classic mechanics

Question

1. Write the Lagrangian for a free particle in one dimension
2. Find the equation of motion using the Lagrange equations

3.What you can tell about the physics involved in the problem from the equation of

movement? Are there conserved amounts?

4.Knowing that the particle has position zero at t = 0 and that its velocity is v0 at t = 0, solve the equation of movement. What kind of motion is expressed by the solution?

Answer 1

1. L=1/2m(dx/dt)^2
2. The Euler-Lagrange equations states that d/dt(dL/(dx/dt))-dL/dx=0. dL/(dx/dt) gives us m(dx/dt), and d/dt(dL/(dx/dt)) gives us m(d^2 x / dt^2), i.e., ma (mass times acceleration), and since there is no potential, V(x) = 0, this implies that dL/dx = 0, so we obtain newton's law ma=0.
3. Either the particle has no mass, or no acceleration, or neither. Momentum is conserved.
4. ma = 0 -> v = C -> x = Ct+D (C is a constant, and D is a constant). x = 0 at t=0 implies 0=0+D -> D=0 -> x = Ct -> v=C. So if v=v0 at t=0, v always equals v0 -> C=v0, so our equation of motion is x=v0*t. This is constant motion.