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How does one know that a system (of differential equation and initial value constraints) is time-invariant (perhaps by inspection...)? What are the implications of a system with this property (esp. When applied to a forced system)?

Thank you.

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You check it directly by seeing if equations change under time translation, and if the system is Lagrangian, then the implication is conservation of energy. –  Alexei Averchenko Nov 14 '11 at 9:26
    
@AlexeiAverchenko: Thanks! Could you possibly give a simple example to illustrate how an equation "doesn't change" under time translation? Thanks again. –  Gerda Nov 14 '11 at 10:11
    
Can you tell us more about your background? Roughly, an ODE $\frac{\mathrm{d}}{\mathrm{d}t}u = f(u, t)$ does not depend on time if $f(u, t_1) = f(u, t_2)$ for any two times $t_1$ and $t_2$. What's great about those systems is that you can consider $f$ a vector field and then it turns out that various kinds of singularities (points where $f$ is zero or not defined) determine the qualitative nature of the system, which can then be studied using topological tools. If there exists a Lagrangian that defines the system in question, you can also define (conserved) energy. –  Alexei Averchenko Nov 14 '11 at 12:03
    
@AlexeiAverchenko: Thanks! :) –  Gerda Nov 14 '11 at 19:44

1 Answer 1

A system can be treated as a single ODE for vector-valued function. The comment-answer by Alexei Averchenko explains the concept of time-invariance and some consequences of having this structure:

An ODE $du/dt =f(u,t)$ does not depend on time if $f(u,t_1)=f(u,t_2)$ for any two times $t_1$ and $t_2$. What's great about those systems is that you can consider $f$ a vector field and then it turns out that various kinds of singularities (points where $f$ is zero or not defined) determine the qualitative nature of the system, which can then be studied using topological tools. If there exists a Lagrangian that defines the system in question, you can also define (conserved) energy.

Concerning the part

What are the implications of a system with this property (esp. When applied to a forced system)?

The implication for the force is that it's constant in time. Some such forces are conservative (have a potential function associated with it): gravitational or electrostatical. Others are not, e.g., friction. In the former case, there is a conserved "energy"; in the latter you might be able to write down a Liapunov function (a quantity that does not increase with time).

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