Partial derivatives in analytic mechanics

[Simonelis]

In ordinary multivariable calculus the following

situation is common:

We have some letters, say, x, y, z and call them variables.

We have some relations, say, there is only one f(x,y,z)=0.

Then you have to choose your dependent variable and two (three

variables minus one relation) independent variables. Ordinary

taylor series up to first order for infinitesimal changes in all

variables gives f1*dx+f2*dy+f3*dz=0, where f1, f2, f3 are derivatives

of f which are taken "explicitly", right from f(x,y,z)=0. Now the

following questions can be answered:
1) If y is the dependent variable, what is its partial derivative with
respect to x? the answer is -f1/f2, because you hold z: dz=0, calculate
dy/dx and formally take the limit.

2) if z is the dependent variable, what is its partial derivative with
respect to y? the answer is -f2/f3, procedure being the same.

Everything works just perfect for calculus and thermodynamics.

But in mechanics we often take derivatives with respect to derivatives.

For example, I have a set of variables q_1,...q_N and time t.

Supposedly I have to add to these their time derivatives (which I denote by upper case)

Q_1,...Q_N. Now I have a set of 2*N+1 variables and no relations (the lagrangian would

be a relation, but then the number of variables would be 2*N+2)

I am confused how to interpret partial derivatives:

1) q_i with respect to q_j
2) Q_i with respect to Q_j
3) q_i with respect to Q_j and Q_j with respect to q_i

Thank you in advance for your comments

 

[jvicens]

.

Thank you for your interesting post. It is true that the concept of partial derivatives can be confusing, especially when dealing with multiple variables and relations. In ordinary multivariable calculus, we typically take partial derivatives with respect to independent variables, but in mechanics, we often need to take derivatives with respect to derivatives.

To answer your questions, let's consider the example you provided with the variables q_1, ..., q_N and their time derivatives Q_1, ..., Q_N. In this case, we have 2*N+1 variables and no relations, so we have a system with 2*N+1 equations. Now, let's consider the partial derivatives you mentioned:

1) q_i with respect to q_j

This would be the partial derivative of q_i with respect to q_j, where all other variables are held constant. In other words, we are looking at the change in q_i when only q_j is varied.

2) Q_i with respect to Q_j

Similarly, this would be the partial derivative of Q_i with respect to Q_j, where all other variables are held constant. This represents the change in Q_i when only Q_j is varied.

3) q_i with respect to Q_j and Q_j with respect to q_i

This is a bit more complicated, as it involves taking the partial derivative of a variable with respect to a derivative. In this case, we are looking at the change in q_i when only Q_j is varied, and the change in Q_j when only q_i is varied. This can be interpreted as the "sensitivity" of q_i to changes in Q_j, and vice versa.

I hope this helps clarify the concept of partial derivatives in mechanics. It is important to keep in mind that these derivatives are taken with respect to independent variables, and in the case of derivatives, we are looking at their sensitivity to changes in other variables. If you have any further questions or need more clarification, please don't hesitate to ask.