I have the following statement in my course notes:(adsbygoogle = window.adsbygoogle || []).push({});

"For a system of N particles moving in 3 spatial dimensions,we can determine the entire motion as a function of timeprovided we know:

[tex] \begin{array}{cc} \mathbf{Q} = (q_1, q_2, ... , q_{3N}}) \\ \mathbf{\dot{Q}} = ({\dot{q}_1, \dot{q}_2, ... , \dot{q}_{3N}}) \end{array} [/tex]

at some given timet

"One does not require Cartesian coordinates here, provided the 3N coordinates [itex] \{q_j\} [/itex] completely define the system position i.e. they span the 3N degrees of freedom. Likewise for the velocity coordinates [itex] \{\dot{q_j}\} [/itex]."

The notes go on to point out that the fact we need only positions and velocities in specifying the state of the system is not a mathematical truth but a fact about nature discovered by Newton, and should be clear from his 2nd law, but much clearer in the Lagrangian and Hamiltonian formalism that we are about to launch into.

Fair enough. I'm just trying to understand why this fact about nature is so "clear." For instance, if I don't know anything about Lagrangians and want to solve the system using Newton's Second Law, the statement in bold above asserts that I can do so, provided I have the 6N required boundary conditions. Here's the question: doesn't this bold statementimplicitly assumethat I alreadyknowthe force laws governing all the forces that act on the system? If not, then how could I possible solve for any equation(s) of motion? What good are 6N boundary conditions if I can't even formulate 3N second-order differential equations to apply them to? For each differential equation, I'd be able to go no further than:

[tex] m\frac{d^{2}q_j}{dt^2} = \ \ ??? [/tex]

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# Generalised Coordinates

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