Compute the canonical momentum

  • #1
Lambda96
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Homework Statement
Compute the canonical momentum ##\pi=\frac{\partial L}{\partial \dot{\textbf{x}}}##
Relevant Equations
none
Hi everyone,

I am not sure if I have calculated task b correctly:

Bildschirmfoto 2024-11-13 um 20.16.55.png


The Lagrange function has the form ##L=\frac{1}{2}m\Vert \dot{\textbf{x}}\|^2+\frac{q}{c} \dot{\textbf{x}} \cdot \textbf{A}-q\phi##

I then formed the canonical momentum ##\pi=\frac{\partial L}{\partial \dot{\textbf{x}}}##


$$\pi=m \dot{\textbf{x}}+\frac{q}{c} \cdot \textbf{A}$$
 
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  • #2
Lambda96 said:
I then formed the canonical momentum ##\pi=\frac{\partial L}{\partial \dot{\textbf{x}}}##

$$\pi=m \dot{\textbf{x}}+\frac{q}{c} \cdot \textbf{A}$$
Should ##\pi## be bold (##\pmb{\pi}##)? Otherwise, it looks good to me.
 
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  • #3
Thank you TSny for your help 👍

Unfortunately I had forgotten to write ##\pmb{\pi}## :smile:
 
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FAQ: Compute the canonical momentum

What is canonical momentum in classical mechanics?

Canonical momentum is a quantity derived from the Lagrangian formulation of classical mechanics. It is defined as the partial derivative of the Lagrangian with respect to the generalized velocity of a system. Mathematically, it is expressed as \( p_i = \frac{\partial L}{\partial \dot{q}_i} \), where \( p_i \) is the canonical momentum corresponding to the generalized coordinate \( q_i \) and its velocity \( \dot{q}_i \).

How do you compute the canonical momentum for a simple harmonic oscillator?

For a simple harmonic oscillator, the Lagrangian \( L \) is given by \( L = \frac{1}{2} m \dot{x}^2 - \frac{1}{2} k x^2 \), where \( m \) is the mass, \( k \) is the spring constant, \( x \) is the displacement, and \( \dot{x} \) is the velocity. The canonical momentum is computed as \( p = \frac{\partial L}{\partial \dot{x}} = m \dot{x} \).

What is the relationship between canonical momentum and mechanical momentum?

Canonical momentum and mechanical momentum are related but can differ in certain contexts. Mechanical momentum is defined as \( p_{mech} = mv \), where \( m \) is mass and \( v \) is velocity. In many cases, especially in conservative systems, the canonical momentum will equal the mechanical momentum. However, in systems with constraints or in the presence of fields, the canonical momentum may include additional terms that account for these effects.

Can canonical momentum be used in quantum mechanics?

Yes, canonical momentum plays a significant role in quantum mechanics. In the context of quantum mechanics, the canonical momentum operator is used in the formulation of the Schrödinger equation and in the context of path integrals. It is typically represented as \( \hat{p} = -i\hbar \frac{\partial}{\partial x} \) in position space, where \( \hbar \) is the reduced Planck's constant.

What are the implications of computing canonical momentum in field theory?

In field theory, canonical momentum is defined for fields rather than particles. The canonical momentum density is given by \( \pi(x) = \frac{\partial \mathcal{L}}{\partial \dot{\phi}(x)} \), where \( \mathcal{L} \) is the Lagrangian density and \( \phi \) is the field variable. The computation of canonical momentum is essential for deriving the equations of motion using Hamiltonian formalism and for quantizing the fields in quantum field theory.

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