- #1

- 8

- 0

- Thread starter pccrp
- Start date

- #1

- 8

- 0

- #2

- 16,773

- 8,013

Another method is to keep the Cartesian coordinates [itex]\vec{r}_i[/itex] and implement the constraints with help of Lagrange multipliers. Then you can vary the [itex]\delta \vec{r}_i[/itex] independently. This leads to equations of motion, where additional forces from the constraints are taken into account.

- #3

UltrafastPED

Science Advisor

Gold Member

- 1,912

- 216

The Principle of Virtual Work tells us that constraints do no work.

D'Alembert's Principle permits the impressed forces of dynamical systems to be handled in the same way as constraints - the application of this is to find the unknown trajectories caused by the impressed forces via the calculus of variations - thanks to Euler and Lagrange.

For a system in equilibrium the independence of the coordinates tells us that each of the terms must be zero; for dynamics it is only the sum which must be zero.

- #4

- 8

- 0

But why in dynamic situations only the sum must be zero, and not each member of the sum?

- #5

UltrafastPED

Science Advisor

Gold Member

- 1,912

- 216

In the dynamical case you have a difference which must be zero ... but now the expression depends upon the trajectories via Newton's second law of motion: so yes, each term must go to zero, and this is the variational principle which is used to find those trajectories.

Sorry if my earlier statement was misleading and unclear.

- Last Post

- Replies
- 2

- Views
- 3K

- Last Post

- Replies
- 1

- Views
- 403

- Last Post

- Replies
- 10

- Views
- 8K

- Replies
- 3

- Views
- 2K

- Last Post

- Replies
- 12

- Views
- 2K

- Replies
- 1

- Views
- 1K

- Last Post

- Replies
- 4

- Views
- 2K

- Replies
- 2

- Views
- 2K

- Replies
- 1

- Views
- 2K

- Replies
- 3

- Views
- 777