LOG#204. Mechanics and light.

Gravitational or electrical forces are inverse squared laws:



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




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



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)


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


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


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



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

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

    \[\sum_i F_i=ma_i\]




If F_i=0, then p_i=constant!

Kinetic energy for non-relativistic particles read


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


and it yields a uniform motion with solution





The first term is precisely:

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

And thus,

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


    \[U=-F\cdot \Delta x\]



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:



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


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


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


i.e., the Helmholtz free energy.

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