# Partial derivatives in analytic mechanics

1. Oct 30, 2011

### 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