Solving a 2nd-Order ODE for Conservation of Energy

In summary, the conservative second-order ODE has two integrate functions f(x) and f(x). The potential energy is V(x)=-\int^{x_0} f(\xi) \xi. If we substitute in the function values and solve for V(x), we find that the system satisfies conservation of energy \frac{1}{2}x^{2}+V(x)=E.
  • #1
Jazradel
3
0

Homework Statement


Consider a mechanical system describe by the conservative 2nd-order ODE
[itex]\frac{\partial^{2}x}{\partial t^{2}}=f(x)[/itex]
(which could be non linear). If the potential energy is [itex]V(x)=-\int^{x}_{0} f(\xi) d \xi[/itex], show that the system satisfies conservation of energy [itex]\frac{1}{2}x^{2}+V(x)=E[/itex] (E is a constant).

Homework Equations


As above.

The Attempt at a Solution


I've missed almost everything we've done on ODEs, so I don't really have any idea how to being. Even knowing what to call the problem, or a link to some notes/worked examples/relevant textbook would be great. I think the start is to define:
[itex]x(t)=\begin{array}{c}
x_{1}(t) \\
x_{2}(t) \\
\end{array}[/itex]
Then sub into the first equation:
[itex]\frac{\partial^{2} x_{1}}{\partial t^{2}}=f_{1}(x_{1},x_{2})[/itex]
[itex]\frac{\partial^{2} x_{2}}{\partial t^{2}}=f_{2}(x_{1},x_{2})[/itex]
Now I think I should use the chain rule, and integrate equation 2, but I can't see how.
 
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  • #2
Jazradel said:

Homework Statement


Consider a mechanical system describe by the conservative 2nd-order ODE
[itex]\frac{\partial^{2}x}{\partial t^{2}}=f(x)[/itex]
(which could be non linear). If the potential energy is [itex]V(x)=-\int^{x_0} f(\xi) \xi[/itex], show that the system satisfies conservation of energy [itex]\frac{1}{2}x^{2}+V(X)=E[/itex] (E is a constant).
I think you mean [itex]\frac{1}{2}\dot{x}^2 + V(x) = E[/itex]. Just integrate the original equation with respect to x. You'll want to use the chain rule to evaluate the integral on the LHS.
 
  • #3
I just checked, it's definitely [itex]\frac{1}{2}x^{2}+V(x)=E[/itex]. I did have to fix the V(X) and the domain of the integration.

You're saying do this?
[itex]\int f(x) dx = \int \frac{\partial^{2}x}{\partial t^{2}} dx[/itex]
[itex] \int \frac{\partial^{2}x}{\partial t^{2}} dx = \int \frac{\partial}{\partial t} ( \frac{\partial x}{\partial t} ) dx [/itex]
The problem is I have no idea how to apply the chain rule to this case.
 
  • #4
Jazradel said:
I just checked, it's definitely [itex]\frac{1}{2}x^{2}+V(x)=E[/itex].
That can't be right. It would only hold if V(x)=-1/2 x2+V0. The first term is supposed to be the kinetic energy, so it needs to depend on v2, not x2.
You're saying do this?
[itex]\int f(x) dx = \int \frac{\partial^{2}x}{\partial t^{2}} dx[/itex]
[itex] \int \frac{\partial^{2}x}{\partial t^{2}} dx = \int \frac{\partial}{\partial t} ( \frac{\partial x}{\partial t} ) dx [/itex]
The problem is I have no idea how to apply the chain rule to this case.
You want to use
[tex]\frac{\partial}{\partial t} = \frac{\partial x}{\partial t} \frac{\partial}{\partial x}[/tex]
 
  • #5
Ah thanks, that should be great help.

You can view the assignment here http://www.maths.utas.edu.au/People/Forbes/KYA314Ass2in2011.pdf . Question 2. (a) is the one in question. I think I've typed it correctly though.

Edit: I have confirmed it is not a typo.
 
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  • #6
That's a typo for sure.
 
  • #7
For problems like these, "conservation of energy" arises from using "quadrature" on the differential equation.

That is, if x''= f(x), a function of x, only, we can let v= x' and then use the chain rule:
[tex]\frac{d^2x}{dt^2}= \frac{dv}{dt}= \frac{dv}{dx}\frac{dx}{dt}= v\frac{dv}{dx}[/tex]
so that our equation x''= f(x) becomes v v'= f(x) where the differentiation is now with respect to x:
[tex]v\frac{dv}{dx}= f(x)[/tex]
[tex]v dv= f(x)dx[/tex]
[tex]\frac{1}{2}v^2= \int_0^x f(x)dx+ C[/tex]
[tex]\frac{1}{2}v^2- \int_0^x f(x)dx= C[/tex]
 

FAQ: Solving a 2nd-Order ODE for Conservation of Energy

1. What is a 2nd-Order ODE?

A 2nd-Order ODE (Ordinary Differential Equation) is a mathematical equation that relates a function and its derivatives. In the context of conservation of energy, it is used to describe the relationship between the position, velocity, and acceleration of a system.

2. How is conservation of energy related to 2nd-Order ODEs?

Conservation of energy is a fundamental principle in physics that states energy cannot be created or destroyed, only transformed from one form to another. This principle can be mathematically represented using a 2nd-Order ODE, which helps to describe the change in energy of a system over time.

3. Why is it important to solve 2nd-Order ODEs for conservation of energy?

Solving 2nd-Order ODEs for conservation of energy allows us to understand and predict the behavior of a system over time. By solving these equations, we can determine how the energy of a system will change and whether it will remain constant or dissipate over time.

4. What are some commonly used methods for solving 2nd-Order ODEs for conservation of energy?

There are several methods for solving 2nd-Order ODEs, including separation of variables, substitution, and variation of parameters. In the context of conservation of energy, the most commonly used methods are the Euler-Lagrange equations and the Hamiltonian equations.

5. Are there any limitations or assumptions when solving 2nd-Order ODEs for conservation of energy?

Yes, there are some limitations and assumptions when using 2nd-Order ODEs to represent conservation of energy. These equations assume that the system is closed (no external forces acting on it) and that energy is conserved. Additionally, some equations may only be applicable to certain types of systems, such as conservative or non-conservative systems.

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