I Is there a way in Lagrangian mechanics to incorporate the inertial response from initial momentum when using initial conditions instead of BCs?

[ahmadphy]

In standard Lagrangian mechanics we usually determine the particle’s path by minimizing the action between fixed endpoints — i.e.using boundary conditions (initial and final positions). In contrast Newtonian mechanics evolves motion from initial conditions (position and velocity).
My question is:
Is there a formulation of Lagrangian mechanics that allows solving for the path based on initial conditions where the effect of initial momentum is treated as an inertial response that interacts with external forces?
More specifically:
Suppose a particle with some initial velocity enters a region with a force field.
Can the Lagrangian framework be modified or reinterpreted to account for how the particle’s initial momentum influences its deflection — almost like the momentum contributes to the dynamics similarly to how a force would?
Is there a way to view the Euler-Lagrange term as a kind of inertial “resistance” or contribution that plays a role in computing a local resultant with the field force?
I’m trying to understand whether there exists a variational or dynamical formalism that blends the initial-condition approach of Newtonian mechanics with the Lagrangian framework, perhaps using momentum as a proxy for how a particle “resists” external forces at each point in time.
Any insights, references, or similar formalisms would be much appreciated.

 

[PeroK]

Lagrangian mechanics is equivalent to Newton's laws and gives general solutions. You can specify any initial conditions.

 

[ahmadphy]

You're absolutely right Lagrangian mechanics can indeed handle initial conditions just like Newton's laws, since they're mathematically equivalent. My question was more about (interpretation): Is there a way to explicitly reframe the Euler-Lagrange equations so that the role of momentum feels more like an active inertial resistance to forces, rather than just a time derivative in the equations?
For example:
- In Newtonian mechanics, initial velocity directly shapes how a particle responds to forces (e.g. a fast-moving particle resists deflection more).
- Can we similarly isolate momentum's role in the Lagrangian framework, perhaps by rewriting the EL equations to emphasize the time derivative of p as a kind of inertial term opposing external forces?

Lagrangian mechanics is equivalent to Newton's laws and gives general solutions. You can specify any initial conditions.

 

[weirdoguy]

role of momentum feels more like an active inertial resistance to forces

Where are you getting this from? Is it your own interpretation?

 

[ahmadphy]

Where are you getting this from? Is it your own interpretation?

Yes, that’s my interpretation of the concept. I think of momentum as a kind of resistance to change not in an active sense like a force, but as something that makes it harder to change the state of motion of an object. Essentially, the more momentum an object has (the faster it’s moving or the more mass it has), the more difficult it is to alter its motion. So, when an object is already moving, it resists changes to that motion because of its momentum, and changing it requires more effort or force, especially if you want to do so quickly

 

[weirdoguy]

Yes, that’s my interpretation of the concept.

Well, then there lies the answer to your question.

the more difficult it is to alter its motion.

And what does that mean quantitatively? Where does it show in the standard math?

 

[ahmadphy]

And what does that mean quantitatively? Where does it show in the standard math?

When I said "the more difficult it is to alter its motion", I was referring to the relationship between momentum and force. In the standard mathematical framework, this idea shows up through Newton's Second Law and the impulse-momentum theorem.
Impulse-Momentum Theorem quantifies this resistance to change: the change in momentum (impulse) is equal to the applied force over the time the force acts.
For the same force, a moving object with larger momentum will require a larger force or a longer duration of force to achieve the same momentum change. However, if you apply the force over a shorter time, the force needs to be larger to achieve the same momentum change in that brief time.

 

[weirdoguy]

In the standard mathematical framework, this idea shows up through Newton's Second Law and the impulse-momentum theorem.

Yes, and this is already built in E-L equations

 

[ahmadphy]

Yes, and this is already built in E-L equations

but does E-L equations Express initial momentum as a distinct boundary contribution rather than just an integration constant and Decompose the dynamics to show inertial effects and applied forces as explicit counterparts?

 

[Orodruin]

The EL equations only express the equation of motion. As this is a differential equation you will need to specify a sufficient set of boundary conditions. Initial position and momentum works just fine.