Lorentz Force - Lorentz Force and Analytical Mechanics

Lorentz Force and Analytical Mechanics

See also: Momentum

The Lagrangian for a charged particle of mass m and charge q in an electromagnetic field equivalently describes the dynamics of the particle in terms of its energy, rather than the force exerted on it. The classical expression is given by:

where A and ϕ are the potential fields as above. Using Lagrange's equations, the equation for the Lorentz force can be obtained.

Derivation of Lorentz force from classical Lagrangian (SI units)
For an A field, a particle moving with velocity v = ṙ has potential momentum, so its potential energy is . For a ϕ field, the particle's potential energy is .

The total potential energy is then:

and the kinetic energy is:

hence the Lagrangian:

Lagrange's equations are

(same for y and z). So calculating the partial derivatives:

\begin{align}\frac{d}{dt}\frac{\partial L}{\partial \dot{x}} & =m\ddot{x}+q\frac{d A_x}{dt} \\ & = m\ddot{x}+ \frac{q}{dt}\left(\frac{\partial A_x}{\partial t}dt+\frac{\partial A_x}{\partial x}dx+\frac{\partial A_x}{\partial y}dy+\frac{\partial A_x}{\partial z}dz\right) \\ & = m\ddot{x}+ q\left(\frac{\partial A_x}{\partial t}+\frac{\partial A_x}{\partial x}\dot{x}+\frac{\partial A_x}{\partial y}\dot{y}+\frac{\partial A_x}{\partial z}\dot{z}\right)\\ \end{align}

equating and simplifying:

\begin{align} F_x & = -q\left(\frac{\partial \phi}{\partial x}+\frac{\partial A_x}{\partial t}\right) + q\left \\ & = qE_x + q \\ & = qE_x + q_x \\ & = qE_x + q(\mathbf{\dot{r}}\times\mathbf{B})_x \end{align}

and similarly for the y and z directions. Hence the force equation is:

The potential energy depends on the velocity of the particle, so the force is velocity dependent, so it is not conservative.

The relativistic Lagrangian is

The action is the relativistic arclength of the path of the particle in space time, minus the potential energy contribution, plus an extra contribution which quantum mechanically is an extra phase a charged particle gets when it is moving along a vector potential.

Derivation of Lorentz force from relativistic Lagrangian (SI units)

The equations of motion derived by extremizing the action (see matrix calculus for the notation):

are the same as Hamilton's equations of motion:

both are equivalent to the noncanonical form:

This formula is the Lorentz force, representing the rate at which the EM field adds relativistic momentum to the particle.

Read more about this topic:  Lorentz Force

Famous quotes containing the words force, analytical and/or mechanics:

    “If writers were too wise, perhaps no books would get written at all. It might be better to ask yourself “Why?” afterwards than before. Anyway, the force from somewhere in Space which commands you to write in the first place, gives you no choice. You take up the pen when you are told, and write what is commanded. There is no agony like bearing an untold story inside you.”
    —Zora Neale Hurston (1891–1960)

    “I have seen too much not to know that the impression of a woman may be more valuable than the conclusion of an analytical reasoner.”
    —Sir Arthur Conan Doyle (1859–1930)

    “It is only the impossible that is possible for God. He has given over the possible to the mechanics of matter and the autonomy of his creatures.”
    —Simone Weil (1909–1943)