Energy and motion from ##R^6 \rightarrow R##

  • #1
We know that energy is a function of space and velocity and it’s constant (in ideal case) though time.
So ## E(\vec{x}(t) , \vec{\dot{x}}(t)) = E_0##

where ##\vec{x} , \vec{\dot{x}} \in \mathbb{R}^3##.
So my function is ##E : \mathbb{R}^6 \rightarrow \mathbb{R}##.

Then there is my question: my teacher said that the motion will not be in a 6-dimensional space but on a 5-dimensional *surface*.
I don’t understand that statement (and maybe I understood it wrongly).
Anybody can help me?
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  • #2
That fact ## E(\vec{x}(t) , \vec{\dot{x}}(t)) = E_0## is a consequence of ##E= K(\vec{\dot{x}})+ U(\vec{x})##
  • #3
$$ f(x,y) =x^2 +y^2=R^2 $$
What kind of object is defined by f(x,y)=constant?
  • #4
nasu said:
$$ f(x,y) =x^2 +y^2=R^2 $$
What kind of object is defined by f(x,y)=constant?
Wow, thanks, you're right.

1. What is the relationship between energy and motion?

The relationship between energy and motion is that energy is required to cause or change motion. In other words, energy is the ability to do work, and work is the force applied to an object to cause it to move.

2. How is energy converted into motion?

Energy can be converted into motion through various processes such as chemical reactions, heat transfer, or electrical currents. For example, a car converts the chemical energy from gasoline into motion through the engine.

3. What is the significance of ##R^6 \rightarrow R## in relation to energy and motion?

The notation ##R^6 \rightarrow R## refers to the six-dimensional space in which energy can exist and be transformed into motion. This concept is important in physics as it helps us understand the different forms of energy and how they can be converted into motion.

4. How does motion affect energy?

Motion can affect energy in various ways. For instance, an object in motion has kinetic energy, which is the energy it possesses due to its motion. Motion can also change the potential energy of an object by either increasing or decreasing its height or position.

5. Can energy and motion be created or destroyed?

The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. Similarly, the law of inertia states that an object in motion will remain in motion unless acted upon by an external force. Therefore, energy and motion cannot be created or destroyed, only changed or transferred.

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