Conservation of mechanical energy

Interactive simulation of conservation of mechanical energy, allowing you to explore the motion of a particle along a path with up to three height levels, with optional friction and a spring, observing the exchange between kinetic, gravitational potential, and elastic potential energy.

The system is modeled using Lagrangian mechanics, with the particle constrained to the trajectory y(x). The kinetic energy depends on the slope of the curve, T = (1/2)mẋ²[1+y'(x)²], and the potential energy combines gravitational and elastic terms — the latter activated only when the object is in contact with the spring, via a Heaviside step function. The Euler-Lagrange equations yield a nonlinear second-order differential equation with no exact analytical solution, which the simulation rewrites as a system of two first-order ODEs (position and velocity).

Kinetic friction is included phenomenologically, with constant magnitude μc·m·g always opposing the velocity, restricted to the flat central region of the trajectory via a rectangle function (Heaviside Pi). The complete system is integrated numerically using the fourth-order Runge-Kutta method, with a time step proportional to the "Sim. speed" parameter — yielding a numerical error on the order of 0.025% under the default configuration.

Marco P. M. de Souza, Cristiane M. Oliveira, Rhakny P. Peixoto