Lagrange 101

Classical Mechanics Level 2

A particle moves according to the following Lagrangian .

$\large{L = \frac{1}{2} m \, (\dot{x}^2 + \dot{y}^2 ) - m g \, y}$

What are the accelerations of $x$ and $y$ ?

\large{(\ddot{x},\ddot{y}) = (0,- \sqrt{g})}

\large{(\ddot{x},\ddot{y}) = (0,-g)}

\large{(\ddot{x},\ddot{y}) = (0,g)}

\large{(\ddot{x},\ddot{y}) = (-g,0)}

1 solution

Brian Moehring
Jul 12, 2018

Note that $\frac{\partial}{\partial t} \frac{\partial L}{\partial \dot{r}} = \frac{\partial L}{\partial r}$

Evaluating each side: $\frac{\partial}{\partial t} \frac{\partial L}{\partial \dot{r}} = \frac{\partial}{\partial t}\left(m\dot{x}, m\dot{y}\right) = \left(m\ddot{x}, m\ddot{y}\right)$ $\frac{\partial L}{\partial r} = \left(0,-mg\right)$ and setting them equal to one another $m\left(\ddot{x},\ddot{y}\right) = m\left(0, -g\right) \implies \boxed{\left(\ddot{x},\ddot{y}\right) = \left(0, -g\right)}$

Lagrange 101

1 solution

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