Modern Physics - Invariability of Newton's 2nd Law under a GT?

[PFStudent]

Homework Statement

Show that the form of Newton's Second Law is invariant under:
(a). a Galilean Transformation (GT) in 1-Dimension.
(b). a Galilean Transformation (GT) in 2-Dimensions.
(c). a Galilean Transformation (GT) in 3-Dimensions.

Homework Equations

Newton's Second Law.
$${{\sum_{}^{}}{\vec{F}}} = {m{\vec{a}}}{\,}{\,}{\text{[N.II.L.]}}$$

GT for 1-D
$${{x}^{\prime}} = {{x}-{vt}}$$

$${{y}^{\prime}} = {y}$$

$${{z}^{\prime}} = {z}$$

$${{t}^{\prime}} = {t}$$

GT for 2-D
$${{x}^{\prime}} = {{x}-{{v}{\left({\frac{x}{\sqrt{{x^2}+{y^2}}}}\right)}{t}}}$$
$${{y}^{\prime}} = {{y}-{{v}{\left({\frac{y}{\sqrt{{x^2}+{y^2}}}}\right)}{t}}}$$
$${{z}^{\prime}} = {z}$$

$${{t}^{\prime}} = {t}$$

GT for 3-D
$${{x}^{\prime}} = {{x}-{{v}{\left({\frac{x}{\sqrt{{x^2}+{y^2}+{z^2}}}}\right)}{t}}}$$
$${{y}^{\prime}} = {{y}-{{v}{\left({\frac{y}{\sqrt{{x^2}+{y^2}+{z^2}}}}\right)}{t}}}$$
$${{z}^{\prime}} = {{z}-{{v}{\left({\frac{z}{\sqrt{{x^2}+{y^2}+{z^2}}}}\right)}{t}}}$$
$${{t}^{\prime}} = {t}$$

The Attempt at a Solution

I'm not sure exactly where to begin here. Particularly, how to proceed in the: 2-D and 3-D; cases since I have extra variables to deal with in those GT equations.

Thanks,

-PFStudent

 

[Gokul43201]

How would you proceed in the 1-D case?

 

[PFStudent]

Hey,

How would you proceed in the 1-D case?

Well, in the 1-D case we are only considering motion along one axis, where ${\vec{v}}$ is a constant, hence the following Galilean Transformation,
$${{x}^{\prime}} = {{x}-{vt}}$$

$${{y}^{\prime}} = {y}$$

$${{z}^{\prime}} = {z}$$

$${{t}^{\prime}} = {t}$$

So, my guess is that by beginning with the following,
$${{x}^{\prime}} = {{x}-{vt}}$$

I can proceed as follows,
$${{x}^{\prime}} = {{x}-{vt}}$$

$${{\frac{d}{dt}}{\Bigl[{{x}^{\prime}}\Bigr]}} = {{\frac{d}{dt}}{\Bigl[{{x}-{vt}}\Bigr]}}$$

$${{\frac{d{{x}^{\prime}}}{dt}}} = {{\frac{dx}{dt}}-{v}}$$

$${{\frac{d}{dt}}{\left[{\frac{d{{x}^{\prime}}}{dt}}\right]}} = {{\frac{d}{dt}}{\left[{{\frac{dx}{dt}}-{v}}\right]}}$$

$${\frac{{{d}^{2}}{{x}^{\prime}}}{d{{t}^{2}}}} = {{\frac{{{d}^{2}}{x}}{d{{t}^{2}}}}-{0}}$$

$${\left({{a}^{\prime}}\right)} = {\left({a}\right)}$$

$${{a}^{\prime}} = {a}$$

So, I guess by showing that,
$${{a}^{\prime}} = {a}$$
this implies that N.II.L. is invariant under a (1-D) Galilean Transformation? If so, why?

Thanks,

-PFStudent

 

[PFStudent]

Hey,

Any help on this, still not sure if I'm on the right path here.

Thanks,

-PFStudent

 

[paranormal]

I would like to first clarify that Newton's Second Law is a fundamental principle in classical mechanics, which describes the relationship between force, mass, and acceleration. It states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.

Now, in regards to the invariability of Newton's Second Law under a Galilean Transformation (GT), we must first understand what a GT is. A GT is a mathematical transformation that describes the relationship between the position and time coordinates of an event in one reference frame to the position and time coordinates of the same event in another reference frame. It is a fundamental concept in classical mechanics and is based on the principle of relativity, which states that the laws of physics should be the same in all inertial reference frames.

In the case of a 1-D GT, the equations given in the problem show that the position and time coordinates are shifted by a constant velocity v. This means that the physical laws, including Newton's Second Law, will remain the same in both reference frames, as long as the objects being observed are not accelerating.

In the case of 2-D and 3-D GTs, the equations become more complex, but the underlying principle remains the same. The GT equations show that the position and time coordinates are shifted by a velocity v that is dependent on the direction of motion. However, this does not affect the fundamental relationship between force, mass, and acceleration described by Newton's Second Law. As long as the objects being observed are not accelerating, the net force acting on them will remain the same in both reference frames.

In summary, Newton's Second Law is invariant under a GT in all dimensions as long as the objects being observed are not accelerating. This is a fundamental principle in classical mechanics and plays a crucial role in our understanding of the physical world.