What are the equations of motion in Lagrangian mechanics?

[Pyroadept]

Homework Statement

I'm confused. Some websites say it is dL/dx = d/dt dL/dv,
whereas others say it is the equations of acceleration, velocity and displacement derived from this, which would require integration, yes?

Homework Equations

The Attempt at a Solution

 

[hbweb500]

The Lagrangian immediately yields the equations of motion for a system. Try a harmonic oscillator for example. Then you have:

$$L = T - V = \frac{1}{2}m \dot{x}^2 - \frac{1}{2}k x^2$$

$$\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{x}} \right) = \frac{\partial L}{\partial x}$$

$$\frac{d}{dt} \left( m \dot{x} \right) = -k x$$

$$m \ddot{x} = -k x$$

Which you will recognize as the routine force law for a spring.

So by plugging in a specific L, we are immediately granted the differential equations of motion for each of our coordinates. From there, we need to solve the differential equations to get the velocity and displacements.