1. Theoretical Mechanics: Lagrangian dynamics
Martikainen Jani-Petri

2. Often simpler way to present
Learning goals
Constraints
generalized coordinates
(D’Alembert’s principle)
Lagrange equations
Often simpler way to present
Newton’s laws

3. Constrained motion Often we have constraints on particles Examples?
The number of degrees of freedom is reduced
Want to work only with independent coordinates
Constraints imply forces of constraint on the particles to ensure that constraint is satisfied
Forces of constraints can be complicated and you might not be interested in them. Get rid of them!
We will now learn how to achieve these goals simultaneously

4. Example of a constraint

5. Examples of constraints
Here forces of constraint become too much!

6. Holonomic constraints
N-particles  n=3N degrees of freedom
Holonomic constraints depend on coordinates and (possibly) time
Non-holonomic constraints can also exist: mass sliding on a sphere….why?
Answer: no geometric constraint at all times since particle will leave the surface eventually
Non-holonomic nonintegrable constraints

7. Holonomic constraints
Examples

8. Examples of other constraints?
Non-holonomic: z_j>0 …for example (other examples?)
No explicit time dependence: scleronomic constraint (constant length of a pendulum)
Rheonomic constraint: Opposite to scleronomic (pendulum whose length changes in a predetermined way length-kt^2=0 for example)
Non-holonomic

9. Constraint:sometimes you have it, sometimes…

10. Constraints
Constraints in the lecture….

11. Generalized coordinates
3N-k=n-k independent degrees of freedom
Choose that completely specify the system
These are generalized coordinates
Note: if the constraint depends on time…just add the dependence in.

12. Choosing generalized coordinates
Choose wisely

13. Virtual displacement What is it?
Infinitesimal and instantaneous displacement of coordinates consistent with constraints
General differential of course

14. D’Alembert’s principle
“Reaction forces, or forces of constraint, do no work under a virtual displacement.”
For i:th particle
We have
The middle term drops out due to D’Alembert’s principle
If xi are independent, each term must vanish. Generally more complicated…see notes.
Note: this means also that for a dropping particle kinetic energy ONLY comes from gravity and not forces of constraints…!!

15. D’Alembert’s principle implies what?
See notes elsewhere…

16. Lagrange’s equations
In terms of Kinetic energy T and generalized forces Q
For conservative forces: L=T-V (V=potential energy)

17. Examples
On the blackboard or somewhere…

18. Recipe Choose generalized coordinates Degrees of freedom
How many degrees of freedom (n-k) you have
Think about the system! Here you make your life easier!
How are the positions related to generalized coordinates? Form kinetic energy…

19. Recipe (continues) Potential: L=T-V Calculate derivatives:
Lagrange’s equations:
Enjoy!

20. Puzzle Usually energy conservation, for example,
Can you think of a conservation law as a constraint?
Conservation of energy for example?
Is there a difference?
Usually energy conservation, for example,
implies non-holonomic constraint.
It mixes coordinates and velocities.

21. Next lecture Calculus of variations Hamilton’s principle
Forces of constraint
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