# Partial derivative of Lagrangian with respect to velocity

In summary, the conversation discusses the derivation of the simple equation in classical mechanics, $\frac{\partial L}{\partial \dot{q}}=p$. It is shown that this equation can be derived using two different approaches, resulting in different values for the derivative. The conversation concludes that the use of the product rule is necessary in order to obtain the correct derivative.

I came across a simple equation in classical mechanics,
$$\frac{\partial L}{\partial \dot{q}}=p$$
how to derive that?
On one hand,
$$L=\frac{1}{2}m\dot{q}^2-V$$
so, $$\frac{\partial L}{\partial \dot{q}}=m\dot{q}=p$$
On the other hand,
$$L=\frac{1}{2}m\dot{q}^2-V=\frac{1}{2}m\dot{q}\dot{q}-V=\frac{1}{2}\dot{q}p-V$$
$$\frac{\partial L}{\partial \dot{q}}=\frac{1}{2}p$$
which is half value from the first derivation.

I came across a simple equation in classical mechanics,
$$\frac{\partial L}{\partial \dot{q}}=p$$
how to derive that?
On one hand,
$$L=\frac{1}{2}m\dot{q}^2-V$$
so, $$\frac{\partial L}{\partial \dot{q}}=m\dot{q}=p$$
On the other hand,
$$L=\frac{1}{2}m\dot{q}^2-V=\frac{1}{2}m\dot{q}\dot{q}-V=\frac{1}{2}\dot{q}p-V$$
$$\frac{\partial L}{\partial \dot{q}}=\frac{1}{2}p$$
which is half value from the first derivation.

Well, writing $\frac{1}{2}m \dot{q}^2$ as $\frac{1}{2}\dot{q} p$ doesn't change anything; you have to use the product rule:

$\frac{\partial}{\partial \dot{q}} \frac{1}{2} \dot{q} p = (\frac{\partial}{\partial \dot{q}} \dot{q}) \frac{1}{2} p + \frac{1}{2} \dot{q} (\frac{\partial}{\partial \dot{q}} p)$

that is true.

that is true.

Well, if $p = m\dot{q}$, then $\frac{\partial}{\partial \dot{q}} p = m$

## 1. What is a partial derivative of Lagrangian with respect to velocity?

The partial derivative of Lagrangian with respect to velocity is a mathematical operation that calculates the rate of change of the Lagrangian function with respect to changes in velocity. It is an important tool in Lagrangian mechanics, which is a mathematical framework for studying the dynamics of physical systems.

## 2. Why is the partial derivative of Lagrangian with respect to velocity important?

The partial derivative of Lagrangian with respect to velocity is important because it allows us to calculate the equations of motion for a system in terms of its velocities. This allows us to analyze the behavior of a system and predict its future motion.

## 3. How is the partial derivative of Lagrangian with respect to velocity calculated?

The partial derivative of Lagrangian with respect to velocity is calculated by taking the derivative of the Lagrangian function with respect to each velocity component. This can be done using standard rules of calculus such as the chain rule and product rule.

## 4. What does the partial derivative of Lagrangian with respect to velocity tell us about a system?

The partial derivative of Lagrangian with respect to velocity tells us about the forces acting on a system and how those forces will affect the system's motion. This information is important in understanding the behavior of a system and predicting its future motion.

## 5. Can the partial derivative of Lagrangian with respect to velocity be used in other areas of science?

Yes, the partial derivative of Lagrangian with respect to velocity can be used in many areas of science and engineering, such as physics, mechanical and aerospace engineering, and control systems. It is a powerful tool for analyzing and understanding the dynamics of physical systems.