Solving the Constrained Lagrangian Dynamics equation for a simple pendulum

[Azeemsha]

Homework Statement

How to solve the equation of simple pendulum in Cartesian coordinates with Lagrangian multipliers for constrained forces.

Relevant Equations

with KE = 0.5*m*(xdot^2 + ydot^2) and PE = m*g*l*cos(theta) = m*g*l*(y/l)

equation comes to the form after substituting in Lagrangian equation
m*xdotdot + 2*lamda*x = 0

m*ydotdot + m*g*l + 2*lamda*y = 0

constrained to: x2 + y2 =1
mass m = 1

length l = 1

Lamda is Lagrangian multiplier

How to solve these equations of motion with this constraint ?

The equations is of the form of Differential algebraic Equation, of index 3.

 

[jambaugh]

Just typesetting your equations for you.
Energy Equations:
$$KE=\frac{1}{2}m\left(\dot{x}^2 + \dot{y}^2\right),\quad PE=m g \ell \cos(\theta)=m g \ell\left(\frac{y}{\ell}\right)$$
Lagrangian equation:
$$m \ddot{x} + 2\lambda x = 0,\quad m \ddot{y}+mg\ell + 2\lambda y = 0$$
Constraints:
$$x^2 + y^2 = 1, \quad m=1, \quad \ell = 1$$

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(one) Method of solution: The Lagrangian equations can be integrated by multiplying by an appropriate integrating factor. Example: $\ddot{x}+\alpha x = 0 \to \ddot{x}\dot{x}+\alpha x \dot{x} = 0$ and $\ddot{x}\dot{x} = \frac{d}{dt}\frac{\dot{x}^2}{2}$. [edit: added->] Similarly, $x \dot{x} = \frac{d}{dt}\frac{x^2}{2}$.

Similarly for $\ddot{y}+\beta y + \gamma$ you can make the substitution: $u = \beta y +\gamma\to \dot{u}=\beta\dot{y}, \ddot{u}=\beta\ddot{y}$. This brings it into the same form as the x equation and you may integrate via the integration factor.

This should help you get started.