Notice that this is a second order differential equation. Given two initial conditions, such as particle position and particle velocity at time zero, all subsequent states of the system, i.e. positions and velocities and all quantities which are functions of these, may be solved for deterministically. For a system of N particles, this corresponds to 3N 2nd order ODE's, and requires 6N initial conditions for its solution (in 3 dimensions).
The total energy of a system of particles depends upon the positions and velocities of those particles and may be broken down into a kinetic energy, which depends only on the particle velocities, and a potential energy, which depends on particle positions.
We will only be concerned with Conservative Systems, in which we can express the forces on the particles in terms of the gradient of potential energy:
In Newtonian mechanics, a change in kinetic energy K is directly related to a change in the potential energy U, and the total energy E is conserved.
The equation F=ma now becomes:
which is the Lagrangian equation of motion. This too is a 2nd order ODE. For a system of N particles we have 3N equations and 6N initial conditions (in 3 dimensions).
Generalized momenta are defined as those variables conjugate to the generalized coordinates (positions) in Lagrangian mechanics.
while in polar coordinates (r, theta) the generalized momenta are
The Hamiltonian H is defined as
To get the Hamiltonian equations of motion, we want to differentiate H(p,q) with respect to p and q. This leads to 6N 1st order ODE's for a system of N particles in 3 dimensions. 6N initial conditions are still required.
If we have potential energy U=U(q), a function of position only, and kinetic energy K=K(q), a function of velocities only, and furthermore K is quadratic in q, such as K(q)=aq2, then the momentum p is simply:
Differentiating H(p,q) with respect to p and q gives the following Hamiltonian Equations of Motion: