Flows, vector fields, and differential equations

What is "flow"?

Among the various mathematical systems I have come into contact with, I feel that there are two different perspectives that are most suitable for describing motion and change: flow and transformation group. The former is centered on the object being acted upon, and motion is a function of its change over time; the latter is centered on the transformation itself, and studies the algebra and topological structure of the space composed of various transformations. I think, relatively speaking, the former seems more intuitive to most people. In this article, it will be developed from the perspective of "Flow". In fact, these two ideas have a fundamental connection - this connection is reflected in a basic concept of Lie group theory - Lie Group Action, and the rich theories extended from it.

What is Flow? Very popularly speaking, it represents a rule of motion. Given an initial position x of a point, let it move for a period of time t, and then reach another position y. Then y is a function of the initial position x and the movement time t:

$y=S(t,x)$

If this function S meets some reasonable properties, it is called a flow. Students who have studied differential geometry may feel that this definition is a bit different from the strict definition in mathematics - indeed it is. In differential geometry, the concept of flow needs to be based on manifolds and single-parameter subgroups or integral curves, which is difficult to explain in this way in a blog. I have to give up rigor to a certain extent and start from intuition, hoping to convey the most basic ideas.

Let’s think about it, what properties should a reasonable motion function have? I think there should be at least three points:

Movement is continuous. Physics tells us that there is no so-called "teleportation" in reality. In the above formula, if x is fixed, then $y=S(t,x)$ it is the movement process of the point whose initial position is x. Mathematically, there is no "instant transition", which means that for any x, its motion process $y(t)$ is continuous.
The deformation is continuous. Now suppose we do not consider a point, but an object. Then, the points that were originally neighbors are still neighbors later - strictly speaking, in topology, it means that x and one of its neighbors have each moved for time t, so after the movement, the neighborhood relationship is still maintained - This is equivalent to not changing the topology of the object (i.e. not tearing it apart, but continuous deformation is certainly allowed). Of course, it is not impossible for objects to be torn apart in reality, but this will lead to changes in the topological structure, which is not expressed by general mathematical tools.
Consistency in time. To put it simply, if I first let it move for time t1, and then move for time t2, then it is the same as letting it move for time (t1+t2). Written using the above expression, it is: $S(t_2,S(t_1,x))=S(t_2+t_1,x)$ . This property seems to be taken for granted in physics, but in mathematics, if you randomly give a binary function S, it may not meet this property. This regulation ensures that the S we define will not be physically disordered at least. However, its meaning goes beyond that. We will see later that it algebraically represents a group homomorphism—this mapping plays a central role in Lie algebra.

To sum up, $S(t,x)$ it is a continuous function for All elementary functions are smooth). There is also the consistency condition regarding time. It is particularly emphasized here that we allow t to be positive or negative: taking a negative number for time means letting the point go back along the original path - how to go back. There is a condition implicit in this: two points that are separated at a certain moment can never come together to become one point - otherwise you will not know where to go if you go back - this topology is guaranteed if the topological structure does not change. Here's the thing: Objects can neither be torn apart nor stuck together.

Flow - the unification of transformation groups and motion curves

$S(t,x)$ Looking at it from two aspects, we can get two different understandings. First, fix t:

$T_t(x)=S(t,x)$

It becomes a transformation function about x: transforming a point from one position to another position after time t. Then $T_t$ it's a transformation. Then, different times tt correspond to a different transformation. Moreover, based on the consistency of time, $T_{t_1}$ the transformation is performed first (time t1), and then $T_{t_2}$ the transformation (time t2) is performed, which is equivalent to another transformation $T_{(t_2+t_1)}$ . Mathematically it is $T_{t_2}*T_{t_1}=T_{(t_2+t_1)}$ . If you have a basic understanding of the concept of a group, you can see here that $T_t$ a transformation group is formed from all different times, and $T_t$ the mapping from t to is the isomorphism from the addition group on the real number R to this transformation group mapping. Because $T_t$ it is controlled by one parameter t, there is a special term called "one-parameter group". Due to the commutativity of the additive group, this single parameter transformation group is also commutative - the physical meaning of this commutativity is as I said above: it is the same whether t1 or t2 is taken first.

Therefore, we get the first understanding: flow is an exchangeable single-parameter transformation group that continuously acts on an object. (The so-called "object" here has a special name "manifold" in mathematics. I don't want to expand too much on this point.) In fact, this is a more formal definition of flow.

Looking at it from another perspective, fixing x, we track the movement of this point:

$y_x(t)=S(t,x)$

Then $y_x$ it is the movement process of the point whose initial position (position at t=0) is x—also called a motion curve (curve) or motion trajectory (orbit). Each point has its own motion curve. The so-called flow is the community of all these motion curves, or in other words, the flow is characterized by these motion curves - this is the same as some of our intuitive ideas - we are When painting, I like to draw a few curved lines on the river to represent the flow.

