That is a great question! You have hit upon the N-body problem in physics, a mathematically fascinating and complex area of study. In fact, there are entire graduate-level courses offered for how to solve problems with N > 2 things in it!

The particular problem you have mentioned is the three-body gravitational problem. Although interactions between bodies could be other than gravitational (these are also investigated at the quantum level, but these interactions are instead Coulomb forces, among other interactions that are important at the quantum level), we'll focus on the planets since it's the most familiar. You are indeed correct in pointing that there would be gravitational forces between all bodies involved (i.e., Earth, Jupiter, and the sun). Of course, we have measured what the orbits of the sun, Jupiter, and earth look like (including the other planets as well), but the problem is definitely a lot more interesting than that!

Can this be measured?

The above equation is actually a differential equation for the position of a planet as a function of time. Velocity is the rate of change of your position (i.e., its derivative); acceleration is the rate of change of velocity, and thus acceleration is the double derivative of position. That is

If you solve the differential equation, you'll (hypothetically) find an analytical solution for position as a function time (i.e. an orbit of sorts). With three bodies, you end up with differential equations that are coupled with each other. For example, making planet 3 heavier affects not only its interaction with planet 1 but also with planet 2, and this reflected mathematically with differential equations. I'm not sure how comfortable you are with differential equations, but if you are curious about the math, you can get a taste of it here ; it also has the restricted three body problem, which matches fairly closely with the problem you proposed.

What happens?
It turns out there is no analytical solution to the general case of a three body problem. There have been many studies that look at certain cases. In the more general case, anything could happen. And I mean that quite literally. The path that any arbitrary body in a three-body system is dependent on the initial conditions, and might not even be a stable orbit (i.e., it goes flying off into the distance), which is characteristic of any system that can be described by chaos theory (an entire mathematical branch in its own right). What does chaos in math look like? You can get a taste by looking at the diversity of possible orbits in a three-body problem with this Java applet here . This is still an area of active research. In fact, just last year, some physicists announced the discovery of 13 new solution classes for the three body problem!

The problem you mentioned is an example. For most intents and purposes, the mass of the Earth can be considered negligibly small to either that of the sun or Jupiter. The interactions between the sun and Jupiter will essentially look like that of a two-body problem (assuming there are no other planets in our solar system). The orbit of the Earth depends on where it starts out initially relative to the sun and Jupiter. One example can be found this set of Java applets that has certain initial conditions set for you to see what happens, which happens to have the restricted three-body problem, though with the earth, moon, and a satellite. Perhaps with initial positions closer to what our solar system looks like, you can watch this video here of the the same kind of restricted three-body problem .

The Three-Body Problem Elsewhere
I mentioned that the N-body problem is applicable to the quantum scale, particularly for studying solids at the atomic level. This is indeed what I do everyday! With a solid, things become more complicated. In a solid, you need to consider Coulomb interactions between each of the nuclei with electrons, nuclei with nuclei, and electrons with electrons. Electrons also have additional quantum behaviors (e.g., obey Pauli exclusion) and so add more complication. Imagine having to solve the N-body problem for not just three but 1 X 10 ²⁶ things! That's a lot of things, but still barely one mole of anything! Quite far from modeling a solid. It turns out, this is mathematically intractable and not worth the time to solve. What people have done over the years are very clever approximations and reformulations of the problem that let us imitate a macroscopic solid with just a few hundred atoms. This is a common theme in the N-body problem (i.e., using alternate methods). In fact, problems with four or more bodies become mathematically intractable to solve.

Hope this helps!
Best,

You are absolutely right in thinking that there would be gravity between all of the interacting bodies (sun, earth and Jupiter). All objects take part in the gravitational force as long as they have mass. When dealing with more than two objects, the gravitational force between each must be added together along their defined directions. For example if you line up Jupiter, Earth, and the sun all in a line, earth would feel two opposing forces, one from the sun (from the right) and one from Jupiter (from the left).

The equation for calculating the gravitational force is defined as:
F = G x m₁ x m₂ / r²
where G is the gravitational constant, m₁ and m₂ are the mass of the two interacting bodies you are looking at, and r is the distance between the two objects.

It is important to note the dependence of the force on the mass of the two bodies interacting, as well as the distance between them. Much higher masses give higher gravitational forces, where larger distances give lower gravitational forces.

I challenge you to use this equation and look up some values for the masses of the planets and the sun, as well as their distances to each other. When you have the three individual forces calculated (between
1. earth and sun,
2. earth and jupiter,
3. jupiter and sun),
compare their magnitudes and directions and see which ones are largest and most relevant.

I hope this helps you to realize why the planets stay in orbit around the sun and planets do not get pulled too strongly by other planets that would drastically change their orbit around the sun.

Thanks for the great question!