Constant of motion in velocity dependent motion

[vaibhavtewari]

I have little confusion, we know that if Lagrangian is from from variable $\theta$ then conjugate momenta $P_{\theta}$ is a constant of motion. When it comes to velocity dependent potential like $L=1/2mv^2+qv\times B$ how will this differ ?

 

[vanhees71]

This cannot be right since you ad a scalar and a vector. The correct Lagrangian for the nonrelativistic motion of a point particle in an external electromagnetic field is given by

$$L=\frac{m}{2} \vec{v}^2-q \Phi(\vec{x})+q \frac{\vec{v}}{c} \cdot \vec{A}(\vec{x}).$$

where $\Phi$ is the scalar potential, and $\vec{A}$ is the vector potential of the electromagnetic field. I've used Gaussian (or Heaviside-Lorentz) units.

 

[vaibhavtewari]

You are right, thanks. I actually resolved what I was asking.