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.

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