# Lagrangian problem setup question.

• djones64
In summary: Your name]In summary, the user is seeking help with finding Lagrange's equations of motion for a particle moving on a curve with a specific constraint. The Lagrangian function is defined and the partial derivatives are used to obtain the equations of motion. The equations are then simplified and the user is advised to use a numerical method to solve for x and y.
djones64

## Homework Statement

Consider a particle of mass m moving in a vertical x-y plane along a curve y = a*cos((2$\pi$x)/$\lambda$). Consider its motion in terms of two coordinates x and y.
Find Lagrange's equations of motion with undetermined multipliers.

## Homework Equations

y = a*cos((2$\pi$x)/$\lambda$);
(partial L/partial x) - d/dt(partial L/partial x dot) + $\lambda$*(partial L/partial x) (sorry, couldn't figure out how to write the formula)

## The Attempt at a Solution

All problems that we have been given of this type have had two independent equations for x and y. In this one y depends on x. I attempted to use the parameters x=x and $\dot{x}$=$\dot{x}$. The calculations are too long to post, but the answer I am getting for x double dot is crazy. I can see from the picture that y=a at x=0 and at x=$\lambda$. Is there a way to use that to get my independent x parameter? If this problem belongs in the advanced physics forum, please let me know. Thank you in advance.

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Thank you for your post. I am a scientist and I would be happy to help you with this problem.

Firstly, I would like to clarify that the Lagrange's equations of motion with undetermined multipliers are used to determine the equations of motion for a system with constraints. In this case, the constraint is the curve y = a*cos((2\pix)/\lambda), which restricts the particle's motion to a specific path.

To find the Lagrange's equations of motion, we need to define the Lagrangian function L, which is given by L = T - V, where T is the kinetic energy of the particle and V is the potential energy. In this case, the kinetic energy can be written as T = (1/2)*m*(x dot^2 + y dot^2) and the potential energy is V = m*g*y, where g is the acceleration due to gravity.

Next, we need to determine the partial derivatives of the Lagrangian function with respect to x, y, x dot, and y dot. This will give us the Lagrange's equations of motion:

(d/dt)(partial L/partial x dot) - (partial L/partial x) = 0
(d/dt)(partial L/partial y dot) - (partial L/partial y) = 0

Substituting the expressions for T and V into the Lagrangian function and taking the partial derivatives, we get:

(d/dt)(m*x dot) - m*g*(a*sin(2\pix/\lambda)) = 0
(d/dt)(m*y dot) - m*g = 0

Simplifying these equations, we get:

m*x double dot - (2*pi*m*a/\lambda)*sin(2\pix/\lambda) = 0
m*y double dot - m*g = 0

These are the Lagrange's equations of motion for the given system. To solve for x and y, you will need to use an appropriate numerical method, such as the Euler method or the Runge-Kutta method. I hope this helps.

## 1. What is a Lagrangian problem setup?

A Lagrangian problem setup is a mathematical framework used in classical mechanics to determine the equations of motion for a system of particles. It involves defining a Lagrangian function, which is the difference between the kinetic and potential energies of the system, and then using the Euler-Lagrange equations to find the equations of motion.

## 2. How is a Lagrangian problem setup different from other methods?

A Lagrangian problem setup differs from other methods, such as Newtonian mechanics, in that it is based on the principle of least action. This means that the system will follow a path that minimizes the action, which is the integral of the Lagrangian along the path.

## 3. What types of systems can be analyzed using a Lagrangian problem setup?

A Lagrangian problem setup can be used to analyze a wide range of physical systems, including particles, rigid bodies, and continuous systems such as fluids or fields. It can also be applied to systems with constraints, such as a pendulum swinging on a fixed point.

## 4. How is a Lagrangian problem setup useful in physics?

A Lagrangian problem setup is useful in physics because it provides a more general and elegant approach to solving problems in classical mechanics. It also allows for the incorporation of constraints and is often easier to use for systems with complex geometries.

## 5. What are some real-world applications of a Lagrangian problem setup?

A Lagrangian problem setup has many real-world applications in physics and engineering. It is commonly used in spacecraft trajectory planning, robotics, and the design of control systems. It is also used in the study of fluid dynamics, quantum mechanics, and general relativity.

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