Generalized coordinates in Lagrangian mechanics

In summary, Lagrange's equations remain the same regardless of the units of the coordinates. This means that the generalized coordinates in Lagrangian mechanics do not have to be traditional length and angle units, but can also be measured in different dimensions such as energy or length squared. An example of this is taking a length and dividing it by the speed of light to create a coordinate with units of time. However, angles are dimensionless, so they do not pose an issue.
  • #1
ShayanJ
Insights Author
Gold Member
2,810
604
In some texts about Lagrangian mechanics,its written that the generalized coordinates need not be length and angles(as is usual in coordinate systems)but they also can be quantities with other dimensions,say,energy,[itex]length^2[/itex] or even dimensionless.
I want to know how will be the Lagrange's equations in such coordinates?
Could you give an example whose proper coordinates are as such?
Thanks
 
Physics news on Phys.org
  • #2
Lagranges equations are unchanged regardless of the units of the coordinates. You can take any problem with lengths and angles and simply do an arbitrary change of variables to get a coordinate in any units you like. E.g. Take any length and divide by the speed of light and you have a new coordinate with units of time.
 
  • #3
tbf, angles are dimensionless so there's no issue there.
 

1. What are generalized coordinates in Lagrangian mechanics?

Generalized coordinates refer to a set of independent variables used to describe the state of a mechanical system in Lagrangian mechanics. These coordinates are chosen to be convenient for the specific system being studied, and can represent position, orientation, or other relevant parameters.

2. Why are generalized coordinates used in Lagrangian mechanics?

Generalized coordinates simplify the mathematical description of a mechanical system in Lagrangian mechanics by reducing the number of variables needed. This allows for a more efficient and elegant formulation of the equations of motion, making it easier to analyze and solve complex systems.

3. How are generalized coordinates related to the Lagrangian of a system?

The Lagrangian of a system is a function of the generalized coordinates and their time derivatives. It represents the total kinetic and potential energy of the system and is used to derive the equations of motion using the principle of least action.

4. Can any set of coordinates be used as generalized coordinates?

No, the choice of generalized coordinates is not arbitrary and must be carefully selected to accurately describe the system. They should be independent, and the Lagrangian should be a well-defined function of these coordinates and their time derivatives.

5. How do generalized coordinates differ from Cartesian coordinates?

Generalized coordinates are not limited to the traditional x, y, z coordinates used in Cartesian coordinates. They can be any set of independent variables that accurately describe the state of the system. This allows for a more flexible and efficient approach to solving complex mechanical systems in Lagrangian mechanics.

Similar threads

Replies
7
Views
901
Replies
5
Views
818
  • Mechanics
2
Replies
43
Views
3K
Replies
3
Views
858
  • Mechanics
Replies
2
Views
693
Replies
3
Views
937
Replies
25
Views
1K
Replies
8
Views
1K
Replies
20
Views
8K
Back
Top