# Question 6.9 Taylor: Classical Mechanics

• Lujz_br
In summary, the conversation discusses finding the equation of a path joining the origin to point P(1,1) in the xy plane, by using the Euler-Lagrange equation. The equation is y'' = y, with the solution y = senh(x)/senh(1). The conversation also mentions that there should be two solutions and the general solution is y = A1 ex + A2 e-x.
Lujz_br

## Homework Statement

Hello, I solved others but not 6.9:
Find the equation of the path joining the origin O to point P(1,1) in the xy plane that makes the integral ∫(y'2 +yy' + y2) dx stationary.
∫ from O to P. y' = dy/dx

## Homework Equations

I need use ∂f/∂y = d/dx (∂f/∂y') (euler-lagrange equation) with f = y'2 +yy' + y2
∂f/∂y = y' + 2y
∂f/∂y' = 2y' + y

and d/dx (∂f/∂y') = 2 y'' + y', go to euler-lagrange equation we get:
y' + 2y = 2 y'' + y'
this is equivalent to:
y'' = y (eq 1)

y = ex is solution of eq. 1, but it don't fit (0,0) and (1,1)
I see the solution at final of the book: y = senh(x)/senh(1)
Ok, is solution of eq. 1 and fit points (0,0) and (1,1).

There are any thing more? I feel I don't get good answer without look at the final of the book.
Thanks! Luiz

You have a second-order differential equation, so you should have two solutions. You found one. What's the other one? The general solution will be a linear combination of the two solutions.

Lujz_br
Ok, 1st is y = A1 ex 2nd is y = A2 e-x and general solution is y = A1 ex + A2 e-x which go to y = senh(x)/senh(1).
Ok, remember this (two solutions) is fine way to get the right answer... :)

## 1. What is the significance of Question 6.9 in Taylor's Classical Mechanics?

Question 6.9 in Taylor's Classical Mechanics is a famous problem that tests students' understanding of the concept of energy conservation and the use of the Lagrangian method to solve mechanical problems.

## 2. How difficult is Question 6.9 in Taylor's Classical Mechanics?

The difficulty of Question 6.9 in Taylor's Classical Mechanics can vary depending on the individual's understanding of the concepts involved. However, it is considered to be a challenging problem and often requires a strong grasp of the principles of classical mechanics to solve.

## 3. Can you explain the steps to solve Question 6.9 in Taylor's Classical Mechanics?

To solve Question 6.9 in Taylor's Classical Mechanics, one must first identify the given parameters, such as the masses and initial conditions. Then, the Lagrangian function must be constructed and the equations of motion derived. Finally, the equations can be solved to find the desired quantities.

## 4. What are some common mistakes students make when attempting Question 6.9 in Taylor's Classical Mechanics?

Some common mistakes students make when attempting Question 6.9 in Taylor's Classical Mechanics include not properly identifying the given parameters, making errors in constructing the Lagrangian function, and incorrectly solving the equations of motion.

## 5. Are there any tips for successfully solving Question 6.9 in Taylor's Classical Mechanics?

Some tips for successfully solving Question 6.9 in Taylor's Classical Mechanics include carefully reading and understanding the problem statement, practicing with similar problems to gain familiarity with the concepts, and double-checking all calculations and equations. It is also helpful to break down the problem into smaller, more manageable steps.

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