This function $S(t,x)$ identifies the transformation group and the motion curve - they are two different sides of the same thing. At this point, we have taken the first step towards our goal - ultimately, we are going to connect the transformation group and the vector field - this is the core of Lie groups and Lie algebras.

Flow and vector fields

Now that we have it $y_x(t)$ , by deriving it, we can get the velocity of this point at each moment. The whole trend is a collection of all these curves, so that at every point on the manifold, we can find a curve passing through it and thus mark the velocity at that point. (I emphasize here that for a given flow, the curve passing through a certain point is unique. Can you think about why?) So, we assign a velocity to each point, which is the "velocity field" ). Each velocity is a tangent vector on the curve, so more generally, we call it a "vector field". Here, we see that any flow can establish a corresponding vector field through the velocity of the motion curve. And it can be proved that this vector field is continuous.

What about the other way around? Given a continuous vector field, can we find a flow corresponding to it? There are three aspects here

(Existence), can we find a flow whose velocity field is equal to the given vector field?
(uniqueness), if present, is this stream unique?
(Continuity), $S(t,x)$ is this flow a continuous function with respect to x and t.

This question is a very profound one, and its answer is directly related to the existence, uniqueness, and continuity of solutions to ordinary differential equations in a general sense. The answer is that this is partially true. It is any vector field defined on a manifold. For any point on the manifold, a "local manifold" (open submanifold) containing it can always be found, as well as a flow defined on this local area, such that the flow The velocity field of and the given vector field are locally equal. To put it simply, qualified flows "exist locally" everywhere. Moreover, they are unique in a sense, that is, two qualified "local flows", which are equal in the part where their definition domains overlap. If the given vector field is continuous, then the derived flow will also be continuous.

I am not going to give a strict proof, which can be found in many sources on differential manifolds. Here, I hope to use a popular process to introduce how to construct this flow. We think of the vector field as a large map marked with a lot of very dense signs - telling you how fast and in which direction you should drive after reaching this point. So, when you start from somewhere, you first look at the nearby signs, adjust the car to the indicated speed and direction, drive forward for a short distance and then see the next sign, continue to adjust the speed and direction, and continue like this. The process of driving your car forms a motion trajectory, and the speed at each point is consistent with the indication at that point. Imagine an extreme process, with infinitely dense signage, and the driver is continuously adjusting the speed at every moment, then a motion curve consistent with the vector field will be obtained. We said above that the flow is the collective of all these motion curves, so we start driving from different places, and finally we can construct the entire flow.

Sometimes, the definition domain of the vector field may not be complete, so the car cannot drive indefinitely (otherwise it may drive out), and at this time only a "local flow" can be given. If a vector field has a global flow, it is called a complete vector field.

In this way, we can know how a transformation is made: just follow the instructions and do it step by step. The accumulation of these small steps will form the final transformation effect. What kind of instructions there are, there will be what kind of transformation. In Lie group theory, mathematicians gave the vector field a name: infinitestimal generator - which means that thousands of miles of transformation are born in small steps. In mathematics, the relationship between "thousands of miles" and "steps" is the connection between Lie groups and Lie algebras.

Why don't we describe the transformation directly, rather the vector field that generates it? It's very simple. In many cases, the overall evolution is not easy to describe directly, but small steps of progress are easy to grasp. In many problems, we know that the "divide and conquer" strategy can greatly simplify the problem. From transformation groups to vector fields is the ultimate embodiment of this strategy. A simple example, for example, we want to express a transformation process that does not change the size of an object. If the so-called "incompressibility" is directly expressed by a transformation matrix, it is a rather complex nonlinear constraint, but if it is expressed by a vector field, We only need to restrict the vector field to some finite-dimensional subspace - which is a much simpler linear constraint. There are many, many more such examples.

connection with differential equations

Finally, let's look back at the above problem of "deriving flow from vector fields". We know that the so-called velocity field is the derivative of tt, so this problem can be written as:

Given a vector field $V(x)$ , find $S(t,x)$ such that $dS(t,x)/dt=V(x)$ , and $S(0,x)=x$

This is the initial value problem of ordinary differential equations in a general sense. The answer to this question is related to the answer to the existence, uniqueness and continuity of solutions to ordinary differential equations. Given a vector field, it is equivalent to giving an ordinary differential equation. If xx is given, then the curve formed $y_x(t)$ is the solution to the above differential equation, and the flow $S(t,x)$ is the whole of all these solutions. We know that the solution to a differential equation is usually given in integral form, so the "motion curve" mentioned above has a formal scientific name in mathematics as "integral curve" (Integral curve).

In physics, the "integral curve" is also easy to understand. It is the path formed by accumulating the instructions of the "speed sign". The process of generating the integral curve is the process of "accumulating steps to reach a thousand miles". Moreover, this is not just an image thinking. In practical problems, the numerical solution of differential equations is the best embodiment of this process.

Flows, vector fields, and differential equations

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