# Generalized Momentum is a linear functional of Velocity?

• I
• chmodfree
In summary: I mean it maps \dot{Q} from tangent bundle to the real number field as a functional... No problem now, thank you.
chmodfree
Generalized momentum is covariant while velocity is contravariant in coordinate transformation on configuration space, thus they are defined in the tangent bundle and cotangent bundle respectively.
Question: Is that means the momentum is a linear functional of velocity? If so, the way to construct this functional?

If the Lagrangian is quadratic in the coordinate velocities then it is linear at the point of the tangent bundle and can thus be expressed as the result of a linear mapping by a generalized inertia tensor, $P_a = M_{ab}(Q,t)\dot{Q}^b$

$P_a = \frac{\partial \mathcal{L}(Q,\dot{Q},t)}{\partial \dot{Q}^a} = M_{ab}\dot{Q}^b$ with $M$ not depending on any $\dot{Q}$ if and only if the Lagrangian was quadratic in these velocities.

chmodfree
jambaugh said:
If the Lagrangian is quadratic in the coordinate velocities then it is linear at the point of the tangent bundle and can thus be expressed as the result of a linear mapping by a generalized inertia tensor, $P_a = M_{ab}(Q,t)\dot{Q}^b$

$P_a = \frac{\partial \mathcal{L}(Q,\dot{Q},t)}{\partial \dot{Q}^a} = M_{ab}\dot{Q}^b$ with $M$ not depending on any $\dot{Q}$ if and only if the Lagrangian was quadratic in these velocities.
Oh I see, that means the contraction $P_a\dot{Q}^a$ is a real.

chmodfree said:
Oh I see, that means the contraction $P_a\dot{Q}^a$ is a real.
I don't know about real... I imagine someone might cook up a Lagrangian with complex inertia, but the contraction is a scalar.

bhobba
jambaugh said:
I don't know about real... I imagine someone might cook up a Lagrangian with complex inertia, but the contraction is a scalar.
I mean it maps $\dot{Q}$ from tangent bundle to the real number field as a functional... No problem now, thank you.

## What is generalized momentum?

Generalized momentum is a concept in classical mechanics that describes the motion of a physical system in terms of its position, velocity, and mass. It is a vector quantity that represents the product of an object's mass and its velocity.

## What is a linear functional?

A linear functional is a mathematical operator that maps a vector space to its associated scalar field. In the context of generalized momentum, it is a function that takes in a velocity vector and outputs a scalar value representing the momentum of a physical system.

## How is generalized momentum related to velocity?

Generalized momentum is a linear functional of velocity, meaning that it is a function of an object's velocity that is proportional to its momentum. This relationship allows us to describe the motion of a physical system in terms of its velocity rather than its position.

## Why is generalized momentum important in physics?

Generalized momentum is an important concept in physics because it allows us to describe and analyze the motion of physical systems in a more general and abstract way. It is also a fundamental quantity in many physical laws, such as Newton's laws of motion and the principle of conservation of momentum.

## How is generalized momentum used in practical applications?

Generalized momentum is used in various practical applications, such as in the design and analysis of mechanical systems, spacecraft trajectories, and particle accelerators. It is also used in the development of mathematical models and simulations to predict the behavior of complex physical systems.

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