# Solving Classical Mechanics #3d - Momentum p in Jaccobi Function

• Cosmossos
In summary, the conversation is about comparing the use of momentum in the Euler-Lagrange and Jaccobi functions. The difference is that the Euler-Lagrange equations are second-order while the canonical equations are first-order. The example of a 1D simple harmonic oscillator is used to illustrate this difference. The question is how to relate the two equations of motion for this problem.
Cosmossos

## Homework Statement

The problem is number 3d in the following file:
http://phstudy.technion.ac.il/~wn114101/hw/wn2010_hw06.pdf

## The Attempt at a Solution

I think the difference comes from the using of the momentum p. In the Jaccobi function, we use only coordinate x and its derivatives.
Is it correct?

Last edited by a moderator:
And also number 4a ii.
How Do I do that?
thank you

O.k
I made some progress.
In 4, i need only help with a)ii
thanks

The Euler-Lagrange equations of motion are second order differential equations while the canonical equations of motion are first order differential equations. So in the case of the 1D simple harmonic oscillator (with $m=k=1$),

$$H=\frac{p^2}{2}+\frac{x^2}{2}$$

the canonical equations come from:

$$\dot{p}=-\frac{\partial H}{\partial x}=-x$$

$$\dot{x}=\frac{\partial H}{\partial p}=p$$

so how can you relate these to the Euler-Lagrange equations of motion:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{x}}\right)-\frac{\partial L}{\partial x}=0$$

for the same problem?EDIT: Fixed the E-L eom so that its coordinates match the Hamiltonian & canonical equations of motion.

Last edited:

## 1. What is classical mechanics?

Classical mechanics is a branch of physics that studies the motion of objects under the influence of forces. It uses mathematical equations and principles to describe and predict the behavior of objects in motion.

## 2. What is momentum in classical mechanics?

Momentum is a measure of an object's motion, specifically its mass and velocity. It is defined as the product of an object's mass and its velocity, and it is a vector quantity, meaning it has both magnitude and direction.

## 3. What is the Jaccobi function in classical mechanics?

The Jaccobi function, also known as Hamilton's characteristic function, is a mathematical function used in classical mechanics to describe the motion of a system. It is derived from the Hamiltonian function, which represents the total energy of a system.

## 4. How is momentum related to the Jaccobi function?

In classical mechanics, momentum is related to the Jaccobi function through the Hamiltonian equations of motion. These equations describe how the momentum of a system changes over time, and they are derived from the Jaccobi function.

## 5. How is classical mechanics used in real-world applications?

Classical mechanics has many real-world applications, including predicting the motion of planets and satellites, designing structures and machines, and understanding the behavior of fluids and gases. It is also the basis for many other branches of physics, such as thermodynamics and electromagnetism.

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