Ação S = ∫L dt
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
About this calculator
The 'Action S = ∫L dt' calculator is used to approximate the physical action, a core concept in Lagrangian mechanics. The action is defined as the integral of the Lagrangian function (L) over time. In practical problems, this integral can be approximated using the formula S ≈ L·Δt, where Δt represents the time interval in the analyzed step.
The calculator works by requiring the Lagrangian function (L) and the time interval (Δt). For each step, the value of L is multiplied by Δt and summed across steps to compute the total action. This method is useful in numerical simulations or when the exact integral is analytically intractable.
This tool is valuable in theoretical physics, engineering, and trajectory optimization. For example, it can help determine the path of a pendulum or a satellite's orbit. However, ensuring the Lagrangian function is well-defined and the Δt is sufficiently small to maintain approximation accuracy is crucial.
Frequently asked questions
Why is physical action important?
Action is key in Hamilton's principle, which states that physical systems follow paths that minimize the action. This allows deriving motion equations using variational calculus.
How is the formula S ≈ L·Δt applied?
The formula is used in numerical discretizations, dividing time into small Δt intervals. Each L·Δt term is summed to compute the total action, with higher accuracy for smaller Δt.
Can I use this calculator for non-differentiable functions?
Not recommended. Lagrangian mechanics requires the Lagrangian to be smooth and differentiable for variational calculus. Discontinuous functions may yield inaccurate results.
How to provide the Lagrangian function?
The Lagrangian must be expressed in terms of position, velocity, and time, as L(q, q̇, t). Example: For a free-falling particle, L = (1/2)mv² - mgh.