A Explicit non-holonomic equations of motion

AI Thread Summary
In the discussion on explicit non-holonomic equations of motion, participants explore the challenges of formulating these equations compared to holonomic cases. It is noted that while Lagrange multipliers are essential for addressing non-holonomic constraints, existing literature, including Landau and Lifshitz, may not provide comprehensive guidance. The focus is on finding a general form for the equations of motion, as non-holonomic constraints are local and cannot be resolved by merely selecting independent coordinates. Participants express a desire for a more explicit formulation beyond the standard Euler-Lagrange equations. The conversation emphasizes the complexity of integrating non-holonomic constraints into the motion equations.
andresB
Messages
625
Reaction score
374
In the holonomic case, we can put the Lagrangian in the Lagrange equations to obtain the explicit form of the equations of motion. From Greenwood's classical dynamics book, the equations are
1661995443042.png


Are there such general equations for the non-holonomic case?
 
Physics news on Phys.org
Have a look in Landau Lifshitz vol. 1, who gets the non-holonomic constraints right. You have to introduce Lagrange multipliers in the right way!
 
vanhees71 said:
Have a look in Landau Lifshitz vol. 1, who gets the non-holonomic constraints right. You have to introduce Lagrange multipliers in the right way!

Long time without reading Landau, and I have to say that its treatment of the non-holonomic constraint seems disappointingly scarce.

In any case, I'm not looking for the Euler-Lagrange+ lagrange multipliers equations, they are in every book. Instead I'm lookinf for the final general form of the equation of motion.
 
But these are the final general form of the equation of motion. Non-holonomic constraints are local constraints, and you cannot satisfy them by simply choosing a set of independent coordinates as for holonomic constraints.
 
vanhees71 said:
But these are the final general form of the equation of motion. Non-holonomic constraints are local constraints, and you cannot satisfy them by simply choosing a set of independent coordinates as for holonomic constraints.

But the Lagrange equations are just a step in the final solution of the problem. They have to be solved togheter with the non-holonomic constraint equations. I know how to do it in specific examples that can be found in the standard books, but I would be surprised if no general explicit formulat exist.
 
Thread 'Gauss' law seems to imply instantaneous electric field'
Imagine a charged sphere at the origin connected through an open switch to a vertical grounded wire. We wish to find an expression for the horizontal component of the electric field at a distance ##\mathbf{r}## from the sphere as it discharges. By using the Lorenz gauge condition: $$\nabla \cdot \mathbf{A} + \frac{1}{c^2}\frac{\partial \phi}{\partial t}=0\tag{1}$$ we find the following retarded solutions to the Maxwell equations If we assume that...
Hello! Let's say I have a cavity resonant at 10 GHz with a Q factor of 1000. Given the Lorentzian shape of the cavity, I can also drive the cavity at, say 100 MHz. Of course the response will be very very weak, but non-zero given that the Loretzian shape never really reaches zero. I am trying to understand how are the magnetic and electric field distributions of the field at 100 MHz relative to the ones at 10 GHz? In particular, if inside the cavity I have some structure, such as 2 plates...
Back
Top