Uniquely Defined Accelerations

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In summary: In any case, the summary is that knowing the positions and velocities in a given moment uniquely defines the accelerations at that moment.
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LukasMont
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From Landau's book, here's the following extract:

"If all the co-ordinates and velocities are simultaneously specified, it is known from experience that the state of the system is completely determined and that its subsequent motion can, in principle, be calculated. Mathematically this means that, if all the co-ordinates q and q˙ are given at some instant, the accelerations q¨ at that instant are uniquely defined".

I don't understand how that can be the case. Knowing the positions and velocities in a given moment allow me to calculate the new positions at a dt time afterwards, if accelerations are all zero. If not, I'll need to know the accelerations from some other source, like knowing the forces acting on the system, etc. I don't see how merely having positions and accelerations in a given moment gives you the accelerations in that moment as well.
 
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I think in this case Landau has a little more in the back of his mind. I would interpret "from experience" to mean something like all the classical forces we know depend on at most the first derivative of position. In that sense, any acceleration at a particular time is determined by the position or velocity at that time.
 
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  • #3
Haborix said:
I think in this case Landau has a little more in the back of his mind. I would interpret "from experience" to mean something like all the classical forces we know depend on at most the first derivative of position. In that sense, any acceleration at a particular time is determined by the position or velocity at that time.
@Haborix, that was my interpretation as well...but I thought it would be better to bring the topic into discussion, since sometimes in Landau's books he makes this fast, "simple" conclusions which are actually deeper than his tone may convey. Thanks for your input!
 
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LukasMont said:
@Haborix, that was my interpretation as well...but I thought it would be better to bring the topic into discussion, since sometimes in Landau's books he makes this fast, "simple" conclusions which are actually deeper than his tone may convey. Thanks for your input!
That's definitely a good instinct to have when you go through Landau, but I think in this instant he's just getting the reader primed to see how these facts fall out naturally from extremizing the action.
 

Related to Uniquely Defined Accelerations

1. What is a uniquely defined acceleration?

A uniquely defined acceleration is a type of acceleration that is specific and precise, with no ambiguity. It is a measurement of how quickly an object's velocity changes over time, and is typically expressed in units of meters per second squared (m/s²).

2. How is a uniquely defined acceleration different from other types of acceleration?

Unlike other types of acceleration, which may be relative or dependent on external factors, a uniquely defined acceleration is an absolute value that is independent of any external reference frames. This means that it will be the same regardless of the observer's perspective.

3. What are some examples of uniquely defined accelerations?

Some examples of uniquely defined accelerations include the acceleration due to gravity (9.8 m/s²), the acceleration of a car on a straight road, and the acceleration of a falling object. These values are constant and do not change based on the observer's frame of reference.

4. How are uniquely defined accelerations measured?

Uniquely defined accelerations are typically measured using instruments such as accelerometers, which can detect changes in an object's velocity over time. These measurements can then be used to calculate the acceleration using the equation a = Δv/Δt, where a is the acceleration, Δv is the change in velocity, and Δt is the change in time.

5. Why are uniquely defined accelerations important in science?

Uniquely defined accelerations are important in science because they allow for precise and consistent measurements of an object's motion. They also provide a universal standard for comparing and analyzing different types of motion, making it easier to understand and predict the behavior of objects in the physical world.

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