Analytical Classical Dynamics: Newton's Laws

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[SOLVED] Analytical Classical Dynamics: Newton's Laws

Homework Statement

Consider a system of N mutually interacting point objects. Let the ith object have mass mi and position vector ri. Suppose that the jth object exerts a central force fij on the ith. In addition, let the ith object be subject to an external force Fi. Here, i and j take all possible values. Find expressions for the rate of change of the total linear momentum P and the total angular momentum L of the system.

Homework Equations

The Attempt at a Solution

In an inertial frame, Newton’s second law of motion applied to the ith object yields
$$m_{i}$$$$\frac{d^{2}r_{i}}{dt^{2}}$$=$$\sum$$$$f_{ij}$$+$$\sum$$$$F_{i}$$

I know that the total momentum of the system is written
P=$$\sum$$$$m_{i}$$$$\frac{dr_{i}}{dt}$$

and
$$\frac{dP}{dt}$$=f

The total angular momentum of the system is written
L=$$\sum$$$$l_{i}$$

yielding
$$\frac{dL}{dt}$$=$$r_{i}$$x$$f_{ij}$$=0

These are for the central force only excluding the equation of motion. What I don't know is how the external force is applied to the change in momentum equations. For a central force, the change in angular momentum equals zero. Does the external force affect its angular momentum? Is the change in linear momentum equation simply the summation of forces on the ith particle?

 

[BigJDubs]

Answer:The total linear momentum of the system is given by P=\summ_{i}\frac{dr_{i}}{dt}and the rate of change of the total linear momentum is given by \frac{dP}{dt}=\summ_{i}\frac{d^2r_{i}}{dt^2}+\sumF_{i}The total angular momentum of the system is given by L=\suml_{i}and the rate of change of the total angular momentum is given by \frac{dL}{dt}=\sumr_{i}xf_{ij}+\sumr_{i}xF_{i}