Euler vs. Lagrange: Non-Holonomic Friction & Hamilton's Equations

[rogeralms]

Homework Statement

The difference between the Euler and Lagrange equations and when to use each.
How to set up Lagrange when the energies are non-holonomic such as friction.
What is the difference between curly delta and plain derivative, eg Hamilton's equation

Homework Equations

Euler's equation vs. Lagrange equation and Hamilton's equation

The Attempt at a Solution

These are general conceptual questions. Please see the attached sheet for more detail. I just need a little more understanding rather than help on a specific problem. Links to sites giving more detailed explanations would be greatly appreciated.[/B]

 

[dipole]

1) The equation is called the Euler-Lagrange equation - your question doesn't make much sense, unless you're referring to Euler's Equations in the context of rotating reference frames.

2) Non-holonomic constraints are a very advanced subject actually, and generally you can't deal with them. With friction, you have to come up with some model, but often $\vec{F} = -|\dot{\vec{r}}|^{\alpha}\hat{r}$ is a good approximation.

3) The total derivative of a function is defined in terms of partial derivatives. Example:

$\frac{df(x,y,z,t)}{dt} = \frac{\partial f(x,y,z,t)}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial f(x,y,z,t)}{\partial y}\frac{\partial y}{\partial t} + \frac{\partial f(x,y,z,t)}{\partial z}\frac{\partial z}{\partial t} + \frac{\partial f(x,y,z,t)}{\partial t}$

 

[rogeralms]

Thanks for your response. It turned out I was so confused that I did not state the questions correctly. Please see attachment for answers which will clear up the confused questions.

 

[BvU]

I can comment on the last question: What it says there is that q(t) is the function that minimizes the action integral.
So you are not looking for a variable value that minimizes something, for which you use a differential (usually indicated with a $d$), but for a path from t1 to t2 for which you use this variational notation $\delta$