I have contact form in footer that always displays and also some pages, like Create Account, can have captcha too, so it's dynamically and I'm using next way with jQuery:
html:
<div class="g-recaptcha" id="g-recaptcha"></div>
<div class="g-recaptcha" id="g-recaptcha-footer"></div>
javascript
<script src="https://www.google.com/recaptcha/api.js?onload=CaptchaCallback&render=explicit&hl=en"></script>
<script type="text/javascript">
var CaptchaCallback = function(){
$('.g-recaptcha').each(function(){
grecaptcha.render(this,{'sitekey' : 'your_site_key'});
})
};
</script>