Discussion Overview
The discussion revolves around the formulation of the Lagrangian in terms of generalized coordinates and velocities, specifically exploring cases where time is not explicitly present in the Lagrangian, yet velocities appear outside the kinetic term. Participants are investigating the implications for the Hamiltonian and its relationship to the total energy of a system, particularly in the context of electromagnetic fields.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant seeks a toy model where the Lagrangian depends on transformed variables with velocities appearing outside the kinetic term and time not being explicit.
- Another participant provides a standard example of a Lagrangian for a particle in an external potential, noting that time is typically not explicit in such cases.
- Concerns are raised about the lack of velocity dependence in potential energy terms in standard examples.
- Discussion includes a specific Lagrangian for motion in an electromagnetic field, where velocities appear in the interaction term.
- Participants explore the relationship between the Hamiltonian and total energy, questioning when they may differ, especially in the presence of velocity terms in potential energy.
- One participant suggests that for the Hamiltonian to not equal the total energy, explicit time may be necessary.
- Another participant presents a Lagrangian where the Hamiltonian does not equal the total energy due to the presence of velocity in the potential term.
- Clarifications are made regarding the definition of total energy in the context of static electromagnetic fields and the implications of Noether's theorem.
- Discussions touch on the gauge dependence of canonical momentum and its implications for the physical interpretation of the Hamiltonian.
- Participants express differing views on whether the Hamiltonian can be considered the total energy of a particle in certain scenarios.
Areas of Agreement / Disagreement
Participants express multiple competing views regarding the relationship between the Hamiltonian and total energy, particularly in the context of time dependence and the presence of velocity terms in potential energy. The discussion remains unresolved as to whether a model can exist where the Hamiltonian does not equal the total energy while meeting the specified criteria.
Contextual Notes
Limitations include the dependence on specific definitions of energy and the assumptions regarding the nature of the potential energy terms. The discussion also highlights the complexity of gauge dependence in the context of electromagnetic fields.