Lagrangian Mechanics Solver
Lagrangian Mechanics Solver computes 1D/2D motion from user-defined Lagrangian, visualizes trajectories with detailed inputs.
Lagrangian Mechanics Overview
The solver uses the Euler-Lagrange equation to derive equations of motion:
\\[
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) – \frac{\partial L}{\partial q_i} = 0
\\]
Where:
- \\(L = T – V\\): Lagrangian (kinetic minus potential energy)
- \\(q_i\\): Generalized coordinates (\\(x\\) in 1D; \\(x, y\\) in 2D)
- \\(\dot{q}_i\\): Time derivatives (velocities)
Example: 1D Harmonic Oscillator
\\(T = \frac{1}{2} m \dot{x}^2\\), \\(V = \frac{1}{2} k x^2\\), \\(m = 1.0 \, \text{kg}\\), \\(k = 1.0 \, \text{N/m}\\), initial: \\(x_0 = 1.0 \, \text{m}, \dot{x}_0 = 0.0 \, \text{m/s}\\)
Equation: \\(m \ddot{x} + k x = 0\\)
Example: 2D Pendulum
\\(T = \frac{1}{2} m ( \dot{x}^2 + \dot{y}^2 )\\), \\(V = m g y\\), \\(m = 1.0 \, \text{kg}\\), \\(g = 9.81 \, \text{m/s}^2\\), initial: \\(x_0 = 1.0, y_0 = 0.0, \dot{x}_0 = 0.0, \dot{y}_0 = 0.0\\)