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 $\delta$

 

[paranormal]

I can provide some insight into the differences between Euler and Lagrange equations and when to use each. Both of these equations are used to describe the motion of a system, with Euler's equation being used for conservative systems and Lagrange's equation being used for non-conservative systems.

Euler's equation is based on the principle of least action, which states that the path taken by a system between two points is the one that minimizes the action (a measure of the system's energy). This equation is typically used for systems with conservative forces, where the total energy remains constant.

On the other hand, Lagrange's equation is based on the principle of virtual work, which states that the work done by non-conservative forces (such as friction) is equal to the change in kinetic energy of the system. This equation is more versatile and can be used for both conservative and non-conservative systems. It is particularly useful for systems with non-holonomic constraints, such as friction, where the path of the system is restricted.

When dealing with non-holonomic systems, such as those with friction, it is important to include all forces in the Lagrangian function. This will result in extra terms in the Lagrange equation, which take into account the effects of friction on the system's motion.

The difference between curly delta and plain derivative, as seen in Hamilton's equation, is that curly delta represents a partial derivative, while the plain derivative represents a total derivative. This means that curly delta only considers the changes in the variables that are explicitly included in the equation, while the plain derivative considers all possible variables that may be affecting the system. Hamilton's equation is used to describe the evolution of a system's state over time, taking into account both the system's position and momentum.

In summary, Euler's equation is used for conservative systems, Lagrange's equation is used for both conservative and non-conservative systems, and Hamilton's equation is used to describe the evolution of a system's state over time. When dealing with non-holonomic systems, it is important to include all forces in the Lagrangian function to accurately describe the system's motion. I hope this helps clarify the differences between these equations and their uses. For more detailed explanations and examples, I recommend checking out resources such as textbooks or online lectures on classical mechanics.