Simple existence/uniqueness proof for Newton's 2nd?

  • Thread starter Fredrik
  • Start date
  • Tags
    Proof
In summary, the conversation discusses the existence and uniqueness of solutions for a second order differential equation. Picard's theorem is mentioned as a way to solve a system of first order differential equations, and a link to the theorem is provided. The conversation also discusses the use of boundary conditions and a general technique for solving ODEs, as well as an example of a problem that can arise when using boundary conditions.
  • #1
Fredrik
Staff Emeritus
Science Advisor
Gold Member
10,877
422
Is there a simple proof of the existence and uniqueness of a solution of [itex]x''(t)=f(x'(t),x(t),t)[/itex], where f is an appropriately "nice" function? (Given an initial condition [itex]x(t_0)=x_0,\ x'(t_0)=v_0[/itex], or a boundary condition [itex]x(t_0)=x_0,\ x(t_1)=x_1[/itex]).
 
Physics news on Phys.org
  • #2
It's called Picard's theorem. It relates to first order differential equations, but through using vectors can solve a system of them. Any second order differential equation can be turned into a system of first order differential equations by the following (using your differential equation as the starting point)

[tex] z_1(t)=x(t)[/tex] and [tex]z_2(t)=x'(t)[/tex] we get

[tex]z_1'(t)=z_2(t)[/tex]

and

[tex] z_2'(t)=f(z_1(t), z_2(t),t)[/tex]

Here's a wikipedia link to the theorem
http://en.wikipedia.org/wiki/Picard–Lindelöf_theorem
 
  • #3
Thanks. I've seen references to Picard's theorem and I own a book that proves it, but I thought it was more general than that. I was hoping that there would exist a theorem with a particularly simple proof that covers the cases that are interesting for physics, but I guess not.

What theorem do we use when we've been given boundary conditions instead of an initial condition? There's still a unique solution, right?
 
  • #4
When you're given boundary conditions it's a lot tougher. I forgot to mention a general technique for solving odes:
http://en.wikipedia.org/wiki/Method_of_variation_of_parameters

But you can have problems. Let's take the easy equation [tex]y''+y=0[/tex]. So [tex]y=Acos(x)+Bsin(x)[/tex]. Let's furthermore take the boundary conditions [tex]y(0)=0[/tex] and [tex]y(\pi)=0[/tex]. Then all we get is that the solution is something of the form [tex]y=Asin(x)[/tex]

If instead we had [tex]y(0)=0[/tex] and [tex]y(\pi)=1[/tex] there would be no solution.
 
  • #5


Yes, there is a simple proof for the existence and uniqueness of a solution to Newton's second law of motion. This proof is known as the Picard-Lindelöf theorem, also known as the Cauchy-Lipschitz theorem.

The theorem states that if a function f(x'(t), x(t), t) satisfies certain conditions, such as being continuous and locally Lipschitz, then for any given initial or boundary conditions, there exists a unique solution to the differential equation x''(t) = f(x'(t), x(t), t).

To prove this, we can use the method of successive approximations. We start with an initial guess for the solution, say x_0(t) = x_0. Then we can use this initial guess to find a better approximation, x_1(t), and continue this process until we reach a solution that satisfies the given conditions.

By the continuity and Lipschitz conditions, we can show that the sequence of approximations converges to a unique solution, which satisfies the initial or boundary conditions. This proves the existence and uniqueness of a solution to Newton's second law of motion.

In conclusion, the Picard-Lindelöf theorem provides a simple and elegant proof for the existence and uniqueness of a solution to Newton's second law of motion, making it a fundamental result in the field of differential equations.
 

1. What is Newton's 2nd law of motion?

Newton's 2nd law of motion states that the force exerted on an object is equal to its mass multiplied by its acceleration. This means that the more massive an object is, the more force is needed to accelerate it.

2. Why is it important to prove the existence and uniqueness of Newton's 2nd law?

Proving the existence and uniqueness of Newton's 2nd law is important because it provides a solid foundation for understanding and predicting the behavior of objects in motion. It also allows for the development of more complex laws and equations in physics.

3. What is a simple existence/uniqueness proof for Newton's 2nd law?

A simple existence/uniqueness proof for Newton's 2nd law can be demonstrated by conducting a simple experiment where an object of known mass is pushed with a known force, and its resulting acceleration is measured. This can be repeated multiple times with different objects and forces to show the consistency and universality of the law.

4. How does the existence/uniqueness proof for Newton's 2nd law relate to the scientific method?

The existence/uniqueness proof for Newton's 2nd law follows the principles of the scientific method by using empirical evidence and experimentation to support a hypothesis. The repeated successful testing of the law provides strong evidence for its existence and uniqueness.

5. Are there any limitations to the existence/uniqueness proof for Newton's 2nd law?

While the existence/uniqueness proof for Newton's 2nd law is well-supported by empirical evidence, it may not hold true in extreme circumstances such as at the quantum level or in the presence of strong gravitational fields. Additionally, it only applies to objects in motion and does not account for other factors such as friction or air resistance.

Similar threads

  • Differential Equations
Replies
5
Views
590
  • Differential Equations
Replies
1
Views
698
  • Differential Equations
Replies
1
Views
609
  • Differential Equations
Replies
1
Views
676
  • Differential Equations
Replies
4
Views
1K
Replies
7
Views
1K
  • Differential Equations
Replies
8
Views
2K
  • Differential Equations
Replies
5
Views
1K
  • Classical Physics
Replies
11
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
178
Back
Top