# Help with a mechanical lagrangian problem

• black_hole
In summary, the given equation of motion L = 1/2mv2 - mgz is solved using the Euler-Lagrange equation. The equations of motion are found to be v = -vt + v0 and x = -1/2gt2 + v0t. To implement the constraints, z(τ) = 0 when τ≠0, it is suggested that z = 1/2gt2 - vozt = 0 when t = (z * v0z)/g = τ. The value of S is found using the definition of action, S = ∫Ldt = 1/2m∫v2dt - mg∫zdt. It is noted that the
black_hole

## Homework Statement

We are given L = 1/2mv2 - mgz.

a) Find the equations of motion.
b) Take x(0) [vector] = 0; v(0) [vector] = v0 [vector] ; v0z > 0 and find x(τ) [vector] and v(τ) [vector], such that z(τ) = 0;  τ≠0.
c) Find S.

## Homework Equations

Euler-Lagrange equation and definition of action.

## The Attempt at a Solution

a) Using the Euler-Lagrange equation:

∂L/∂x - d/dt(∂L/∂v) = 0
-mg - d/dt(mv) = 0
-mg = d/dt(mv)
F = -dp/dt

b) If F = -dp/dt, then a = -g. So;

v = -vt + v0 and x = -1/2gt2 + v0t

Hmmm, that's where I'm stuck; I'm not sure how to implement the last two constraints. Maybe if z(τ) = 0 but τ≠0 then z = 1/2gt2 - vozt = 0 when t = (z * v0z)/g = τ ?
(I'm not explicitly given v0z...)

c) S = ∫Ldt = 1/2m∫v2dt - mg∫zdt

If my above expressions for v and z are correct then I presume I can use those tot find the integrals...?

Hi black_hole!

You seem to be mixing up x and z.

This is a ball being thrown up (v0z > 0).
They want to know when the ball is back on the ground...
Or rather where the ball is at that time and what its velocity is as a function of its initial velocity v0.

## 1. What is a mechanical Lagrangian problem?

A mechanical Lagrangian problem is a mathematical problem that involves applying the principles of Lagrangian mechanics to analyze and solve mechanical systems. It is commonly used in physics and engineering to study the dynamics of various physical systems.

## 2. How is a mechanical Lagrangian problem different from a regular mechanical problem?

A mechanical Lagrangian problem differs from a regular mechanical problem in that it uses the concept of energy to describe the dynamics of a system, rather than using forces and accelerations. It also takes into account the constraints and boundary conditions of the system, making it a more comprehensive approach to solving mechanical problems.

## 3. What are the steps involved in solving a mechanical Lagrangian problem?

The first step is to define the system and its constraints. Then, the Lagrangian function is formulated using the system's kinetic and potential energy. Next, the Euler-Lagrange equations are used to derive the equations of motion. These equations can then be solved using various mathematical techniques to find the solution to the problem.

## 4. What types of systems can be solved using the mechanical Lagrangian method?

The mechanical Lagrangian method can be applied to a wide range of systems, including simple pendulums, rigid bodies, and even complex systems such as satellites and planetary orbits. As long as the system can be described using energy and constraints, the mechanical Lagrangian method can be used to solve it.

## 5. What are the advantages of using the mechanical Lagrangian method?

The mechanical Lagrangian method offers several advantages compared to other methods of solving mechanical problems. It provides a more comprehensive understanding of the system's dynamics, it is independent of the choice of coordinates, and it is often more efficient in solving complex systems. It also allows for the use of conservation laws, making it a powerful tool for analyzing mechanical systems.

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