# Problem of inter continental ballistic missiles

Rohin.T.Narayan
Hello guys I have a problem with projectile motion. Suppose we launch an Inter Continental Ballistic Missile from one point to another on Earth's surface ( For example from Tokyo to California ) how do we describe the kinematics of the missile.[ The problem is that the "g" vector is rotating and also we cannot choose a linear co-ordinate system. Any ideas ?

## Answers and Replies

Staff Emeritus
I would suggest using Lagrangian mechanics to solve the problem. Basically you chose a convenient coordinate system (lattitude, longitude, and height? or perhaps Euler angles?), and then you write the Lagrangian in that coordinate system as a function of your chosing variables, and their time derivatives.

For this simple problem, the Lagrangian L of the missile will be the kinetic energy T in an earth-centered inertial frame minus the potential enregy V in an ECI frame.

Then you use Lagrange's equations to get the equations of motion for the missile.

There's an overview at the Wikipedia

http://en.wikipedia.org/wiki/Lagrangian_mechanics

it may not be clear enough if you are not familiar with the subject. You may have to consult a textbook if you want a really detailed explanation. The quick overview is that you have a function L, called the Lagrangian which is written in the form

L(x, x', t), where x is is a coordinate, x' is it's time derivative, and t is time.

Then Lagrange's equations give you the equations of motion directly from the Lagrangian

$$\frac{d}{dt}\left(\frac{\partial L}{\partial x'}\right) =\frac{\partial L}{\partial x}$$

A simple example - in cartesian coordinates in a potential V with only one coordinate x

L(x,x') = .5*m*x'^2 - V(x)

(note that this is kinetic energy minus potential energy).

Then

d/dt(m*x') = -$\partial V/\partial x$

For systems with more than one coordinate, there is one Lagrange's equation for each independent coordiante (variable).