Constant of motion in velocity dependent motion

AI Thread Summary
The discussion focuses on the concept of constants of motion in the context of Lagrangian mechanics, particularly when dealing with velocity-dependent potentials. It clarifies that when the Lagrangian is derived from a variable like θ, the conjugate momentum P_θ is a constant of motion. The correct Lagrangian for a point particle in an electromagnetic field is presented, incorporating both scalar and vector potentials. The confusion regarding the addition of scalar and vector components in the Lagrangian is addressed and resolved. This highlights the importance of understanding the formulation of Lagrangians in varying contexts, especially in electromagnetic scenarios.
vaibhavtewari
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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 ?
 
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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.
 
You are right, thanks. I actually resolved what I was asking.
 

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