Lagrangian and Hamiltonian Mechanics
According to Newton's laws, the incremental work dW done by a force f on a particle moving an incremental distance dx, dy, dz in 3-dimensional space is given by the dot product
Now suppose the particle is constrained in such a way that its position has only two degrees of freedom. In other words, there are two generalized position coordinates X and Y such that the position coordinates x, y, and z of the particle are each strictly functions of these two generalized coordinates. We can then define a generalized force F with the components FX and FY such that
The total differentials of x, y, and z are then given by
Substituting these differentials into (1) and collecting terms by dX and dY, we have
Comparing this with (2), we see that the generalized force components are given by
Now, according to Newton's second law of motion, the individual components of force for a particle of mass m are
Substituting into the expression for FX gives
and similarly for FY. Notice that the first product on the right side can be expanded as
and similarly for the other two products. Since x and X are both strictly functions of t, it follows that partial differentiation with respect to t is the same as total differentiation, and so the order of differentiation in the right-most term of (4) can be reversed (because partial differentiation is commutative). Hence (4) can be written as
Substituting this (and the corresponding expressions for the other two products) into equation (3), we get
Variations in x,y,z and X at constant t are independent of t (since each of these variables is strictly a function of t), so we have
Making these substitutions into (5) gives
Each term now contains an expression of the form r( r/ s), which can also be written as (r2/2)/ s, so the overall expression can be re-written as
The quantity inside the square brackets is simply the kinetic energy, conventionally denoted by T. Thus the generalized force FX, and similarly the generalized force FY, can be expressed as
These are the Euler-Lagrange equations of motion, which are equivalent to Newton's laws of motion. (Notice that if X is identified with x in equation (5), then FX reduces to Newton's expression for fx, and likewise for the other components.)
If the total energy is conserved, then the work done on the particle must be converted to potential energy, conventionally denoted by V, which must be purely a function of the spatial coordinates x,y,z, or equivalently a function of the generalized configuration coordinates X,Y, independent of t. (This is a simple example of Noether's insight that any conserved quantity, such as energy, corresponds to a symmetry, i.e., the independence of a system with respect to a particular variable, such as time.) Thus we have
Comparing this with equation (2), we see that
and therefore the Euler-Lagrange equations (6) for conservative systems can be written as
Rearranging terms, we have
Furthermore, since V is purely a function of the configuration variables, independent of their rates of change, we can just as well substitute (T-V) in place of T on the right sides of these equations, so in terms of the parameter L = T - V these equations can be written simply as
The quantity L is called the Lagrangian. This derivation was carried out for a single particle moving with two degrees of freedom in three-dimensional space, but the same derivation can be applied to collections of any number of particles. For a set of N particles there are 3N configuration coordinates, but the degrees of freedom will often be much less, especially if the particles form rigid bodies. Letting q1, q2, .., qn denote a set of generalized configuration coordinates for a conservative physical system with n degrees of freedom, the equations of motion of the system are
where L is the Lagrangian of the system, i.e., the difference between the kinetic and the potential energies, expressed in terms of the generalized coordinates and their time derivatives. These equations are usually credited jointly to Euler along with Lagrange, because although Lagrange was the first to formulate them specifically as the equations of motion, they were previously derived by Euler as the conditions under which a point passes from one specified place and time to another in such a way that the integral of a given function L with respect to time is stationary. (Roughly speaking, "stationary" means that the value of the integral does not change for incremental variations in the path.) This is a fundamental result in the calculus of variations, and can be applied to fairly arbitrary functions L (i.e., not necessarily the Lagrangian). For a derivation of the Euler conditions, see Relatively Straight.
To illustrate the application of these equations, consider a simple mass-spring system, consisting of a particle of mass m on the x axis attached to the end of a massless spring with spring constant k and null point at x = 0. For any position x, the spring exerts a force equal to F = kx, and the potential energy is the integral of force with respect to displacement. Similarly the kinetic energy is the integral of the inertial force F = ma with respect to displacement. Thus the kinetic and potential energies of the system are
Therefore the Lagrangian of the system is
The partial derivatives are
Substituting into Lagrange's equation, we get the familiar equation of harmonic motion for a mass-spring system
Of course, this simply expresses Newton's second law, F = ma, for the particle. It's also equivalent to the fact that the total energy E = T + V is constant, as can be seen by differentiating E with respect to t and then dividing through by dx/dt.
The equivalence between the Lagrangian equation of motion (for conservative systems) and the conservation of energy is a general consequence of the fact that the kinetic energy of a particle is strictly proportional to the square of the particle's velocity. Of course, in terms of the generalized parameters, it's possible for the kinetic energy to be a function of both q and 
(see, for example, Time for a Rocking Chair), but since the transformation dx = (x/q)dq between x and q is equivalent to dx/dt = (x/q)dq/dt, it follows that for a fixed configuration the kinetic energy is proportional to the squares of the generalized velocity parameters. Therefore, in general, we have
where we've made use of the fact that the potential energy V (for conservative systems) is independent of . Now, the total energy is E = T + V = 2T - L, so the conservation of energy can be expressed in the form
The two terms on the right hand side can be expanded as
Substituting into the previous equation and dividing through by (applying analytic continuation to remove the singularity when = 0), we see that the conservation of energy implies
which is just Lagrange's equation of motion. Of course, the same derivation applies to any number of particles, and their generalized coordinates.
The correspondence between the conservation of energy and the Lagrangian equations of motion suggests that there might be a convenient variational formulation of mechanics in terms of the total energy E = T + V (as opposed to the Lagrangian L = T - V). Notice that the partial derivative of L with respect to x' is the momentum of the particle. In general, given the Lagrangian, we can define the generalized momenta as
(The partial of V is zero, so it's inclusion and sign in this definition is a matter of convention.) Thus to each generalized configuration coordinate qj there corresponds a generalized momenta pj. In our simple mass-spring example, the total energy H = T + V in terms of these conjugate parameters is
The function H(q,p) is called the Hamiltonian of the system. Taking the partial derivatives of H with respect to p and q, we have
Notice that, in this example, p/m equals q' (essentially by definition), and k equals -p' (by the equation of motion). In general it can be shown that, for any conservative system with generalized coordinates qj and the corresponding momenta pj, if we express the total energy H in terms of the qj and pj, then we have
These are Hamilton's equations of motion. Although they are strictly equivalent to Lagrange's and Newton's equations, the equations of Hamilton have proven to be more suitable for adaptation to quantum mechanics. The Lagrangian and Hamiltonian formulations of mechanics are also notable for the fact that they express the laws of mechanics without reference to any particular coordinate system for the configuration space. Of course, in their original forms, they assumed an absolute time coordinate and perfectly rigid bodies, but with suitable restrictions they can be adapted to relativistic mechanics as well.
In quantum mechanics, a pair of conjugate variables qj, pj, such as position and momentum, generally do not commute, which means that the operation consisting of a measurement of qj followed by a measurement of pj is different than the operation of performing these measurements in the reverse order. This is because the eigenstates corresponding to the respective measurement operators are incompatible. As a result, the system cannot simultaneously have both a definite value of qj and a definite value of pj. See Fourier Transforms and Uncertainty for more on this topic.