Computational and Biological Learning Tea Talk

The Principle of Least Action

Alexander Terenin

https://avt.im/ · @avt_im

The Principle of Least Action

$$ \frac{\d x_t}{\d t} = f_t(x_t) $$

Where do the equations of motion come from?

Principle of Least Action

$$ \htmlClass{fragment}{ S(q) = \int_{t_0}^{t_1} L_t(q_t,\dot{q}_t) \d t } \qquad \htmlClass{fragment}{ \delta S(q) = 0 } $$

$L$: Lagrangian

$\delta S(q) = \left.\frac{\d}{\d\eps} S(q + \eps v)\right|_{\eps=0}$

What does this tell us?

Letting $q^\eps = q + \eps v$, write $$ \htmlClass{fragment}{ \delta S(q) } \htmlClass{fragment}{ = \left.\frac{\d}{\d\eps} \int_{t_0}^{t_1} L(q_t^\eps, \dot{q}_t^\eps) \d t\right|_{\eps=0} } \htmlClass{fragment}{ = \left.\int_{t_0}^{t_1} \frac{\d}{\d\eps}L(q_t^\eps, \dot{q}_t^\eps) \right|_{\eps=0} \d t } $$ and, letting $L^\eps = L(q_t^\eps, \dot{q}_t^\eps)$, the total derivative is $$ \htmlClass{fragment}{ \frac{\d L^\eps}{\d\eps} } \htmlClass{fragment}{ = \frac{\d t}{\d\eps} \frac{\pd L^\eps}{\pd t} + \frac{\d q^\eps}{\d\eps} \frac{\pd L^\eps}{\pd q^\eps} + \frac{\d \dot{q}^\eps}{\d\eps} \frac{\pd L^\eps}{\pd \dot{q}^\eps} } \htmlClass{fragment}{ = v \frac{\pd L^\eps}{\pd q^\eps} + \dot{v} \frac{\pd L^\eps}{\pd \dot{q}^\eps} . } $$

Euler-Lagrange Equations

Hence, choosing variations which vanish at the boundary, $$ \htmlClass{fragment}{ \delta S(q) } \htmlClass{fragment}{ = \int_{t_0}^{t_1} v \frac{\pd L}{\pd q} + \dot{v} \frac{\pd L}{\pd \dot{q}} \d t } \htmlClass{fragment}{ = \int_{t_0}^{t_1} v \del{\frac{\pd L}{\pd q} - \frac{\d}{\d t} \frac{\pd L}{\pd \dot{q}} }\d t } \htmlClass{fragment}{ = 0 } $$ using integration by parts, we obtain the differential equation $$ \htmlClass{fragment}{ \frac{\d}{\d t} \frac{\pd L}{\pd \dot{q}} - \frac{\pd L}{\pd q} = 0 } $$ by applying the fundamental lemma of variational calculus.

Newton's Second Law

Consider the Lagrangian $$ \htmlClass{fragment}{ L(q,\dot{q}) } \htmlClass{fragment}{ = \frac{1}{2}m\dot{q}^2 - V(q) } $$ and write $\htmlClass{fragment}{\frac{\pd L}{\pd q}}\htmlClass{fragment}{ = - \frac{\pd V}{\pd q}}\htmlClass{fragment}{ = F}$ along with $\htmlClass{fragment}{\frac{\d}{\d t} \frac{\pd L}{\pd \dot{q}}}\htmlClass{fragment}{ = \frac{\d}{\d t}m\dot{q}}\htmlClass{fragment}{ = ma}$. Then $$ \htmlClass{fragment}{ \frac{\d}{\d t} \frac{\pd L}{\pd \dot{q}} - \frac{\pd L}{\pd q} = 0 } \qquad \htmlClass{fragment}{ \implies } \qquad \htmlClass{fragment}{ F = ma . } $$

Properties of Euler-Lagrange Equations

$$ \frac{\d}{\d t} \frac{\pd L}{\pd \dot{q}} - \frac{\pd L}{\pd q} = 0 $$

Noether's Theorem: symmetries $\iff$ conservation laws

$\quad\leadsto$ conservation of phase volume, conservation of energy, ..

Contact Dynamics

(a) Find the contact time $\small t_c$

(b) Calculate true trajectory

Collisions lead to discontinuous jumps in velocity

Contact Dynamics

$$ \htmlClass{fragment}{ \delta S(q,\dot{q}) = 0 } \qquad \htmlClass{fragment}{ S(q,\dot{q}) = \int_{t_0}^{t_1} L(q_t,\dot{q}_t) \d t } $$

Action integral leads to two components:

(i) smooth dynamics between contact times

(ii) transfer of momentum at contact events

Contact Dynamics

Use neural ODEs to design networks that can model contacts

Contact Dynamics

Principled and accurate handling of discontinuity

Contact Dynamics

Principled and accurate handling of discontinuity