# Constant of motion in velocity dependent motion

• vaibhavtewari
In summary, the conversation discusses the relationship between Lagrangian, variable theta, and conjugate momenta in motion. It also addresses the correct Lagrangian for nonrelativistic motion in an external electromagnetic field, which includes scalar and vector potentials. The use of Gaussian or Heaviside-Lorentz units is mentioned as well.
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 ?

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.

## 1. What is a constant of motion in velocity dependent motion?

A constant of motion in velocity dependent motion is a physical quantity that remains unchanged or constant throughout the motion of a system, even if the velocity of the system changes. It is a fundamental concept in physics that helps us understand and analyze the behavior of objects in motion.

## 2. How do we determine the constant of motion in velocity dependent motion?

The constant of motion in velocity dependent motion can be determined by using mathematical equations and principles, such as the conservation of energy, momentum, and angular momentum. These equations help us identify and calculate the quantities that remain constant in a system despite changes in its velocity.

## 3. Why is the constant of motion important in velocity dependent motion?

The constant of motion is important because it helps us predict and understand the behavior of objects in motion. It allows us to make accurate calculations and predictions about the motion of a system, and helps us identify any changes or anomalies that may occur during the motion.

## 4. Can the constant of motion change during the motion of a system?

No, the constant of motion cannot change during the motion of a system. This is because it is a fundamental quantity that remains unchanged throughout the motion, regardless of any changes in the velocity of the system. Any changes in the motion would indicate a violation of the laws of physics.

## 5. How does the concept of constant of motion apply in real-world scenarios?

The concept of constant of motion applies in various real-world scenarios, such as the motion of planets in our solar system, the movement of objects in a pendulum, and the behavior of particles in a chemical reaction. It helps us understand and predict the behavior of these systems and allows us to make accurate calculations and predictions about their motion.

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