# Generalised Momentum: Defs & Meaning

• Incand
In summary, there are two different ways to define generalized momentum, either by taking the partial derivative of the Lagrangian with respect to the time derivative of the coordinate ##q## or the kinetic energy ##T## with respect to the same. However, in most simple mechanical systems, the potential energy ##V## is not dependent on velocity. In more complex systems, such as those with magnetic fields and electric charges, the potential energy may depend on velocity and therefore the definition using ##T## may not be applicable. The correct definition of generalized momentum is always given by the partial derivative of the Lagrangian with respect to the time derivative of the coordinate ##q##.
Incand
I have two books that define generalised momentum differently. Either
##p_i = \frac{\partial L}{\partial \dot q_i}##
or
##p_i = \frac{\partial T}{\partial \dot q_i}##.
Is this since defining generalised momentum only make sense when the potential energy is independent of a coordinate ##q## and hence the above definitions are equal? Or is one of these more general than the other?

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The differentiations should be done with respect to the time derivative of ##q##, not ##q## itself. Is ##T## the kinetic energy? In most simple mechanical systems the potential energy is not dependent on velocity.

Incand
hilbert2 said:
The differentiations should be done with respect to the time derivative of ##q##, not ##q## itself. Is ##T## the kinetic energy? In most simple mechanical systems the potential energy is not dependent on velocity.

Thanks, missed the dots. Added them now. Yes ##T## is the kinetic energy. Is it possible that the potential energy ##V## depends on ##\dot q##? Is one of these definitions correct in that case?

If there are magnetic fields and electric charges in the system, the potential energy depends on velocities. Then you can't use the definition where you differentiate ##T##.

Incand
Cheers! That was a good example!

The momentum canonically conjugated to the generalized coordinate ##q^i## is defined by
$$p_i=\frac{\partial L}{\partial \dot{q}^i},$$
and nothing else!

Incand

## What is generalised momentum?

Generalised momentum is a concept in physics that describes the motion of an object in terms of its mass, velocity, and direction. It is a vector quantity that takes into account the direction and speed of an object's motion.

## How is generalised momentum different from regular momentum?

Regular momentum, also known as linear momentum, only takes into account an object's mass and velocity. Generalised momentum, on the other hand, takes into account the object's position, orientation, and motion in a larger space.

## What is the equation for calculating generalised momentum?

The equation for generalised momentum is p = m*v, where p is the generalised momentum, m is the mass of the object, and v is the velocity of the object. It can also be written as p = m*r*v, where r is the position vector of the object.

## Why is generalised momentum important in physics?

Generalised momentum is important because it allows us to describe and analyze the motion of objects in a more comprehensive way. It is especially useful in systems involving multiple objects or objects with complex movements and orientations.

## How is generalised momentum used in real-life applications?

Generalised momentum is used in various fields, such as mechanics, engineering, and astrophysics, to analyze and predict the motion and behavior of objects. It is also used in practical applications such as rocket propulsion systems and calculating the trajectories of spacecraft.

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