# Constant of motion in velocity dependent motion

1. Jun 19, 2011

### 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 ?

2. Jun 20, 2011

### 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.

3. Jun 20, 2011

### vaibhavtewari

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