<aside> 🫴 The Lagrangian:

$$ \boxed{L=T-V}=\frac 12 m\dot x^2-V(x) $$

</aside>

• its derivative with respect to $x$:

  $$ \frac{\partial L}{\partial x}=-\frac{\partial V}{\partial x} $$

• By Newton’s 2nd

  $$ \frac{\partial L}{\partial x}=-\frac{\partial V}{\partial x}=m\ddot x=\frac{\text d}{\text dt}\left ( \frac{\partial L}{\partial \dot x} \right ) $$

<aside> 🕊️ Lagrange’s equation:

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

</aside>

The rate of change of momentum is equal to the force

Simple example

💼 Case: Consider a mass on a spring $V=\frac 12 kx^2$

1. 🍎 Newton’s method:

   $$ m\ddot x=F=-\frac{\partial V}{\partial x}=-kx $$

   • $m\ddot x=-kx$ corresponds to a simple harmonic motion with frequency $\omega=\sqrt{k/m}$

2. ⚡ Energy conservation

   $$ E=\frac 12 m\dot x^2+\frac 12 kx^2 = \text{constant} $$

   • Thus we can write

     $$ \frac{\text d E}{\text dt}=(m \ddot{x}+kx)\dot x=0 $$

     we get either $\dot x=0$ or $m\ddot x=-kx$

3. 🍋 Lagrange’s method

   $$ L=\frac 12 m\dot x^2-\frac 12 kx^2 $$

   • so

     $$ \begin{aligned} \frac{\text d}{\text dt}\left ( \frac{\partial L}{\partial \dot x}\right )&=\frac{\partial L}{\partial x} \\ \frac{\text d}{\text dt}(m\dot x)&=-kx \\ m\ddot x&=-kx \end{aligned} $$

Classical mechanics in 3 dimensions

In three dimensions $V=V(x,y,z)$ and $T=\frac 12 m(\dot x^2+\dot y^2+\dot z^2)$ and we obtain three Lagrange’s equations:

$$ \begin{aligned} \frac{\text d}{\text dt}\left ( \frac{\partial L}{\partial \dot x}\right )&=\frac{\partial L}{\partial x} \\ \frac{\text d}{\text dt}\left ( \frac{\partial L}{\partial \dot y}\right )&=\frac{\partial L}{\partial y} \\ \frac{\text d}{\text dt}\left ( \frac{\partial L}{\partial \dot z}\right )&=\frac{\partial L}{\partial z}

\end{aligned} $$

or alternatively in a more compact form for $x_i=x,y,z$ :

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

Real and virtual displacements

• 💻 A virtual displacement $\delta \vec r_i$ is a displacement consistent with the forces of constraints but without reference to the applied forces and the evolution of the system with time
• 🚓 A real displacement $\text d\vec r_i$ involves knowledge of all the forces acting on the system.

The differences are

💻 Virtual displacement

• Occurs at a constant time

$$ \delta \vec r_i=\sum^n_{j=1}\frac{\partial \vec r_i}{\partial q_j}\delta q_j $$

virtual displacement of the generalized coordinates with small change $\{ \delta q_j \}$.

• Can have values consistent with the forces of constraint not just those associated with the actual evolution of the system

🚓 Real displacement

• Occurs during a time interval $\text dt$

$$ \text d \vec r_i=\left [ \sum^n_{j=1}\frac{\partial \vec r_i}{\partial q_j}\dot q_j+\frac{\partial \vec r_i}{\partial t} \right ]\text dt $$

real displacement in time $\text dt$

• Only follows the actual evolution of the system