====== Lagrangian Mechanics ======

In **Lagrangian mechanics**, the dynamics of a system are described by the difference between kinetic and potential energy. This difference is called the Lagrangian:

$$ L = T - V $$

Where \( T \) is the total kinetic energy of the system, i.e., the energy related to the motion of all particles in the system, and \( V \) is the total potential energy, meaning the energy related to the positions of different particles within a force field.

In **Newtonian mechanics**, you understand the dynamics of a system by calculating all the vector forces, while in **Lagrangian mechanics**, you use scalar energy to determine the path that minimizes the action. The time integral of the Lagrangian is called the **action**:

$$ S = \int_{t_1}^{t_2} L \, dt $$

where \( t \) represents time. According to the **Principle of Least Action**, a system will follow the path between any two points that minimizes the value of the action.

To find the **equation of motion** of the system, you need to substitute the Lagrangian into the **Euler-Lagrange equation** as follows:

$$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0 $$

where \( q_i \) is a type of **generalized coordinate**, and \(\dot{q}_i\) is the **generalized velocity**, which is the first time derivative of the generalized coordinate. In **Cartesian coordinates**, \(i = (x, y, z)\), and for each coordinate, you will get an equation of motion.

===== Simple Harmonic Oscillator =====

The **Lagrangian** of a simple harmonic oscillator is a good example to calculate. For a mass \( m \) attached to a spring (with spring constant \( k \)) oscillating along the x-axis, the Lagrangian is:

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

Substituting this into the **Euler-Lagrange equation**, we get the **equation of motion**:

$$ m\ddot{x} + kx = 0 \Rightarrow \ddot{x} \propto -x $$

In other words, the **acceleration** is proportional to the **displacement** and in the opposite direction of the displacement.