How can we Find a Trajectory With Only two Initial Conditions?

When reading through lagrange mechanics I found myself with the notion that all you need to know of a system is its cordinates at a given a time and its velocities. When you know this you can predict the entire trajectory. But why don't you need to know the acceleration too? and the third derivative? I didn't grasp this, it seems to that same positions and velocities with different accelerations will render different paths. Cna someone help me?
- Matheus Araujo (age 19)
Brasilia, DF, Brasil

Hi Matheus,

One way of specifying the path of an object would be to specify its position, its derivative of position with respect to time, and all the higher derivatives with respect to time at any one point in time. Given all this information, you could just expand the position in a Taylor series about that point in time to find the position at any other time.

A much more efficient and powerful method would be to use just the initial position and velocity, as well as your knowledge of the rest of the system. For example, if you know all the forces (or potential energies) influencing the particle, then you can use Newton's rules (or equivalently Lagrangian/Hamiltonian mechanics) to  solve for the position and velocity at later times.

Why specifically do these laws predict trajectories? At some point, the question "why" only has the answer "because experiment says so." Hopefully someone smarter than myself can enlighten us both, but I can only say "experiment shows it works."

Cheers,

David Schmid

 

A good example is the trajectory of a thrown ball in the earth's gravitational field.  You know that the horizontal acceleration is zero and that the vertical acceleration is -g, the acceleration due to gravity.  That's all you need to solve the equations when you know the initial position and velocity.   We neglected air resistance in this simple example but the effect of that  can be added later.  LeeH

 

(published on 09/30/2013)