Mechanical Vibration/Lagrange form Applied

Why Use Lagrange Form?
The largest benefit of using the Lagrange form is the deriving the equation of motion is easier for complex systems.

To start out we will start out applying the Lagrange formulation on a spring and mass system defined within the givens. We will
 * 1) Determine the equation of motion
 * 2) Plot the equation of motion

Givens
$$ k=1000 $$

$$ m=10kg $$

Lagrangian Form \& Energy Equations
$$ \frac{d}{dt}(\frac{dT}{\dot{x}})-\frac{dT}{dx} +\frac{dU}{dx}=0$$

where T equals the kinetic energy of the system

and

U equals the potential energy in the system.

$$T=1/2*k*x^2$$

$$U=1/2*m*g*h$$ or

$$U=1/2*m*g*-\delta x$$

work
The derivative of dT with respect to x is: $$\frac{dT}{dx}=k*x$$