1 Lagrangian
1.1 Lagrangian Equation
The Lagrangian equation is a scalar representation of a physical system’s position in phase space, with units of energy, and changes in the Lagrangian reflect the movement of the system in phase space. To expand upon this initial statement, the idea of Lagrangian mechanics is to convert the state of the system into generalized coordinates \(\boldsymbol{q}\) and changes in those coordinates \(\dot{\boldsymbol{q}}\). These coordinates represent a simplified phase space for the system.
For a system of \(N\) particles in \(\mathbb{R}^3\) the configuration space of the system is a smooth manifold \(Q\subseteq\mathbb{R}^{3N}\) where each configuration \(q(t):=(q_{1}(t), \dots, q_{N}(t))\in Q\) specifies the admissible spacial positions of each of the particles at a given instant of time. A trajectory of the system is a smooth function \(q:[t_{0},t_{1}]\to Q\) describing the evolution of the configuration over time. The tangent bundle \(TQ\subseteq \mathbb{R}^{6N}\) of the configuration space contains tuples \((q,\dot{q})\in TQ\) representing the particle positions and velocities.
The Lagrangian equation is a smooth function \(\mathcal{L}:TQ\times \mathbb{R}\to \mathbb{R}\).
The action [[functional|functional]] of the trajectory can therefore be defined as the integral of the Lagrangian along the path \[ \mathscr{A}[q]=\int_{t_{0}}^{t_{1}}{\mathcal{L}(q(t), \dot{q}(t),t)}~d{t}. \] The laws of motion in Lagrangian Mechanics are derived form the postulate that among all trajectories between two given configurations, the actual one that will be taken by the system must be a critical point (often but not necessarily a local minimum) of the action functional. This leads to the Euler-Lagrange Equation.