# Lagrangian Dynamics

## Notation

• position vector of particle of mass
• is velocity
• is linear momentum
• is kinetic energy
• is the unit vector perpendicular to (the unit position vector)

## Equations / Theorems

### Galilean Transformation

Transforms from one inertial frame to another moving at constant velocity relative to it

Then,

### Conservation laws

Remember, in this case we have and thus $× ### Lagrange's equations Equations of motion in terms of generalised coordinates. • equivalent to Newton's laws • constraint equation have been eliminated • constraint forces do not appear - they don't contribute to Remember that the generalized forces are simply the forces projected onto the generalized coordinates . #### Derivation We restrict our attention to functions of the form , where denotes the generalised coordinates. We assume Newton's Laws to be true in this derivation. Since we can apply the "cancellation of dots" which just says that for the case where is not a function of , then we have thus, Further, Summing over we see that it's just the kinetic energy as a function of the generalised coordinates, velocities and time. Thus, where the first term is just . Since the constraint forces do now work, the sum over all constraint forces is zero. Therefore we end up with Lagrange Equations #### Using conservative forces If the forces are conservative there exists a potential energy function such that Assume$V = V(\{qj\}, t) then

Since , we may write

where

This is for holonomic constraints.

This applies to systems which also does not conserve their energy, unlike the usual

which is only valid for systems which conserve energy.

##### Examples
• Single particle in Cartesian Coordinate

We have

Substituting into Lagrange's equation for conservative forces

• Single particle in polar coordinates

Substituing into Lagrange's equations wrt.

and wrt.

## Definitions

### Intertial frame

A frame in which Newtons 1st and 2nd laws hold.

### Newtons Laws

#### Newtons 3rd law

##### Weak

for two objects and acting on each other.

##### Strong

Weak assumption AND acts along

due to

where gives us

### Constraint force

Constraint forces do no work in any small instantaneous displacement of the system consistent with the constraints themselves.

Does not mea that the constraint forces can do no work during the actual motion of the system, e.g. a particle constrained to lie on a surface which is itself moving: there may then be a component of the actual particle velocity in the direction of the constraint force, so that work is done.

### Holonomic constraints

#### Example

i.e. it's a algebraic equation between the coordinates, not a differentiable relation and not an inequality.

### Generalised coordinates

Consider a system with coordinates, i.e. 3D with particles each with coordinates, and let these coordinates be denoted by $xi,$ where .

That is, we're just "flattening" the matrix to a vector.

If holonomic constraints, not all are independent, and set of independent variables

We might have dependence between the due to the constraints being imposed, and thus representing the coordinates in the above way is just "removing" the dependence between the .

Hence, we end up with a basis of dimensions.

Our aim is to derive 2nd order differential eq. for the set of generalised coordinates .

Transformation from to is invertible using constraint eq., that is:

cannot be varied independently without violating the constraints, whereas we can vary while still satisfying the constraints.

#### Generalised velocities

If denotes the generalised coordinates, then represents the generalised velocities.

#### Generalised forces

Here may have a component in the direction of a constraint force, due to motion, and so the constraint force may do work, e.q. a body on a surface is utself moving.

"Instantaneous", i.e.

Then a small change consistent with constraints

In a virtual displacement the work done is

since constraint forces do no work.

where