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While reading his lectures, I came to these lines:

On the basis of Newton's second law of motion,which gives the relation between the acceleration of any body & the force acting on it,any problem in mechanics can be solved in principle. ...to determine the motion of few particles,one can use the numerical method ... But there are good reasons to make a further study of Newton's laws. ....there are quite simple cases of motion which can be analyzed not only by numerical methods, but also by direct mathematical analysis. ......there are really very few problems which can be solved exactly by analysis. ....if there are two bodies going around the sun, so that the total number of bodies is three, then analysis cannot produce a simple formula for the motion, and in practice the problem must be done numerically.

Now, what did Feynman want to tell through this passage? What is the difference between numerical methods & mathematical analysis? When are these two used?

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    $\begingroup$ Numerical methods can give you an approximate solution to a problem but tell us next to nothing about the structure of the solution space. Mathematical analysis may not be able to give us anything but trivial solutions, but in many cases it can tell us what the overall structure of the solutions has to look like. And Feynman is correct, there are only about a handful of fully integrable Hamiltonian systems. Any Hamiltonian system that can not be mapped to one of these few classes is not integrable, which naively means that one can not solve these systems in general. $\endgroup$ – CuriousOne Dec 18 '14 at 8:52
  • $\begingroup$ A numerical method attempts to approximately solve questions about a system, such as the solutions to the equations of motion of a system. An analytic solution obtained through mathematical analysis would be an exact solution, but mathematical analysis can yield other answers about a system, whether it be correlation functions, inequalities, or what have you... $\endgroup$ – JamalS Dec 18 '14 at 9:01
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    $\begingroup$ @Qmechanic do you really think putting hyperlinks to google is more useful than just saying that the terms are easily googleable? IMO such links would be disappointing for future readers, which may expect them to lead to some well-structured articles not unfiltered search results, which they can come up with as easily. In fact, such links are not much better than LMGTFY, which, AFAIR, is discouraged on SE. $\endgroup$ – Ruslan Dec 18 '14 at 13:31
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    $\begingroup$ I always first try to find concrete links, and Google search is only a second choice. Yes, I believe that active links and references in general make a post better and more accessible to the reader. In fact, if you think you can improve a post, you are encouraged to do so. I completely agree that lmgtfy.com is discouraged. A pure Google search link, on the other hand, can still serve a useful purpose, e.g. if OP uses an abbreviation that is not obvious to the reader, and there is no obvious page where the abbreviation is explained. $\endgroup$ – Qmechanic Dec 18 '14 at 19:57
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The other answers are correct. I would like to add to them with an example.

Take a spring, with spring constank $k$, with a mass, $m$, at one end and fixed to a large immovable object at the other.

Let the only force acting on the mass be due to the spring and the difference from the equilibrium position to be $x$, which can be positive and negative.

This can be represented by the diagram

enter image description here

Now the force on the mass is equal to $-kx$ and the acceleration, $a$, is given by $-{k\over m}x$ so

$$ a = {d^2x \over dt^2} = -{k \over m} x$$

because the acceleration is ${d^2x \over dt^2}$. Thus, we get the second order differential

$$ {d^2x \over dt^2} = -{k \over m} x$$

which can be solved analytically or with mathematical analysis to give

$$ x = A sin\left( \sqrt{k \over m} t + \phi\right)$$

where $A$ is the amplitude and $\phi$ is a phase factor. $A$ and $\phi$ depend on the initial conditions.

Now the powerful thing about this solution, which is provided by mathematical analysis, is that it will be correct no matter what the values of $k$, $m$, $A$ and $\phi$ will be, provided that the model we have set up is correct. Furthermore, we can differentiate the equation for $x$ with respect to time to find $v$ and $a$ as functions of time.

[note, of course, this is a case of SHM, or simple harmonic motion, and very well know. Also the solution presented here is correct, but not unique]

If, however, we take a three body system of sun, earth and moon, we cannot find a simple equation to describe the motion of these bodies. In particular, it is not possible to predict (even qualitatively) what the motion is likely to be like for all different combination of masses that could be chosen for sun, earth and moon and all different initial positions and all different intial velocities.

It is possible, however, to use numerical techniques to simulate the motion of any combination of masses with any particular intial positions and velocities. The simplest one to describe is the Euler method. Time moves forward in discrete small steps $\delta t$. For every quantity, e.g. $x_{earth}$ the $x$ position of the earth, we find the rate of change of $x_{earth}$ with time $dx_{earth}\over dt$ and use it to find the new value after the small time step of $\delta t$.

$$ x_{earth}(t+\delta t) = x_{earth}(t) + {dx_{earth}\over dt}\delta t$$

There are better methods, such as Euler Cromer and Runge Kutta; they work on the same principle of moving forward in discrete time steps.

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A multi-body problem consisting of $N$ objects requires $N$ coupled differential equations that need to be solved simultaneously (if you want to find the objects trajectories in time with known initial positions)

When you solve $x + 2y = 3 \text{and} x + y = 2$, this is what is known as mathematical analysis. The exact solution can be found: $$x = 1, y = 1$$.

However, when it comes to more complicated problems, especially when experimental data is involved - there is not always an exact solution. This requires a computer to run an iterative-type calculation to provide the best solution that best matches the data, for example using least-squares to provide the closest estimate to a solution.

This is known as numerical method.

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This is a usual term about solving differential equations. By "analytic" (or mathematical analysis), we mean finding an algebraic expression like $y=f(t)$ which satisfy the desired differential equation. But sometimes we solve the equation only at some special points. The latter method is called "numerical".

Since the Newton's law (and other principal equations in physics) is a differential equation, we use these terms for it.

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