# Testing the second type of equations of Lagrangian Mechanics

I'm new to classical mechanics and learned a new cool thing, second type equations of Lagrangian Mechanics. So, I was just testing if it really works or not. So, I made a question myself to test it, but I think that I've some problem here.

First, assume the $$x$$ coordinate of the ball of mass $$m$$. Initially, it's at $$x=0$$, under the influence of Gravity it should fall (assuming the setup is in such a way that Gravity is acting from downwards.)

So, the cool equation says that :

$$\displaystyle{\boxed{\boxed{ \dfrac{d}{dt} \left ( \dfrac{\partial L}{\partial {x'} } \right) - \dfrac{\partial L}{\partial x} = 0 }} \quad \dots \quad (*)}$$ Where, $$L = K.E. - P.E. = ( mg \sin \theta x) - (mg \sin \theta h)\quad \dots \quad (1)$$

However using the Euler-Lagrange equation (*) with the Lagrangian (1) does not give me a sensible answer, so my question is where is my mistake?

So, I've calculated K.E. here as: $$K = \frac 1 2 m v^2 = \frac 1 2 m (2(g \sin \theta) x) \quad \dots \quad (\text{As }v^2 = u^2 + 2as)$$

and $$P = mgh = mg\sin \theta(h - x )$$

The issue might be to do with using $$v^2=2as$$ in the kinetic energy (as suggested in comments) but I don't see why this is improper.

So, Why I can't substitute the Velocity equations in that Lagrangian Equation (That Kinetic energy part).

• Suggestion: Replace second type of equations of Lagrangian Mechanics with Lagrange equations of second kind. – Qmechanic Apr 6 at 18:17
• Comments are not for extended discussion; this conversation has been moved to chat. – ACuriousMind Apr 6 at 18:38
• @alephzero This is being discussed on meta. – Blue Apr 6 at 19:35

1. OP's calculation (v7) violates the rule is that one is not allowed to use equations of motion in the Lagrangian $$L(q,v,t)$$ before all differentiations in Lagrange equations have been carried out. This is mainly because generalized positions $$q^i$$, generalized velocities $$v^j$$, and time $$t$$ are independent variables in the Lagrangian $$L(q,v,t)$$, cf. e.g. this Phys.SE post.