A learn's puzzle on transformation equation

[atreess]

Hi, I'm a new learner of CM and I was stuck at the very beginning. TT
The transformation equation is ri=ri(q1,q2,...,qn,t)
I wonder why t should be in the bracket.
Aren't r determined by these independent coordinates？ Isn't t already contained in them?

 

[gabbagabbahey]

Hi, I'm a new learner of CM and I was stuck at the very beginning. TT
The transformation equation is ri=ri(q1,q2,...,qn,t)
I wonder why t should be in the bracket.
Aren't r determined by these independent coordinates？ Isn't t already contained in them?

Certainly the $q_{i}$ can have some time-dependence, and that gives $r_{i}$ implicit time-dependence. But, you can also choose to define some new coordinates $r_{i}$ with explicit dependence on time.

For example, consider a particle in 1-D motion described by $x(t)=v_0t$. There is no reason you can't define a new variable $q(x,t)=3x+a_0 t^2$, if you have some use for that transformation.

 

[atreess]

Thanks very much for your reply!
But how about the degrees of freedom? A system with n degrees of freedom just needs n independent coordinates. If adding t, there must be one coordinate(q1, q2,..., or t) which is dependent.

 

[atreess]

I guess that t represents the change of the constraint with time. Is it right?

 

[PJC01]

Hello TT,

I can understand your confusion with the transformation equation. Let me try to clarify it for you.

The transformation equation, as you mentioned, is ri = ri(q1,q2,...,qn,t). This equation represents the transformation of a point in space, ri, based on a set of independent coordinates q1, q2, ..., qn, and time t. This equation is commonly used in physics and mathematics to describe the motion or transformation of objects in space and time.

The reason why t is included in the bracket is because it is considered as an independent variable in this equation. While time may seem like a component of the coordinates, it is actually a separate variable that affects the transformation of the point ri. In other words, the value of t can change the position of the point ri in space.

To answer your question, t is not already contained in the independent coordinates. It is an additional variable that is necessary to fully describe the transformation of the point ri.

I hope this explanation helps you understand the transformation equation better. Keep learning and exploring the fascinating world of transformation equations!

Best,