LagrangianModelle

LagrangianModelle, or Lagrangian models, are dynamical models derived from Lagrangian mechanics. The Lagrangian L is the difference between kinetic energy T and potential energy V. In generalized coordinates q_i with velocities q̇_i, the equations of motion follow from the Euler–Lagrange equations: d/dt(∂L/∂q̇_i) − ∂L/∂q_i = 0. This approach yields coordinate-free formulations that are well suited to systems with constraints.

Constraints can be incorporated through reduced coordinates or by introducing Lagrange multipliers for holonomic or non-holonomic constraints. The framework naturally handles conservative forces; non-conservative forces are added as generalized forces Q_i. For systems with dissipation, a Rayleigh dissipation function R can be used, leading to d/dt(∂L/∂q̇_i) − ∂L/∂q_i + ∂R/∂q̇_i = Q_i.

Dissipative effects can be included with a Rayleigh dissipation function R, leading to d/dt(∂L/∂q̇_i) − ∂L/∂q_i + ∂R/∂q̇_i = Q_i. For field theories, the Lagrangian is replaced by a Lagrangian density, and the Euler–Lagrange equations yield the field equations; Noether's theorem relates symmetries to conservation laws.

LagrangianModelle are widely used in mechanical engineering, robotics, vehicle dynamics, biomechanics, and physics. They underpin classical systems such as pendulums and double pendulums, as well as complex robotic arms. In aerospace and acoustics, Lagrangian formulations support reduced-order models and multibody dynamics.

Advantages include compact representations and convenient treatment of constraints and energy, while limitations include the challenge of deriving an appropriate L and the potential complexity in non-conservative or non-holonomic cases. Numerical methods such as variational integrators preserve structure over long simulations.