x_0\omega^2\cos^2(\omega t)\]

and then

    \[E=\dfrac{1}{2}v(t)^2+\dfrac{1}{2}x(t)^2=\dfrac{1}{2}mx_0^2\omega^2=constant\]

as before!

Light can NOT be described with classical NEWTONIAN mechanics. It took several decades an roughly speaking several centuries to code electromagnetic laws into a single set of equations. Maxwell wrote the synthesis of our current electromagnetic knowledge of light:

  1. Gauss law for the electric field: \nabla \cdot E=\div E=\rho/\varepsilon_0=4\pi K_C\rho. Equivalently, \phi=\oint_S E\cdot dS=4\pi K_CQ=Q/\varepsilon_0. For point particles, this law provides E_i=K_CQx_i/r_i^3=Qu_i/4\pi\varepsilon_0 r_i^2. Moreover, F_i=qE_i, and the gauge field E_i=-\nabla_i\varphi. \varphi=V=Q/4\pi\varepsilon_0 r_i is the potential.
  2. Faraday’s law: \nabla\times E=-\partial_t B, or equivalently \oint_\Gamma E\cdot dl=-\partial_t\oint_C B\cdot dS.
  3. Gauss law for the magnetic field (no magnetic monopoles in standard electromagnetism): \nabla\cdot B=0, or \oint_SB\cdot dS=\phi_B=0.
  4. Ampere’s law: \nabla\times B=j/\varepsilon_0c^2. This original Ampere’s law does not conserve electric charge, so Maxwell added an extra term, the displacement current, yielding

        \[\nabla\times B=j/\varepsilon_0c^2+(\partial_t E)/c^2\]

    .

The combination of the 4 equations above produces wave-like equations for E, B:

    \[\dfrac{1}{c^2}\partial_t^2 E_i-\nabla^2 E_i=\left(\dfrac{1}{c^2}\partial_t^2-\nabla^2\right) E_i=0\]

    \[\dfrac{1}{c^2}\partial_t^2 B_i-\nabla^2 B_i=\left(\dfrac{1}{c^2}\partial_t^2-\nabla^2\right) B_i\]

Plane wave solutions are allowed, with \sin(\omega t-k_i x^i) or generally

    \[E_i\sim c B_i\sim e^{\omega t-k_i x^i}\]

Wave speed is given by

    \[\dfrac{1}{c^2}=\varepsilon_0\mu_0\]

so Maxwell cleverly pointed out that light should be an electromagnetic wave! Furthermore, E\perp B\perp v in general. Light can also be polarized. Polarization or fluctuations in the directions of (E, B) is due to the transverse character of the electromagnetic waves. Maxwell’s equations unify E,B into a single framework. All the electromagnetic phenomena from a common dynamics. Special relativity allows to condense Maxwell equations into \partial_\mu F^{\mu\nu}=j^\nu and \varepsilon_{\mu\nu\sigma\tau}\partial^\nu F^{\sigma\tau}=0. Clifford algebra simplify these equations into a single \partial F=j. Differential forms also allows for such a simplification. Maxwell equations have a new invariance beyond galilean invariance: Lorentz invariance. Essentially, Maxwell equations imply Special Relativity.

In presence of matter, Maxwell equation must be completed with constitutive relations for the electromagnetic fields, plus

    \[\nabla \cdot D=\rho_{free}\]

    \[\nabla \cdot B=0\]

    \[\nabla \times E=-\partial_t B\]

    \[\nabla \times H=j_{free}+\partial_t D\]

and the boundary conditions for

    \[D=\varepsilon_0 E+P\;\;\; P=P(E)\]

    \[H=\dfrac{B}{\mu_0}+M\;\;\; M=M(E)\]

where P, M are the polarization and the magnetization for the (D, H) pair.

Remark: Beyond mechanics and light, today we care about entropic forces,

    \[F_i^{ent}=T\dfrac{\partial S}{\partial q^i}\]

Entropic forces and conservative forces can be added

    \[F_t=F_i^{ent}+F_i^{cons}=T\dfrac{\partial S}{\partial q^i}-\dfrac{\partial U}{\partial q^i}\]

Only in the zero temperature limit, we get the usual conservative terms. The above force can be obtained from

    \[A=U-TS\]

i.e., the Helmholtz free energy.

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